August 10, 2005 Master Review Vol. 9in x 6in { (for ... · August 10, 2005 Master Review Vol. 9in x...

August 10, 2005 Master Review Vol. 9in x 6in – (for Lecture Note Series, IMS, NUS) epidemio

Publishers’ page


CONTENTS

The Basic Epidemiology Models

Herbert W. Hethcote 1

Epidemiology Models with Variable Population Size


Age-Structured Epidemiology Models


v


THE BASIC EPIDEMIOLOGY MODELS I & II: MODELS,

EXPRESSIONS FOR R0, PARAMETER ESTIMATION, AND

APPLICATIONS

Herbert W. Hethcote

Department of Mathematics

University of Iowa

14 Maclean Hall

Iowa City, Iowa 52242, U.S.A.

E-mail: [email protected]

Abstract: Three basic models (SIS endemic, SIR epidemic, and SIR en-demic) for the spread of infectious diseases in populations are analyzedmathematically and applied to specific diseases. Threshold theorems in-volving the basic reproduction number R0, the contact number σ, andthe replacement number R are presented for these models and their ex-tensions such as SEIR and MSEIRS. Values of R0 and σ are estimatedfor various diseases in order to compare the percentages that must bevaccinated in order to achieve herd immunity.

1.1 Introduction 2

1.2 Why do epidemiology modeling? 6

1.3 Definitions, assumptions and model formulations 17

1.3.1 Formulating epidemiology models 20

1.3.2 Three threshold quantities: R0, σ, and R 22

1.4 The basic SIS endemic model 23

1.5 The basic SIR epidemic model 25

1.6 The basic SIR endemic model 29

1.7 Similar models with M and E epidemiological states 31

1.7.1 The SEIR epidemic model 32

1.7.2 The MSEIRS endemic model 34

1.7.2.1 Equilibria and thresholds 36

1.7.2.2 Special cases of the MSEIRS endemic model 37

1


2 H.W. Hethcote

1.8 Threshold estimates using the two SIR models 38

1.9 Comparisons of some directly transmitted diseases 40

1.10 The effectiveness of vaccination programs 42

1.10.1 Smallpox 42

1.10.2 Poliomyelitis 43

1.10.3 Measles 44

1.10.4 Rubella 45

1.10.5 Chickenpox (varicella) 46

1.10.6 Influenza 47

1.11 Other Epidemiology Models with Thresholds 48

References 51

1.1. Introduction

In these first two tutorial lectures, I introduce the terminology, notation,

and standard results for epidemiology models by formulating and analyzing

three basic epidemiology models. These three classic epidemiological mod-

els provide an intuitive basis for understanding more complex epidemiology

modeling results presented in other lectures in this course. These first two

lectures are based on introductory material in previous publications [80, 75]

with the addition of some new material. The threshold for many epidemi-

ology models is the basic reproduction number R0, which is defined as the

average number of secondary infections produced when one infected indi-

vidual is introduced into a host population where everyone is susceptible

[45]. For many deterministic endemic models, an infection can get started

in a fully susceptible population if and only if R0 > 1. Thus the basic re-

production number R0 is often considered as the threshold quantity that

determines when an infection can invade and persist in a new host pop-

ulation. The role of R0 is demonstrated in the three basic models. Then

thresholds are estimated from data on several diseases and the implications

of the estimates are considered for diseases such as smallpox, polio, measles,

rubella, chickenpox, and influenza.

Epidemiology models with variable population size are considered in my

third tutorial lecture. Infectious disease models that include both time t

and age a as independent variables are more realistic than the three basic

models. This is true because age groups mix heterogeneously, the recovered

fraction usually increases with age, risks from an infection may be related


The Basic Epidemiology Models 3

to age, vaccination programs often focus on specific ages, and epidemio-

logic data is often age specific. Age-structured epidemiological models are

formulated and analyzed in my fourth tutorial lecture.

This course presents the latest results on the theory and applications

of mathematical modeling of infectious disease epidemiology and control.

Mathematicians, statisticians, and epidemiologists are presenting success-

ful examples of mathematical modeling studies. They are also describing

current epidemiological problems and questions about strategies for vacci-

nation and other prevention methods that have been studied using mathe-

matical modeling approaches including deterministic and stochastic models.

Some lectures emphasize new methods in dynamical modeling of infectious

diseases, while others consider new applications of modeling approaches

and new methods for parameter estimation from data. Both mathemati-

cal modelers and public health policy decision makers can benefit from this

course on modeling as a decision making tool for the epidemiology and con-

trol of infectious diseases. Epidemiologists and public health policy makers

will learn more about successful and potential applications of modeling

approaches to understanding disease transmission and using interventions

to reduce disease incidence. Applied mathematicians and statisticians will

learn about challenging new problems in modeling the spread and preven-

tion of diseases.

Despite improved sanitation, antibiotics, and extensive vaccination pro-

grams, infectious diseases continue to be major causes of suffering and mor-

tality. Moreover, infectious disease agents adapt and evolve, so that new

infectious diseases have emerged and some existing diseases have reemerged

[111]. Newly identified diseases include Lyme disease (1975), Legionnaire’s

disease (1976), toxic-shock syndrome (1978), hepatitis C (1989), hepatitis

E (1990), and hantavirus (1993). Popular books have given us exciting

accounts of the emergence and detection of new diseases [66, 134, 136, 146].

The human immunodeficiency virus (HIV), which is the etiological agent

for acquired immunodeficiency syndrome (AIDS), emerged in 1981 and has

become an important sexually-transmitted disease throughout the world.

Drug and antibiotic resistance have become serious issues for diseases such

as tuberculosis, malaria, and gonorrhea. Malaria, dengue, and yellow fever

have reemerged and are spreading into new regions as climate changes oc-

cur. Diseases such as plague, cholera, and hemorrhagic fevers (Bolivian,

Ebola, Lassa, Marburg, etc.) continue to erupt occasionally.

Surprisingly, new infectious agents called prions have recently joined


4 H.W. Hethcote

the previously known agents: viruses, bacteria, protozoa, and helminths

(worms). There is strong evidence that prions are the cause of spongi-

form encephalopathies, e.g. bovine spongiform encephalopathy (BSE, “mad

cow disease”), Creutzfeldt-Jakob disease (CJD), kuru, and scrapie in sheep

[134]. Biological terrorism with diseases such as smallpox or plague has

become a new threat. In the 21st century, we have already encountered

severe acute respiratory syndrome (SARS), a disease that emerged from

a mutation of a wild animal coronavirus. An 2001 outbreak of foot and

mouth disease in the United Kingdom caused great economic hardships

there [157, 60]. Avian influenza has devastated bird populations in SE

Asia; moreover, there is the concern that a new human influenza strain will

arise by recombination of an avian influenza strain with a human influenza

strain. In the future we will undoubtedly face more new infectious disease

challenges. It is clear that human or animal invasions of new ecosystems,

global warming, environmental degradation, increased international travel,

and changes in economic and social patterns will continue to provide op-

portunities for new and existing infectious diseases [121]. The emergence

and reemergence of novel and deadly forms of infectious diseases, global

climate change, and population growth have increased the need for sound

quantitative methods to guide disease interventions.

A model for smallpox was formulated and solved by Daniel Bernoulli in

1760 in order to evaluate the effectiveness of variolation of healthy people

with the smallpox virus [15], however, deterministic epidemiology modeling

seems to have started in the 20th century. Hamer formulated and analyzed

a discrete time model in 1906 in his attempt to understand the recurrence

of measles epidemics [69]. Hamer’s model may have been the first to as-

sume that the incidence (number of new cases per unit time) depends on

the product of the densities of the susceptibles and infectives. Ross was in-

terested in the incidence and control of malaria, so he developed differential

equation models for malaria as a host-vector disease in 1911 [138]. Other

deterministic epidemiology models were then developed in papers by Ross,

Ross and Hudson, Martini, and Lotka [9, 44, 49]. Starting in 1926 Kermack

and McKendrick published papers on epidemic models and obtained the

epidemic threshold result that the density of susceptibles must exceed a

critical value in order for an epidemic outbreak to occur [9, 105, 125].

Mathematical epidemiology seems to have grown exponentially start-

ing in the middle of the 20th century (the first edition in 1957 of Bailey’s

book [9] is an important landmark), so that a tremendous variety of mod-



els have now been formulated, mathematically analyzed and applied to in-

fectious diseases. Reviews of the literature [12, 24, 44, 48, 50, 77, 81, 82, 156]

show the rapid growth of epidemiology modeling. Recent models have

involved aspects such as passive immunity, gradual loss of vaccine

and disease-acquired immunity, stages of infection, vertical transmission,

disease vectors, macroparasitic loads, age structure, social and sexual

mixing groups, spatial spread, vaccination, quarantine, and chemother-

apy. Special models have been formulated for human diseases such

as measles, rubella, chickenpox, whooping cough, diphtheria, smallpox,

malaria, onchocerciasis, filariasis, rabies, gonorrhea, herpes, syphilis, and

HIV/AIDS. The breadth of the subject is shown in the books on

epidemiology modeling [4, 6, 7, 9, 10, 11, 13, 22, 23, 24, 38, 39, 43, 61, 62, 67]

[84, 88, 98, 106, 110, 120, 131, 133, 138, 144, 153, 154].

Table 1 classifies diseases by agent (virus, bacteria, protozoa, helminths,

prions) and method of transmission. This useful classification scheme is

similar to one presented by K. Dietz [45]. The models considered in these

first two tutorial lectures are suitable for viral and bacterial diseases which

are transmitted directly from person to person. More complicated models

with separate epidemiological compartments for nonhuman species must be

used when there is transmission by insects called vectors or a reservoir of

wild or domestic animal infectives.

In recent years epidemiological modeling of infectious disease transmis-

sion has had an increasing influence on the theory and practice of disease

management and control. Mathematical modeling of the spread of infec-

tious diseases has become part of epidemiology policy decision making in

many countries, including the United Kingdom, Netherlands, Canada, and

the United States. Epidemiological modeling studies of diseases such as

gonorrhea, HIV/AIDS, BSE, foot and mouth disease, measles, rubella, and

pertussis have had an impact on public health policy in these countries.

Thus modeling approaches have become very important for decision-making

about infectious disease intervention programs. Recent approaches include

deterministic models, computer simulations, Markov Chain Monte Carlo

models, small world and other network models, stochastic simulation mod-

els, and microsimulations of individuals in a community. These techniques

are often implemented computationally and use data on disease incidence

and population demographics. Sometimes the epidemiology, immunology,

and evolution of a disease must all be considered. For example, some recent

research has studied the rational design of influenza vaccines by considering


6 H.W. Hethcote

Table 1. Classification of infectious diseases by agent and mode of transmission.

agent person→person person→environment reservoir→vector reservoir→person

environment→person vector→person

virus measles arboviruses: rabies

chickenpox yellow fever hantavirus

mumps dengue fever

rubella encephalitis

(SIR smallpox tick fevertype) influenza sandfly fever

poliomyelitis West Nile Virusherpes

HIV (AIDS virus)SARS (coronavirus)

bacteria gonorrhea typhoid fever plague brucellosistuberculosis cholera lyme disease tuleramia

(SIS or pneumonia Legionnaire’s disease anthraxSIRS meningitistype) strep throat

pertussis

protozoa syphillis amebiasis malariatrypanosomiasis

leishmaniasis

helminths dracunculiasis schistomosomiasis trichinosis(worms) filariasis

onchocerchiasis

prions kuru BSE (mad cowdisease)

VCJD (inhumans)

scrapie

the effects on the immunology of influenza immunity in individuals of the

yearly epidemics of influenza A variants, the vaccine composition each year,

and the yearly evolutionary drift of influenza A virus variants.

1.2. Why do epidemiology modeling?

Epidemiology is the study of the distribution and determinants of disease

prevalence in humans. Epidemiologists study both infectious diseases and

chronic diseases such as cancer and cardiovascular disease. One function of

epidemiology is to describe the distribution of the disease, i.e., find out who

has how much of what, where and when. Another function is to identify

the causes or risk factors for diseases in order to find out why everyone

doesn’t have the same thing uniformly. A third function of epidemiology

is to build and test theories. A fourth function is to plan, implement and



evaluate detection, control and prevention programs.

Epidemiological modeling can play an important role in the last two

functions above. Here we focus on modeling infectious diseases in human

populations and do not consider models for chronic diseases such as cancer

and heart disease. In most of this section “epidemiological modeling” refers

to dynamic modeling where the population is divided into compartments

based on their epidemiological status (e.g. susceptible, infectious, recov-

ered) and the movements between compartments by becoming infected,

progressing, recovering or migrating are specified by differential or differ-

ence equations.

Even though vaccines are available for many infectious diseases, these

diseases still cause suffering and mortality in the world, especially in de-

veloping countries. In developed countries chronic diseases such as cancer

and heart disease have received more attention than infectious diseases,

but infectious diseases are still a more common cause of death in the world.

Recently, emerging and reemerging diseases have led to a revived interest

in infectious diseases. The transmission mechanism from an infective to

susceptibles is understood for nearly all infectious diseases and the spread

of diseases through a chain of infections is known. However, the transmis-

sion interactions in a population are very complex, so that it is difficult to

comprehend the large scale dynamics of disease spread without the formal

structure of a mathematical model. An epidemiological model uses a mi-

croscopic description (the role of an infectious individual) to predict the

macroscopic behavior of disease spread through a population.

Mathematical models have become important tools in analyzing the

spread and control of infectious diseases. The model formulation process

clarifies assumptions, variables, and parameters; moreover, models provide

conceptual results such as thresholds, basic reproduction numbers, contact

numbers, and replacement numbers. Mathematical models and computer

simulations are useful experimental tools for building and testing theories,

assessing quantitative conjectures, answering specific questions, determin-

ing sensitivities to changes in parameter values, and estimating key pa-

rameters from data. Understanding the transmission characteristics of in-

fectious diseases in communities, regions and countries can lead to better

approaches to decreasing the transmission of these diseases. Mathemati-

cal models are used in comparing, planning, implementing, evaluating, and

optimizing various detection, prevention, therapy and control programs.

Epidemiology modeling can contribute to the design and analysis of epi-


8 H.W. Hethcote

demiological surveys, suggest crucial data that should be collected, identify

trends, make general forecasts, and estimate the uncertainty in forecasts

[75, 84].

In many sciences it is possible to conduct experiments to obtain in-

formation and test hypotheses. Experiments with the spread of infectious

diseases in human populations are often impossible, unethical or expensive.

Data is sometimes available from naturally occurring epidemics or from the

natural incidence of endemic diseases; however, the data is often incomplete

due to underreporting. This lack of reliable data makes accurate parameter

estimation difficult, so that it may only be possible to estimate a range

of values for some parameters. Since repeatable experiments and accurate

data are usually not available in epidemiology, mathematical models and

computer simulations can be used to perform needed theoretical experi-

ments. Calculations can easily be done for a variety of parameter values

and data sets.

Comparisons can lead to a better understanding of the processes of dis-

ease spread. Modeling can often be used to compare different diseases in

the same population, the same disease in different populations, or the same

disease at different times. Epidemiological models are useful in comparing

the effects of prevention or control procedures. Hethcote and Yorke [88]

use models to compare gonorrhea control procedures such as screening, re-

screening, tracing infectors, tracing infectees, post-treatment vaccination

and general vaccination. Communicable disease models are often the only

practical approach to answering questions about which prevention or con-

trol procedure is most effective. Quantitative predictions of epidemiological

models are always subject to some uncertainty since the models are ideal-

ized and the parameter values can only be estimated. However, predictions

of the relative merits of several control methods are often robust in the

sense that the same conclusions hold over a broad range of parameter val-

ues and a variety of models. Optimal strategies for vaccination can be

found theoretically by using modeling.

An underrecognized value of epidemiological modeling is that it leads

to a clear statement of the assumptions about the biological and sociologi-

cal mechanisms which influence disease spread. The parameters used in an

epidemiological model must have a clear interpretation such as a contact

rate or a duration of infection. Models can be used to assess many quantita-

tive conjectures. Epidemiological models can sometimes be used to predict

the spread or incidence of a disease. For example, Hethcote [73] predicted



that rubella and Congenital Rubella Syndrome will eventually disappear

in the United States because the current vaccination levels using the com-

bined measles-mumps-rubella vaccine are significantly above the threshold

required for herd immunity for rubella. An epidemiological model can also

be used to determine the sensitivity of predictions to changes in parameter

values. After the parameters are identified which have the greatest influ-

ence on the predictions, it may be possible to design studies to obtain better

estimates of these parameters.

Mathematical models have both limitations and capabilities that must

be recognized. Sometimes questions cannot be answered by using epidemi-

ological models, but sometimes the modeler is able to find the right combi-

nation of available data, an interesting question and a mathematical model

which can lead to the answer.

Table 2 gives fifteen purposes of epidemiological modeling and then three

limitations. One cannot infer from the numbers of purposes and limitations

that the advantages overwhelm the limitations since the first limitation is

very broad. More detailed explanations of these purposes and limitations

are given in the paragraphs below.

A primary reason for infectious disease modeling is that it leads to clear

statements of the assumptions about the biological and social mechanisms

which influence disease spread. The model formulation process is valuable

to epidemiologists and modelers because it forces them to be precise about

the relevant aspects of transmission, infectivity, recovery and renewal of

susceptibles. Epidemiological modelers need to formulate models clearly

and precisely using parameters which have well-understood epidemiological

interpretations such as a contact rate or an average duration of infection

[71, 77, 78]. Complete statements of assumptions are crucial so that the

reasonableness of the model can be evaluated in the process of assessing

the conclusions.

An advantage of mathematical modeling of infectious diseases is the

economy, clarity and precision of a mathematical formulation. A model us-

ing difference, differential, integral or functional differential equations is not

ambiguous or vague. Of course, the parameters must be defined precisely

and each term in the equations must be explained in terms of mechanisms,

but the resulting model is a definitive statement of the basic principles

involved. Once the mathematical formulation is complete, there are many

mathematical techniques available for determining the threshold, equilib-

rium, periodic solutions, and their local and global stability. Thus the full


10 H.W. Hethcote

Table 2. Purposes and limitations of epidemiological modeling.

Fifteen Purposes of Epidemiological Modeling

1. The model formulation process clarifies assumptions, variables and parameters.

2. The behavior of precise mathematical models can be analyzed using mathemat-

ical methods and computer simulations.

3. Modeling allows explorations of the effect of different assumptions and formu-

lations.

4. Modeling provides concepts such as a threshold, reproduction number, etc.5. Modeling is an experimental tool for testing theories and assessing quantitative

conjectures.6. Models with appropriate complexity can be constructed to answer specific ques-

tions.7. Modeling can be used to estimate key parameters by fitting data.

8. Models provide structures for organizing, coalescing and cross–checking diversepieces of information.

9. Models can be used in comparing diseases of different types or at different timesor in different populations.

10. Models can be used to theoretically evaluate, compare or optimize various de-

tection, prevention, therapy and control programs.11. Models can be used to assess the sensitivity of results to changes in parameter

values.

12. Modeling can suggest crucial data which needs to be collected.13. Modeling can contribute to the design and analysis of epidemiological surveys.

14. Models can be used to identify trends, make general forecasts, or estimate theuncertainty in forecasts.

15. The validity and robustness of modeling results can be assessed by using rangesof parameter values in many different models.

Three Limitations of Epidemiological Modeling

1. An epidemiological model is not reality; it is an extreme simplification of reality.2. Deterministic models do not reflect the role of chance in disease spread and do

not provide confidence intervals on results.3. Stochastic models incorporate chance, but are usually harder to analyze than

the corresponding deterministic model.

power of mathematics is available for analysis of the equations. Moreover,

information about the model can also be obtained by numerical simula-

tion on digital computers of the equations in the model. The mathematical

analyses and computer simulations can identify important combinations

of parameters and essential aspects or variables in the model. In order to

choose and use epidemiological modeling effectively on specific diseases, one

must understand the behavior of the available formulations and the implica-

tions of choosing a particular formulation. Thus mathematical epidemiology

provides a foundation for the applications [82, 81].

Modelers are able to explore and examine the effects of different as-

sumptions and formulations. For example, they can compare an ordinary



differential equations model, which corresponds to a negative exponential

distribution for the infectious period, with a delay-differential equations

model, which assumes that everyone has the same fixed, constant-length

infectious period. They can examine the impact of assuming homogeneous

mixing instead of heterogeneous mixing. They can study the influence of

different mixing patterns between groups on the spread of a disease in a

heterogeneous population [103, 24, 78]. They can decide if models with and

without an exposed class for people in the latent period behave differently.

The results of exploring different formulations may provide insights for epi-

demiologists and are certainly useful to modelers who are choosing models

for specific diseases. For example, if the essential behaviors (thresholds and

asymptotic behaviors) are the same for a formulation using ordinary differ-

ential equations and a formulation using delay differential equations, then

the modeler would probably choose to use the ordinary differential equa-

tions model since it is simpler.

Epidemiologists need to understand the concept of a threshold which de-

termines whether the disease persists or dies out. They need to know that

the reproduction number is a combination of parameters which gives the

number of secondary cases “reproduced” by a typical infective during the

infectious period in a population where everyone is susceptible. They need

to be aware that this reproductive number is the same as the contact num-

ber, which is the average number of adequate contacts of an infective person

during the infectious period [83]. They need to realize that the average re-

placement number is one at an endemic equilibrium because, in an average

sense, each infective replaces itself by one new infective in an equilibrium

situation. Probably the most valuable contributions of mathematical mod-

eling to epidemiologists are concepts like those above. Mathematical models

serve as a framework to explain causes, relationships and ideas. Mathemat-

ical models are very useful in obtaining conclusions that have an easily

understood interpretation. Helping epidemiologists internalize conceptual

and intuitive understandings of disease processes, mechanisms and model-

ing results is an essential aspect of the work of mathematical modelers.

Epidemiological modeling is an important part of the epidemiologist’s

function to build and test theories. Mathematical and computer simulation

models are the fundamental experimental tools in epidemiology. The only

data usually available are from naturally occurring epidemics or from the

natural incidence of endemic diseases; unfortunately, even these data are not

complete since many cases are not reported. Since repeatable experiments


12 H.W. Hethcote

and accurate data are usually not available in epidemiology, mathematical

and computer simulation models must be used to perform needed theoret-

ical experiments with different parameter values and different data sets. It

is easy in a computer simulation to find out what happens if one or several

parameters are changed.

Another advantage of epidemiological modeling is the availability of

models to assess quantitative conjectures. For example, models can be used

to check the claim that a two-dose vaccination program for measles would

lead to herd immunity whereas a one-dose program would not [73]. With

a model one could see if the AIDS virus would eventually disappear in a

population of homosexual men if half of them became celibate or practiced

safer sex. Mathematical models are a formal, quantitative way of building

and testing theories.

When formulating a model for a particular disease, it is necessary to de-

cide which factors to include and which to omit. This choice often depends

on the particular question that is to be answered. Simple models have the

advantage that there are only a few parameters, but have the disadvantage

of possibly being naive and unrealistic. Complex models may be more re-

alistic, but they may contain many parameter values for which estimates

cannot be obtained. The art of epidemiological modeling is to make suitable

choices in the model formulation so that it is as simple as possible and yet

is adequate for the question being considered. It is important to recognize

both the capabilities and limitations of epidemiological modeling. Many

important questions cannot be answered using a given class of models. The

most difficult problem for a modeler is to find the right combination of

available data, an interesting question and a mathematical model which

can lead to the answer.

By fitting the incidence predicted by a model to actual incidence data

(number of new cases per month), it is possible to estimate key parameters

such as average durations, basic reproduction numbers, contact numbers,

and replacement numbers. This is done for the three basic models covered

in these first two tutorial lectures. Hethcote and Van Ark [83] present meth-

ods for estimating contact numbers for the groups in a multigroup model

for disease transmission. A model can serve as the conceptual and quan-

titative unifier of all the available information. If parameters have already

been estimated from the literature, then the modeling provides a check on

these a priori estimates. Thus the model can be a way of coalescing param-

eter estimates and other pieces of data to see if they fit into a consistent



framework.

Comparisons can lead to a better understanding of the process of dis-

ease spread. It may be enlightening to compare the same disease at different

times, the same disease in different populations or different diseases in the

same population. One method for comparing diseases is to estimate pa-

rameter values for the diseases and then compare the parameter values.

Later in these first two tutorial lectures, this is done for various diseases

such as measles, whooping cough, chicken pox, diphtheria, mumps, rubella,

poliomyelitis and smallpox.

A very important reason for epidemiological modeling is the value of

models for theoretical evaluations and comparisons of detection, preven-

tion, therapy and control programs. Epidemiologists and politicians need to

understand the effects of different policy decisions. Qualitative predictions

of infectious disease models are always subject to some uncertainty since

the models are idealized and the parameter values can only be estimated.

However, quantifications of the relative merits of several control methods

or of control versus no control are often robust in the sense that the same

conclusions hold for a broad range of parameter values and a variety of mod-

els. Control strategies for gonorrhea such as screening, rescreening, infectee

tracing, infector tracing and potential vaccines for both women and men

were compared in Hethcote and Yorke [88]. Various vaccination strategies

for rubella and measles have been compared using modeling [73, 76, 4].

Sometimes an optimization method can be used on some aspect of an

epidemiological model. Hethcote and Waltman [86] used dynamic program-

ming to find an optimal vaccination strategy to control an epidemic at least

cost. Longini et al. [119] determined the best age or social groups to vac-

cinate for Hong Kong and Asian influenza if there were a limited amount

of vaccine available. Hethcote [74] found theoretical optimal ages of vacci-

nations for measles in three geographic locations. A primary conclusion of

that paper was that better data is needed on vaccine efficacy as a function

of age in order to better estimate the optimal age of vaccination. Thus epi-

demiological modeling can be used to identify crucial data that needs to be

collected. See Wickwire [156] for a survey of modeling papers on the control

of infectious diseases.

Predictions or quantitative conclusions of a model are said to be sensi-

tive to a parameter if a small change in the parameter causes a large change

in the outcome. A model is insensitive to a parameter if the outcomes are

the same for a wide range of values of that parameter. An important rea-


14 H.W. Hethcote

son for epidemiological modeling is the determination of the sensitive and

insensitive parameters since this information provides insight into the epi-

demiological processes. Once the sensitive parameters are identified, it may

be possible to make special efforts to gather data to obtain better estimates

of these parameters. Thus the modeling process can help identify crucial

data that should be collected. If the data and information are insufficient

to estimate parameters in a model or to piece everything together, then the

model formulation process can identify the missing information which must

be gathered.

Modeling can enhance an epidemiological survey by providing a frame-

work for the analysis of the survey results. In the design phase modeling

can help identify important questions that need to be asked in order to

have adequate data to obtain results. The statistical aspects of a survey

design are now analyzed before the survey is conducted. Similarly, those

design aspects relevant to parameter estimation in models should be ana-

lyzed before the survey is conducted to see if all significant data are being

collected.

Another purpose of epidemiological modeling is to make forecasts about

the future incidence of a disease. Although people often think of prediction

as the first or only purpose of epidemiological modeling, the purposes given

above are more important. Accurate forecasts are usually not possible be-

cause of the idealization in the models and the uncertainty in the parameter

values. However, possible forecasts under various scenarios can sometimes

be given or trends can be identified. One purpose of epidemiological mod-

eling is to reduce the uncertainty in future incidence projections.

The validation of models and modeling results is difficult since there

are rarely enough good data to adequately test or compare different mod-

els with data [7]. However, when the same concept or result is obtained

for a variety of models and by several modelers, then the confidence in

the validity of the result is increased. For example, three different papers

using different approaches to comparing two rubella vaccination strategies

agree on the criteria determining situations when each strategy is best [76].

Similarly, when a quantitative result holds for a wide range of parameter

values, then this result has some robustness. For example, if the same rel-

ative values of a variety of control procedures are obtained for wide ranges

of parameter values, then the relative values may be robust even though

the absolute values may vary [88].

After discussing the purposes and advantages of epidemiological mod-



eling, it is imperative to also discuss the limitations. Epidemiologists and

policy makers need to be aware of both the strengths and weaknesses of

the epidemiological modeling approach.

The first and most obvious limitation is that all epidemiological mod-

els are simplifications of reality. For example, it is often assumed that the

population is uniform and homogeneously mixing; this is always a simplifi-

cation, but the deviation of reality from this simplification varies with the

disease and the circumstances. This deviation from reality is rarely testable

or measurable; however, it can sometimes be estimated intuitively from an

understanding of the disease epidemiology. As described in the introduc-

tion, part of the reason for the differences between the model’s behavior and

the actual spread of the disease is that transmission is based on the inter-

actions of human beings, and homo sapiens have highly variable behavior

and interactions. People do not behave in reasonably predictable ways like

molecules or cells or particles.

Because transmission models are simplifications with usually unknown

relationships to actual disease spread, one can never be completely certain

about the validity of findings obtained from modeling such as conceptual

results, experimental results, answers to questions, comparisons, sensitiv-

ity results and forecasts. Even when models are made more complicated in

order to better approximate actual disease transmission, they are still ab-

stractions. For example, multigroup models based on risk groups or sexual

activity levels or age structure still assume that the distinctions between

groups are clear and sharp and that the mixing between groups follows some

assumed pattern. The modeler must always exercise judgment in deciding

which factors are relevant and which are not when analyzing a specific

disease or question.

The modeler needs to know that an SIR model is usually adequate since

the behavior is essentially the same for the SEIR model where E is the latent

period class. The notation SEIR for a model means that susceptibles (S)

move to an exposed class (E) (when they are in a latent period; i.e., they

are infected, but not yet infectious), then move to an infectious class (I),

and finally to a removed class (R) when they recover and have permanent

immunity. The modeler must realize that assuming that the infectivity is

constant during the infectious period is reasonable for diseases like measles

with a short infectious period (about 8 days), but is not reasonable for a

disease like the human immunodeficiency virus (HIV) where the average

time from HIV infection to AIDS is about 10 years. To incorporate age


16 H.W. Hethcote

structure, the modeler must decide whether to use a model with time and

age as continuous variables or to use discrete compartments corresponding

to age brackets; the former requires the estimation of birth, death and aging

rate coefficients as functions of time while the latter requires the estimation

of birth, death and transfer rate constants. The modeler needs to decide

whether the disease is epidemic in a short time period so that vital dynamics

(births and deaths) can be excluded or the disease is endemic over a long

time period (many years) so that vital dynamics must be included. The

modeler must be aware that as the complexity of a model is increased so

that it better approximates reality, the number of parameters increases

so that it becomes increasingly difficult to estimate values of all of these

parameters. Thus the modeler must (consciously or unconsciously) make

many decisions regarding the relevant aspects in choosing a model for a

specific disease or question.

When describing the purposes for epidemiological modeling, we have

been thinking primarily of deterministic models, but the purposes also ap-

ply to stochastic epidemiological models. Deterministic models are those

which use difference, differential, integral or functional differential equa-

tions to describe the changes in time of the sizes of the epidemiological

classes. Given the starting conditions for a well-posed deterministic model,

the solutions as a function of time are unique. In stochastic models, there

are probabilities at each time step of moving from one epidemiological class

to another. When these models are simulated with the probabilities cal-

culated using random number generators, the outcomes of different runs

are different so that this approach is often called Monte Carlo simulation;

conclusions are obtained by averaging the results of many computer simu-

lations. Simple deterministic models for epidemics have a precise threshold

which determines whether an epidemic will or will not occur. In contrast,

stochastic models for epidemics yield quantities such as the probability

that an epidemic will occur and the mean time to extinction of the disease.

Thus the approach, concepts and appropriate questions are quite different

for stochastic models [9, 11, 39, 43, 49, 62, 99, 133].

Both deterministic and stochastic epidemiological models have other

limitations besides being only approximations of reality. Deterministic mod-

els do not reflect the role of chance in disease spread. Sometimes parameter

values in deterministic models are set equal to the mean of observed values

and the information on the variance is ignored. A set of initial conditions

leads to exactly one solution in a deterministic model; thus no information



is available on the reliability of or the confidence in the results. When quan-

tities such as the basic reproduction number, contact number, replacement

number, average infectious period, etc. are estimated by fitting output to

data, confidence intervals on these estimates are not obtained. Confidence

intervals also can not be found for comparisons and forecasts. Some un-

derstanding of the dependence on parameter values is obtained through

a sensitivity analysis where the effects of changes in parameter values on

results are found. If the variance of the observed value of a parameter is

high and the results are sensitive to that parameter, then the confidence in

the results would be low. However, the lack of precise confidence intervals

implies that the reliability of the results is uncertain.

Stochastic epidemiological models incorporate chance, but it is usually

harder to get analytic results for these models. Moreover, computational re-

sults are also harder since Monte Carlo simulations require many computer

runs (25 to 100 or more) in order to detect patterns and get quantitative

results. Even for stochastic epidemiological models where parameters are

estimated by fitting the mean of the simulations to data, it may not be

possible to find confidence intervals on these parameter estimates.

1.3. Definitions, assumptions and model formulations

The study of disease occurrence is called epidemiology. An epidemic is an

unusually large, short term outbreak of a disease. A disease is called endemic

if it persists in a population. Thus epidemic models are used to describe

rapid outbreaks that occur in less than one year, while endemic models

are used for studying diseases over longer periods, during which there is

a renewal of susceptibles by births or recovery from temporary immunity.

The spread of an infectious disease involves not only disease-related factors

such as the infectious agent, mode of transmission, latent period, infectious

period, susceptibility and resistance, but also social, cultural, demographic,

economic and geographic factors. The three basic models considered here

are the simplest prototypes of three different types of epidemiological mod-

els. It is important to understand their behavior before considering general

models incorporating more of the factors above.

The population under consideration is divided into disjoint classes whose

sizes change with time t. In the basic epidemiological models presented

here, it is assumed that the population has constant size N, which is suffi-

ciently large so that the sizes of each class can be considered as continuous

variables. Models with variable population size are considered in my third


18 H.W. Hethcote

tutorial lecture. If the model is to include vital dynamics, then it is assumed

that births and natural deaths occur at equal rates and that all newborns

are susceptible. The three basic models assume that the population is ho-

mogeneously mixing, but models for heterogeneously mixing populations

have been formulated and analyzed [83, 78]. A basic concept in epidemi-

ology is the existence of thresholds; these are critical values for quantities

such as the basic reproduction number, contact number, population size,

or vector density that must be exceeded in order for an epidemic to occur

or for a disease to remain endemic. The formulations used here are some-

what different from the most classical formulations [9], since they involve

the fractions of the populations in the classes instead of the numbers in the

classes. These formulations have much more intuitive threshold conditions

involving basic reproduction numbers or contact numbers instead of popu-

lation sizes. See [71, 83] for further comparisons of formulations in terms of

proportions and numbers in the classes.

Compartments with labels such as M, S, E, I, and R are often used

for the epidemiological classes as shown in Figure 1. If a mother has been

infected, then some IgG antibodies are transferred across the placenta, so

that her newborn infant has temporary passive immunity to an infection.

The class M contains these infants with passive immunity. After the mater-

nal antibodies disappear from the body, the infant moves to the susceptible

class S. Infants who do not have any passive immunity, because their moth-

ers were never infected, also enter the class S of susceptible individuals;

that is, those who can become infected. When there is an adequate contact

of a susceptible with an infective so that transmission occurs, then the sus-

ceptible enters the exposed class E of those in the latent period, who are

infected, but not yet infectious. After the latent period ends, the individual

enters the class I of infectives, who are infectious in the sense that they are

capable of transmitting the infection. When the infectious period ends, the

individual enters the recovered class R consisting of those with permanent

infection-acquired immunity.

The choice of which compartments to include in a model depends on

the characteristics of the particular disease being modeled and the pur-

pose of the model. The passively immune class M and the latent period

class E are often omitted, because they are not crucial for the susceptible-

infective interaction. Acronyms for epidemiology models are often based

on the flow patterns between the compartments such as MSEIR, MSEIRS,

SEIR, SEIRS, SIR, SIRS, SEI, SEIS, SI, and SIS. For example, in the



M

?

births withpassive immunity

-transferfrom M

?deaths

S

?

births withoutpassive immunity

-horizontalincidence

?deaths

E -transferfrom E

?deaths

I -transferfrom I

?deaths

R

?deaths

Fig. 1. The general transfer diagram for the MSEIR model with the passively-immuneclass M, the susceptible class S, the exposed class E, the infective class I, and the recov-ered class R.

MSEIR model shown in Figure 1, passively immune newborns first become

susceptible, then exposed in the latent period, then infectious, and then

removed with permanent immunity. An MSEIRS model would be similar,

but the immunity in the R class would be temporary, so that individuals

would regain their susceptibility when the temporary immunity ends. An

MSEIRS endemic model in a population with constant size is formulated

and analyzed in a later section of these first two tutorial lectures.

The simplest model in which recovery does not give immunity is the

SIS model, since individuals move from the susceptible class to the infective

class and then back to the susceptible class upon recovery. If individuals

recover with permanent immunity, then the simplest model is an SIR model.

If individuals recover with temporary immunity so that they eventually

become susceptible again, then the simplest model is an SIRS model [71, 82].

If individuals do not recover, then the simplest model is an SI model. In

general, SIR models are appropriate for viral agent diseases such as measles,

mumps, and smallpox, while SIS models are appropriate for some bacterial

agent diseases such as meningitis, plague, and sexually transmitted diseases,

and for protozoan agent diseases such as malaria and sleeping sickness (see

Table 1).

The three most basic deterministic models for infectious diseases which

are spread by direct person-to-person contact in a population are the SIS

endemic, the SIR epidemic, and the SIR endemic models. We use these

three models to illustrate the essential results and then discuss their exten-

sions to the more general models listed in the previous paragraph. These

simplest models are formulated as initial value problems for systems of

ordinary differential equations and are analyzed mathematically. The pre-

sentation here provides a sound intuitive understanding for the three most

basic epidemiological models for microparasitic infections. Theorems are


20 H.W. Hethcote

stated regarding the asymptotic stability regions for the equilibrium points

and phase plane portraits of solution paths are presented. Parameters are

estimated for various diseases and are used to compare the vaccination

levels necessary for herd immunity for these diseases. Although the three

models presented are simple and their mathematical analyses are elemen-

tary, these models provide notation, concepts, intuition and foundation for

considering more refined models. Other possible refinements are described

briefly in the last section.

1.3.1. Formulating epidemiology models

The horizontal incidence shown in Figure 1 is the infection rate of suscep-

tible individuals through their contacts with infectives, so it is the number

of new cases per unit time. If S(t) is the number of susceptibles at time

t, I(t) is the number of infectives, and N is the total population size, then

s(t) = S(t)/N and i(t) = I(t)/N are the susceptible and infectious frac-

tions, respectively. Let β be the average number of adequate contacts (i.e.

contacts sufficient for transmission) of a person per unit time. Here we as-

sume that the contact rate β is fixed and does not depend on the population

size N or vary seasonally. The type of direct or indirect contact necessary

to be adequate for transmission depends on the specific disease. From these

definitions we see that βI/N = βi is the average number of contacts with

infectives per unit time of one susceptible, and (βI/N)S = βNis is the num-

ber of new cases per unit time due to the S = Ns susceptibles. This form

of the incidence is called the standard incidence, because it is formulated

from the basic principles above [71, 77]. Because this form of the incidence

depends on the frequency I/N of infectives, (βI/N)S is sometimes called

frequency-dependent incidence [87].

The mass action law ηIS = η(Ni)(Ns), with η as a mass action coeffi-

cient, has sometimes been used for the horizontal incidence. The parameter

η has no direct epidemiological interpretation, but comparing it with the

standard formulation shows that β = ηN , so that this form implicitly as-

sumes that the contact rate β increases linearly with the population size.

Naively, it might seem plausible that the population density and hence

the contact rate would increase with population size, but the daily contact

patterns of people are often similar in large and small communities, cities,

and regions. For human diseases the contact rate seems to be only very

weakly dependent on the population size. Using an incidence of the form

ηNvSI/N , data for five human diseases in communities with population



sizes from 1,000 to 400,000 ([6], p. 157, [7], p. 306) implies that v is be-

tween 0.03 and 0.07. This strongly suggests that the standard incidence

corresponding to v = 0 is more realistic for human diseases than the simple

mass action incidence corresponding to v = 1. This result is consistent with

the concept that people are infected through their daily encounters and

the patterns of daily encounters are largely independent of community size

within a given country (e.g. students of the same age in a country usually

have a similar number of daily contacts).

The standard incidence is also a better formulation than the simple

mass action law for animal populations such as mice in a mouse-room or

animals in a herd [40], because disease transmission primarily occurs locally

from nearby animals. For more information about the differences in models

using these two forms of the horizontal incidence, see [63, 64, 65, 71, 83, 127].

See [87] for a recent very detailed comparison of the standard (frequency-

dependent) and mass action incidences. Vertical incidence, which is the

infection rate of newborns by their mothers, is sometimes included in epi-

demiology models by assuming that a fixed fraction of the newborns are

infected vertically [22]. Various forms of nonlinear incidences have been

considered [85, 116, 117, 118]. See [81] for a survey of mechanisms including

nonlinear incidences that can lead to periodicity in epidemiological models.

A common assumption is that the movements out of the M , E, and I

compartments and into the next compartment are governed by terms like

δM , εE, and γI in an ordinary differential equations model. It has been

shown [82] that these terms correspond to exponentially distributed waiting

times in the compartments. For example, the transfer rate γI corresponds

to P (t) = e−γt as the fraction that is still in the infective class t units

after entering this class and 1/γ as the mean waiting time. For measles

the mean period 1/δ of passive immunity is about 6 to 9 months, while

the mean latent period 1/ε is 1-2 weeks and the mean infectious period

1/γ is about 1 week. Another possible assumption is that the fraction still

in the compartment t units after entering is a non-increasing, piecewise

continuous function P (t) with P (0) = 1 and P (∞) = 0. Then the rate of

leaving the compartment at time t is −P ′(t) so the mean waiting time in the

compartment is∫∞

0t(−P ′(t))dt =

∫∞

0P (t)dt. These distributed delays lead

to epidemiology models with integral or integro-differential or functional

differential equations. If the waiting time distribution is a step function

given by P (t) = 1 if 0 ≤ t ≤ τ , and P (t) = 0 if τ ≤ t, then the mean

waiting time is τ and for t ≥ τ the model reduces to a delay-differential


22 H.W. Hethcote

equation [82]. Each waiting time in a model can have a different distribution,

so there are many possible models [77].

Table 3. Summary of notation.

M passively-immune infants

S susceptibles

E exposed people in the latent period

I infectives

R recovered people with immunity

m, s, e, i, r fractions of the population in the classes above

β contact rate

1/δ average period of passive immunity

1/ε average latent period

1/γ average infectious period

R0 basic reproduction number

σ contact number

R replacement number

1.3.2. Three threshold quantities: R0, σ, and R

The basic reproduction number R0 is defined as the average number of

secondary infections that occur when one infective is introduced into a

completely susceptible host population [45]. Note that R0 is also called the

basic reproduction ratio [42] or basic reproductive rate [7]. It is implicitly

assumed that the infected outsider is in the host population for the entire

infectious period and mixes with the host population in exactly the same

way that a population native would mix. The contact number σ is defined

as the average number of adequate contacts of a typical infective during

the infectious period [71, 83]. An adequate contact is one that is sufficient

for transmission, if the individual contacted by the susceptible is an infec-

tive. The replacement number R is defined to be the average number of

secondary infections produced by a typical infective during the entire pe-

riod of infectiousness [71]. Some authors use the term reproduction number

instead of replacement number, but it is better to avoid the name reproduc-

tion number since it is easily confused with the basic reproduction number.

Note that these three quantities R0, σ, and R are all equal at the beginning

of the spread of an infectious disease when the entire population (except

the infective invader) is susceptible. In recent epidemiological modeling lit-

erature, the basic reproduction number R0 is often used as the threshold

quantity which determines whether a disease can invade a population.



Although R0 is only defined at the time of invasion, σ and R are de-

fined at all times. For most models, the contact number σ remains constant

as the infection spreads, so it is always equal to the basic reproduction

number R0. In these models σ and R0 can be used interchangeably and

invasion theorems can be stated in terms of either quantity. But for some

pertussis models [80], the contact number σ becomes less than the basic re-

production number R0 after the invasion, because new classes of infectives

with lower infectivity appear when the disease has entered the population.

The replacement number R is the actual number of secondary cases from

a typical infective, so that after the infection has invaded a population and

everyone is no longer susceptible, R is always less than the basic repro-

duction number R0. Also after the invasion, the susceptible fraction is less

than one, so that not all adequate contacts result in a new case. Thus the

replacement number R is always less than the contact number σ after the

invasion. Combining these results leads to

R0 ≥ σ ≥ R,

with equality of the three quantities at the time of invasion. Note that

R0 = σ for most models, and σ > R after the invasion for all models.

1.4. The basic SIS endemic model

The basic SIS model is the simplest model for diseases in which an infection

does not confer immunity. Thus susceptibles become infected and then

become susceptible again upon recovery. It is an endemic model because

the disease persists, in contrast to the epidemic model in the next section in

which the disease eventually dies out. The SIS model with vital dynamics

(births and deaths) is given by

dS/dt = µN − µS − βIS/N + γI, S(0) = S0 > 0 (1.4.1)

dI/dt = βIS/N − γI − µI, I(0) = I0 > 0

with S(t) + I(t) = N . This SIS model has an inflow of newborns into the

susceptible class at rate µN and deaths in the classes at rates µS and µI.

The deaths balance the births, so that the population size N is constant.

The mean lifetime 1/µ would be about 75 years in the United States. This

model is appropriate for bacterial agent diseases such as gonorrhea, menin-

gitis and streptococcal sore throat.


24 H.W. Hethcote

Dividing the equations in (1.4.1) by the constant total population size

N and using i(t) = I(t)/N and s(t) = S(t)/N = 1 − i(t) yields

di/dt = βi(1 − i) − (γ + µ)i, i(0) = i0 > 0. (1.4.2)

Note that this equation involves the infected fraction, but does not involve

the population size N . Here only nonnegative solutions are considered since

negative solutions have no epidemiological significance. Since (1.4.2) is a

Bernoulli differential equation, the substitution y = i−1 converts it into a

linear differential equation from which the unique solution is found to be

i(t) =

{

e(γ+µ)(σ−1)t

σ[e(γ+µ)(σ−1)t−1]/(σ−1)+1/i0for σ 6= 1

1βt+1/i0

for σ = 1(1.4.3)

where σ = β/ (γ + µ) is the contact number, which is the same as the basic

reproduction number R0. The theorem below follows directly from the

explicit solution.

Theorem 1.4.1: The solution i(t) of (1.4.2) approaches 0 as t → ∞ if

R0 = σ ≤ 1 and approaches 1 − 1/σ as t → ∞ if R0 = σ > 1.

This theorem means that for a disease without immunity with any pos-

itive initial infective fraction, the infective fraction approaches a constant

endemic value if the contact number exceeds 1; otherwise, the disease dies

out. Here the contact number σ remains equal to the basic reproduction

number R0 for all time, because no new classes of susceptibles or infectives

occur after the invasion. For this model the threshold quantity is given by

R0 = σ = β/(γ + µ), which is the contact rate β times the average death-

adjusted infectious period 1/(γ + µ), and the critical threshold value is 1.

Although the model (1.4.1) reduces to a one dimensional initial value prob-

lem (1.4.2), we describe the behavior in the si phase plane, so that it can

be compared with the phase planes for the other two basic models. For this

SIS model solution paths stay on the line s + i = 1 and all paths approach

the point (s, i) = (1/σ, 1 − 1/σ) when R0 = σ > 1 and approach the point

(s, i) = (1, 0) when R0 = σ ≤ 1.

Note that the replacement number R = σs = R0s is 1 at the endemic

equilibrium point, since s = 1/σ = 1/R0 when i = 1−1/σ. This is plausible,

since the prevalence would increase if the replacement number were greater

than 1 and it would decrease if the replacement number were less than 1.

A threshold result for an SI model is obtained from the Theorem by taking

the removal rate γ to be zero in the model. If both the removal rate γ and



the birth and death rate µ are zero, then so that σ = β/ (γ + µ) = ∞,

so that there is no threshold and eventually everyone is infected. This SI

model with γ + µ = 0 is called the “simple epidemic model” considered in

Bailey ([9], p. 22).

The prevalence is defined as the number of cases of a disease at a given

time so that it corresponds to I, but the fractional prevalence would corre-

spond to i. Since the incidence is defined to be the number of new cases per

unit time, it corresponds to the βIS/N term in model (1.4.1). At an en-

demic equilibrium the prevalence is equal to the incidence times the average

duration of infection 1/(γ + µ) since the right side of the second equation

in (1.4.1) is zero at an equilibrium.

The incidence and the prevalence of some diseases oscillate seasonally

in a population. This oscillation seems to be caused by seasonal oscillation

in the contact rate β. For example, the incidence of childhood diseases

such as measles and rubella increase each year in the winter when children

aggregate in schools [118, 158, 46]. If the contact rate β changes with time

t, then β in the models above is replaced by β(t). If β(t) is periodic with

period p, then Hethcote [70] has found the asymptotic behavior of solutions

i(t) of (1.4.2). If the average contact number σ = β/(γ +µ) satisfies σ ≤ 1,

then i(t) damps in an oscillatory manner to 0 for large t. However, if σ > 1,

then i(t) approaches an explicit periodic solution for large t. See [70, 75] for

figures illustrating these two cases. Gonorrhea is an example of a disease

for which infection does not confer immunity, so it is of SIS type. See [88]

for graphs of the seasonal oscillation of reported cases of gonorrhea from

1946 to 1984. Numerous models for gonorrhea transmission dynamics and

control including a seasonal oscillation model are presented in Hethcote and

Yorke [88].

1.5. The basic SIR epidemic model

The basic SIR epidemic model is given by the initial value problem

dS/dt = −βIS/N, S(0) = So ≥ 0,

dI/dt = βIS/N − γI, I(0) = Io ≥ 0,

dR/dt = γI, R(0) = Ro ≥ 0,

(1.5.1)

where S(t), I(t), and R(t) are the numbers in these classes, so that S(t) +

I(t) + R(t) = N . This SIR model is a special case of the MSEIR model in

Figure 1, in which the passively immune class M and the exposed class E


26 H.W. Hethcote

0 0.2 0.4 0.6 0.8 1susceptible fraction, s

0

0.2

0.4

0.6

0.8

1

infectivefraction,i

smax↑= 1

σ

σ = 3

Fig. 2. Phase plane portrait for the classic SIR epidemic model with contact number

σ = 3.

are omitted. This model uses the standard incidence and has recovery at

rate γI, corresponding to an exponential waiting time e−γt. Since the SIR

epidemic model is used for outbreaks occurring in a short time period, this

model has no vital dynamics (births and deaths). Dividing the equations

in (1.5.1) by the constant total population size N yields

ds/dt = −βis, s(0) = so ≥ 0,

di/dt = βis − γi, i(0) = io ≥ 0,(1.5.2)



with r(t) = 1− s(t)− i(t), where s(t), i(t), and r(t) are the fractions in the

classes. The triangle T in the si phase plane given by

T = {(s, i) |s ≥ 0, i ≥ 0, s + i ≤ 1} (1.5.3)

is positively invariant and unique solutions exist in T for all positive time,

so that the model is mathematically and epidemiologically well posed [71].

Here the contact number σ = β/γ is the contact rate β per unit time

multiplied by the average infectious period 1/γ, so it has the proper inter-

pretation as the average number of adequate contacts of a typical infective

during the infectious period. Here the replacement number at time zero is

σso, which is the product of the contact number σ and the initial susceptible

fraction so.

Theorem 1.5.1: Let (s(t), i(t)) be a solution of (1.5.2) in T . If σso ≤ 1,

then i(t) decreases to zero as t → ∞. If σso > 1, then i(t) first increases up

to a maximum value imax = io + so − 1/σ − [ln(σso)]/σ and then decreases

to zero as t → ∞. The susceptible fraction s(t) is a decreasing function and

the limiting value s∞ is the unique root in (0, 1/σ) of the equation

io + so − s∞ + ln(s∞/so)/σ = 0. (1.5.4)

Typical paths in T are shown in Figure 2, and solutions as a function of

time are shown in Figure 3. Note that the hallmark of a typical epidemic

outbreak is an infective curve that first increases from an initial io near

zero, reaches a peak, and then decreases towards zero as a function of

time. For example, a recent measles epidemic in the Netherlands [35] is

shown in Figure 4. The susceptible fraction s(t) always decreases, but the

final susceptible fraction s∞ is positive. The epidemic dies out because,

when the susceptible fraction s(t) goes below 1/σ, the replacement number

σs(t) goes below one. The results in the theorem are epidemiologically

reasonable, since the infectives decrease and there is no epidemic, if enough

people are already immune so that a typical infective initially replaces itself

with no more than one new infective (σso ≤ 1). But if a typical infective

initially replaces itself with more than one new infective (σso > 1), then

infectives initially increase so that an epidemic occurs. The speed at which

an epidemic progresses depends on the characteristics of the disease. See

[75] for more examples of epidemic outbreak curves.

To prove the theorem, observe that the solution paths

i(t) + s(t) − [ln s(t)]/σ = io + so − [ln so]/σ


28 H.W. Hethcote

0 5 10 15 20 25time in days

0

0.2

0.4

0.6

0.8

1

infective fraction

susceptible fraction

σ = 3

Fig. 3. Solutions of the classic SIR epidemic model with contact number σ = 3 and

average infectious period 1/γ = 3 days.

in Figure 2 are found from the quotient differential equation di/ds = −1 +

1/(σs). The equilibrium points along the s axis are neutrally unstable for

s > 1/σ and are neutrally stable for s < 1/σ. For a complete (easy) proof,

see [71] or [75]. One classic approximation derived in [9] is that for small

io and so slightly greater than smax = 1/σ, the difference smax − s(∞) is

about equal to so − smax, so the final susceptible fraction is about as far

below the susceptible fraction smax (the s value where the infective fraction

is a maximum) as the initial susceptible fraction was above it (see Figure

2). Observe that the threshold result here involves the initial replacement

number R = σso and does not involve the basic reproduction number R0.



Fig. 4. Reported number of measles cases in the Netherlands by week of onset andvaccination status during April 1999–January 2000. Most of the unvaccinated cases werepeople belonging to a religious denomination that routinely does not accept vaccination.

The 2961 measles cases included 3 measles-related deaths.

1.6. The basic SIR endemic model

The classic endemic model is the SIR model with vital dynamics (births

and deaths) given by

dS/dt = µN − µS − βIS/N, S(0) = So ≥ 0,

dI/dt = βIS/N − γI − µI, I(0) = Io ≥ 0,

dR/dt = γI − µR, R(0) = Ro ≥ 0,

(1.6.1)

with S(t)+ I(t)+R(t) = N . This SIR model is almost the same as the SIR

epidemic model (1.5.1) above, except that it has an inflow of newborns into

the susceptible class at rate µN and deaths in the classes at rates µS, µI,

and µR. The deaths balance the births, so that the population size N is

constant. Dividing the equations in (1.6.1) by the constant total population


30 H.W. Hethcote


0

0.2

0.4

0.6

0.8

1

infectivefraction,i

R0 = σ = 0.5

Fig. 5. Phase plane portrait for the classic SIR endemic model with contact numberσ = 0.5.

size N yields

ds/dt = −βis + µ − µs, s(0) = so ≥ 0,

di/dt = βis − (γ + µ)i, i(0) = io ≥ 0,(1.6.2)

with r(t) = 1 − s(t) − i(t). The triangle T in the si phase plane given by

(1.5.3) is positively invariant, and the model is well posed [71]. Here the

contact number σ remains equal to the basic reproduction number R0 for

all time, because no new classes of susceptibles or infectives occur after

the invasion. For this model the threshold quantity is given by R0 = σ =




0

0.2

0.4

0.6

0.8

1

infectivefraction,i

1/σ

R0 = σ = 3

Fig. 6. Phase plane portrait for the classic SIR endemic model with contact numberσ = 3, average infectious period 1/γ = 3 days, and average lifetime 1/µ = 60 days. This

unrealistically short average lifetime has been chosen so that the endemic equilibrium is

clearly above the horizontal axis and the spiralling into the endemic equilibrium can beseen.

β/(γ + µ), which is the contact rate β times the average death-adjusted

infectious period 1/(γ + µ).

Theorem 1.6.1: Let (s(t), i(t)) be a solution of (1.6.2) in T . If R0 = σ ≤

1 or io = 0, then solution paths starting in T approach the disease-free

equilibrium given by s = 1 and i = 0. If R0 = σ > 1, then all solution

paths with io > 0 approach the endemic equilibrium given by se = 1/σ and


32 H.W. Hethcote

ie = µ(σ − 1)/β.

Figures 5 and 6 illustrate the two possibilities given in the theorem. If

R0 = σ ≤ 1, then the replacement number σs is less than one when io > 0,

so that the infectives decrease to zero. Although the speeds of movement

along the paths are not apparent from Figure 5, the infective fraction de-

creases rapidly to very near zero, and then over 100 or more years, the

recovered people slowly die off and the birth process slowly increases the

susceptibles, until eventually everyone is susceptible at the disease-free equi-

librium with s = 1 and i = 0. If R0 = σ > 1, io is small, and so is large

with σso > 1, then s(t) decreases and i(t) increases up to a peak and then

decreases, just as it would for an epidemic (compare Figure 6 with Fig-

ure 2). However, after the infective fraction has decreased to a low level,

the slow processes of the deaths of recovered people and the births of new

susceptibles gradually (over about 10 or 20 years) increases the susceptible

fraction until σs(t) is large enough that another smaller epidemic occurs.

This process of alternating rapid epidemics and slow regeneration of sus-

ceptibles continues as the paths approach the endemic equilibrium given in

the theorem. At this endemic equilibrium the replacement number σse is

one, which is plausible since if the replacement number were greater than or

less than one, the infective fraction i(t) would be increasing or decreasing,

respectively.

Theorem 1.6.1 is proved in [71] and in [75] using phase plane methods

and Liapunov functions. For this SIR model there is a transcritical (stability

exchange) bifurcation at σ = 1, as shown in Figure 7. Notice that the iecoordinate of the endemic equilibrium is negative for σ < 1, coincides with

the disease-free equilibrium value of zero at σ = 1, and becomes positive

for σ > 1. This equilibrium given by se = 1/σ and ie = µ(σ − 1)/β is

unstable for σ < 1 and is locally asymptotically stable for σ > 1, while the

disease-free equilibrium given by s = 1 and i = 0 is locally stable for σ < 1

and unstable for σ > 1. Thus these two equilibria exchange stabilities as the

endemic equilibrium moves through the disease-free equilibrium when σ = 1

and becomes a distinct, epidemiologically feasible, locally asymptotically

stable equilibrium when σ > 1.

The following interpretation of the results in the theorem and paragraph

above is one reason why the basic reproduction number R0 has become

widely used in the epidemiology literature. If the basic reproduction number

R0 (which is equal to the replacement number R when the entire population



0 0.5 1 1.5 2 2.5 3−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6BIFURCATION DIAGRAM FOR THE SIR ENDEMIC MODEL

infe

ctiv

e fr

actio

n, i

stable disease−free equilibrium

unstable disease−free equilibrium

unstable endemic equilibrium

stable endemic equilibrium

contact number, σ

Fig. 7. The bifurcation diagram for the SIR endemic model, which shows that thedisease free and endemic equilibria exchange stability when the contact number σ is 1.

is susceptible) is less than one, then the disease-free equilibrium is locally

asymptotically stable and the disease cannot “invade” the population. But

if R0 > 1, then the disease-free equilibrium is unstable with a repulsive

direction into the positive si quadrant, so the disease can “invade” in the

sense that any path starting with a small positive io moves into the positive

si quadrant where the disease persists. Thus for this classic SIR endemic

model and for many other more complex models [42], the behavior is almost

completely dependent on the threshold quantity R0, which determines not

only when the local stability of the disease-free equilibrium switches, but

also when the endemic equilibrium enters the feasible region with a positive

infective fraction. The latter condition is used to obtain expressions for R0

in age-structured models in my fourth tutorial lecture.


34 H.W. Hethcote

1.7. Similar models with M and E epidemiological states

An infected or vaccinated mother transfers some IgG antibodies across the

placenta to her fetus, so that her newborn infant has temporary passive

immunity to an infection. When these passive antibodies are gone (no new

antibodies are produced by the infant), the infant moves from the passively

immune state M to the susceptible state S. Infants who do not have any pas-

sive immunity, because their mothers were neither infected nor vaccinated,

also enter the class S of susceptible individuals, who can become infected.

When there is an adequate contact of a susceptible with an infective so

that transmission occurs, then the susceptible enters the exposed class E of

those in the latent period, who are infected, but not yet infectious. The

incubation period is defined as the period from initial exposure to the ap-

pearance of symptoms. Since a person may becomes infectious before or

after symptoms appear, the incubation period is often different from the la-

tent period. In infectious disease modeling, we are always interested in the

latent period, since we want the period until the person becomes infectious.

After the latent period ends, the individual enters the class I of infectives,

who are infectious in the sense that they are capable of transmitting the

infection. Models with the M and E epidemiological states have behaviors

that are analogous to the models without these states.

Examples of models with the epidemiological states M and/or E are

MSEIR, MSEIRS, SEIR, SEIRS, SIR, SIRS, SEI, SEIS, SI, and SIS. Mod-

els such as MSEIRS, SEIRS, SIRS, SEIS, and SIS with a flow back into the

susceptible class S are always endemic models and never epidemic models,

since there is always a threshold above which the disease remains endemic.

Models of the types MSEIR, SEIR, SIR, SEI, and SI without return flow

into the susceptible class S are endemic if they have a birth and death pro-

cess, and are epidemic if they are for a short time period without a vital

dynamics process. Behaviors for these models are similar to those for the

analogous basic models. We consider an SEIR epidemic model first and

then an MSEIRS endemic model.

1.7.1. The SEIR epidemic model

Since an epidemic occurs in a short time period, we ignore loss of temporary

immunity and the birth and death processes. Therefore we have no flow

from the removed class back to the susceptible class and omit the passively

immune class M into which newborns would move. Hence we consider the



SEIR epidemic model, which has analogous behavior to that of the basic

SIR epidemic model. For this model the initial value problem is

dS/dt = −βSI/N, S(0) = S0 > 0

dE/dt = βSI/N − εE, E(0) = E0 ≥ 0

dI/dt = εE − γI, I(0) = I0 > 0

dR/dt = γI, R(0) = R0 ≥ 0

(1.7.1)

where S(t), E(t), I(t), and R(t) are the numbers in the susceptible, ex-

posed, infectious, and removed classes, respectively, so that S(t) + E(t) +

I(t) + R(t) = N . This model uses the standard incidence, has movement

out of the exposed (latent) class E at rate εE (corresponding to an expo-

nential waiting time e−εt),and has recovery at rate γI (corresponding to

an exponential waiting time e−γt). As explained in the formulation section,

the average latent period is 1/ε and the average infectious period is 1/γ.

Dividing the equations in (1.7.1) by the constant total population size N

yields

ds/dt = −βsi, s(0) = so > 0

de/dt = βsi − εe, e(0) = eo ≥ 0

di/dt = εe − γi, i(0) = io > 0

(1.7.2)

with r(t) = 1 − s(t) − e(t) − i(t), where s(t), e(t), i(t), and r(t) are the

fractions in the classes. The tetrahedron T in the sei phase space given by

T = {(s, e, i) |s ≥ 0, e ≥ 0, i ≥ 0, s + e + i ≤ 1} (1.7.3)

is positively invariant and unique solutions exist in T for all positive time,

so that the model is mathematically and epidemiologically well posed. As

in the basic SIR epidemic model, the contact number σ = β/γ is the contact

rate β per unit time multiplied by the average infectious period 1/γ, so it

has the proper interpretation as the average number of adequate contacts of

a typical infective during the infectious period. Moreover, the replacement

number at time zero is still σso, which is the product of the contact number

σ and the initial susceptible fraction so.

Theorem 1.7.1: Let (s(t), e(t), i(t)) be a solution of (1.7.2) in T . If σso ≤

1, then e(t) and i(t) decrease to zero as t → ∞. If σso > 1, then e(t) + i(t)

first increases up a maximum with value emax + imax = eo + io + so −

[ln(σso)]/σ and then decreases to zero as t → ∞. The susceptible fraction

s(t) is a decreasing function and the limiting value s∞ is the unique root

in the interval (0, 1/σ) of the equation

eo + io + so − s∞ + [ln(s∞/so)]/σ = 0.


36 H.W. Hethcote

Typical paths in the projection of the tetrahedron T on the si face are

similar to those in Figure 2, and solutions as a function of time are similar

to those in Figure 3. Thus we still have the typical epidemic outbreak with

an infective curve that first increases from an initial io near zero, reaches a

peak, and then decreases towards zero as a function of time.

As before, the susceptible fraction s(t) always decreases, but the final

susceptible fraction s∞ is positive. The epidemic dies out because, when

the susceptible fraction s(t) goes below 1/σ, the replacement number σs(t)

goes below one. As before, if enough people are already immune so that a

typical infective initially replaces itself with no more than one new infective

(σso ≤ 1), then there is no epidemic outbreak. But if a typical infective

initially replaces itself with more than one new infective (σso > 1), then

infecteds initially increase so that an epidemic occurs. The speed at which

an epidemic of a particular disease progresses depends on the contact rate

β, the average latent period 1/ε, and the average infectious period 1/γ.

Measles has a latent period of about 7 days and an infectious period of

about 7 days, so the measles epidemic in Figure 4 evolved slowly and lasted

for about 9 months. Because the latent period for influenza is only 1-3 days

and the infectious period is only 2-3 days, an influenza epidemic can sweep

through a city in less than 6 weeks.

The proof of this theorem is similar to that for the SIR epidemic model.

The tetrahedron T is positively invariant, since no direction vectors at a

boundary point are outward. The only equilibrium points, which are along

the s axis where e = 0 and i = 0, are neutrally unstable for s > 1/σ and

are neutrally stable for s < 1/σ. The equation ds/dt = −βsi ≤ 0 implies

that s (t) is non-increasing and s(t) ≥ 0, so that a unique limit s∞ exists as

t → ∞. Since dr/dt = γi ≥ 0 and r(t) is bounded above by 1, the limit r(∞)

exists. Since e(t)+ i(t) = 1−s(t)−r(t), the limit e(∞)+ i(∞) exists. From

d(e+ i)/dt = (βs−γ)i, we see that the limit i(∞) must exist and i(∞) = 0

if s(∞) 6= 1/σ. Alternatively, i(∞) = 0, since otherwise dr/dt = γi/2 for

t sufficiently large, so that r(∞) = ∞, which contradicts r(∞) ≤ 1. Also

e(∞) = 0, since e(∞) > 0 would contradict di/dt = εe−γi. Solution paths

e(t) + i(t) + s(t) − [ln s(t)]/σ = eo + io + so − [ln so]/σ (1.7.4)

are found from the quotient differential equation d(e+ i)/ds = −1+1/(σs).

The theorem conclusions follow from (1.7.4) and the observation that e(t)+

i(t) has a maximum on the solution path when σs = 1. Similar to the

SIR epidemic model, the threshold result involves the initial replacement



number σso and does not involve the basic reproduction number R0.

1.7.2. The MSEIRS endemic model

The transfer diagram for the MSEIRS model is shown in Figure 8. The

MSEIR with permanent immunity is suitable for a directly transmitted vi-

ral disease such as measles, rubella, or mumps. As in the SIR endemic

model, let the birth and death rate constants be µ, so the population size

N(t) remains constant. An MSEIR model with exponentially changing pop-

ulation size is formulated and analyzed in [80]. The numbers of people in

the epidemiological classes are denoted by M(t), S(t), E(t), I(t), and R(t),

where t is time, and the fractions of the population in these classes are

m(t), s(t), e(t), i(t), and r(t). We are interested in finding conditions that

determine whether the disease dies out (i.e. the fraction i goes to zero) or

remains endemic (i.e. the fraction i remains positive).

M

?

births

-δM

?deaths

S

?

births

-λS

?deaths

E -εE

?deaths

I -γI

?deaths

R

?deaths

XXXX

XXXX»»»»»»»»9

ρR

Fig. 8. Transfer diagram for the MSEIRS model with the passively-immune class M,

the susceptible class S, the exposed class E, the infective class I, and the recovered classR.

The birth rate µS into the susceptible class of size S corresponds to

newborns whose mothers are susceptible, and the other newborns µ(N −S)

enter the passively immune class of size M , since their mothers were in-

fected or had some type of immunity. Although all women would be out

of the passively immune class long before their childbearing years, theo-

retically a passively immune mother would transfer some IgG antibodies

to her newborn child, so the infant would have passive immunity. Deaths

occur in the epidemiological classes at the rates µM , µS, µE, µI, and µR,

respectively.

In this MSEIRS epidemiological model, the transfer out of the passively

immune class is δM , the transfer out of the exposed class is εE, the recov-

ery rate from the infectious class is γI, and the rate of loss of immunity


38 H.W. Hethcote

is ρR. The linear transfer terms in the differential equations correspond to

waiting times with negative exponential distributions, so that when births

and deaths are ignored, the mean passively immune period is 1/δ, the mean

latent period is 1/ε, the mean infectious period is 1/γ, and the mean pe-

riod of infection-induced immunity is 1/ρ [82]. These periods would 1/δ =

6 months, 1/ε = 14 days, and 1/γ = 7 days in an MSEIR model for chick-

enpox [142]. For sexually transmitted diseases, it is useful to define both a

sexual contact rate and the fraction of contacts that result in transmission,

but for directly-transmitted diseases spread primarily by aerosol droplets,

transmission may occur by entering a room, hallway, building, etc. that

is currently or has been occupied by an infective. Since there is no clear

definition of a contact or a transmission fraction, they are replaced by a

definition that includes both. An adequate contact is a contact that is suf-

ficient for transmission of infection from an infective to a susceptible. Let

the contact rate β be the average number of adequate contacts per person

per unit time, so that the force of infection λ = βi is the average number

of contacts with infectives per unit time. Then the incidence (the number

of new cases per unit time) is λS = βiS = βSI/N , since it is the number

of contacts with infectives per unit time of the S susceptibles. As described

earlier, this standard form βSI/N for the incidence is consistent with nu-

merous studies which show that the contact rate β is nearly independent

of the population size.

The system of differential equations for the numbers in the epidemio-

logical classes is

dM/dt = µ(N − S) − (δ + µ)M + ρR, (1.7.5)

dS/dt = µS + δM − βSI/N − µS,

dE/dt = βSI/N − (ε + µ)E,

dI/dt = εE − (γ + µ)I,

dR/dt = γI − (ρ + µ)R.

It is convenient to convert to differential equations for the fractions in the

epidemiological classes with simplifications by dividing by the constant

population size N and eliminating the differential equation for s by us-

ing s = 1 − m − e − i − r. Then the ordinary differential equations for the



MSEIR model are

dm/dt = µ(e + i + r) − δm + ρr

de/dt = βi(1 − m − e − i − r) − (ε + µ)e

di/dt = εe − (γ + µ)i

dr/dt = γi − (ρ + µ)r

(1.7.6)

A suitable domain is

D = {(m, e, i, r) : m ≥ 0, e ≥ 0, i ≥ 0, r ≥ 0,m + e + i + r ≤ 1}.

The domain D is positively invariant, because no solution paths leave

through any boundary. The right sides of (1.7.6) are smooth, so that initial

value problems have unique solutions that exist on maximal intervals [68].

Since paths cannot leave D, solutions exist for all positive time. Thus the

model is mathematically and epidemiologically well posed.

1.7.2.1. Equilibria and thresholds

The basic reproduction number R0 for this MSEIRS model is the same as

the contact number σ given by

R0 = σ =βε

(γ + µ)(ε + µ). (1.7.7)

This R0 is the product of the contact rate β per unit time, the average

infectious period adjusted for population growth of 1/(γ + µ), and the

fraction ε/(ε + µ) of exposed people surviving the latent class E. Thus R0

has the correct interpretation that it is the average number of secondary

infections due to an infective during the infectious period, when everyone in

the population is susceptible. The equations (1.7.6) always have a disease-

free equilibrium given by m = e = i = r = 0, so that s = 1. If R0 > 1,

there is also a unique endemic equilibrium in D given by

me =(ε + µ)(γ + µ)(ρ + µ)

δεγ + (ρ + µ)[δε + δγ + εγ + µ(δ + ε + γ) + µ2]

(

1 −1

R0

)

ee =δ(γ + µ)(ρ + µ)


(

1 −1

R0

)

ie =δε(ρ + µ)


(

1 −1

R0

)

re =δεγ


(

1 −1

R0

)

(1.7.8)

where se = 1/R0 = 1/σ. Note that the replacement number σse is 1 at the

endemic equilibrium.


40 H.W. Hethcote

By linearization, the disease-free equilibrium is locally asymptotically

stable if R0 < 1 and is an unstable hyperbolic equilibrium with a stable

manifold outside D and an unstable manifold tangent to a vector into D

when R0 > 1. The disease-free equilibrium can be shown to be globally

asymptotically stable in D if R0 ≤ 1 by using the Liapunov function V =

εe + (ε + µ)i as follows. The Liapunov derivative is V = [βεs− (ε + µ)(γ +

µ)]i ≤ 0, since βε ≤ (ε + µ)(γ + µ). The set where V = 0 is the face of D

with i = 0, but di/dt = εe on this face, so that i moves off the face unless

e = 0. When e = i = 0, dr/dt = −µr, so that r → 0. When e = i = r = 0,

then dm/dt = −δm, so m → 0. Because the origin is the only positively

invariant subset of the set with V = 0, all paths in D approach the origin

by the Liapunov-Lasalle theorem ([68], p. 296). Thus if R0 ≤ 1, then the

disease-free equilibrium is globally asymptotically stable in D.

The characteristic equation corresponding to the Jacobian at the en-

demic equilibrium is a fourth degree polynomial. Using a symbolic alge-

bra program, it can be shown that the Routh-Hurwitz criteria are satis-

fied if R0 > 1, so that the endemic equilibrium (1.7.8) is locally asymp-

totically stable when it is in D. Thus if R0 > 1, then the disease-free

equilibrium is unstable and the endemic equilibrium is locally asymptot-

ically stable. The system (1.7.6) can be defined to be uniformly persis-

tent if lim inft→∞ i(t) ≥ c for some c > 0 for all initial points such that

e(0) + i(0) > 0. The properties of the disease-free equilibrium and The-

orem 4.5 in [151] imply that the system (1.7.6) is uniformly persistent if

R0 > 1. Based on results for the SIR and SEIR models, we expect (but

have not proved rigorously) that all paths in D with some initial latents or

infectives go the endemic equilibrium if R0 > 1. Then we have the usual

behavior for an endemic model, in the sense that the disease dies out below

the threshold, and the disease goes to a unique endemic equilibrium above

the threshold.

1.7.2.2. Special cases of the MSEIRS endemic model

Other similar models also have the endemic threshold property above. The

MSEIR model is similar to the MSEIRS model, but ρ = 0 so that the

immunity after an infection is permanent. This MSEIR model has a simpler

endemic equilibrium, but it has the same basic reproduction number R0

given by (1.7.7). If δ → ∞, then heuristically the M class disappears (think

of people moving through the M class with infinite speed), so that the



MSEIR and MSEIRS models become SEIR and SEIRS models [116] with

the same basic reproduction number R0 given by (1.7.7). If ε → ∞, then

the E class disappears, leading to an MSIR and MSIRS models with R0 =

β/(γ + µ). If an SEIRS model has an ρR transfer term from the removed

class R to the susceptible class S and ρ → ∞, then the R class disappears,

leading to an SEIS model with R0 given by (1.7.7). If ε → ∞, then the E

class disappears and the SEIS model becomes the basic SIS endemic model

with R0 = β/(γ + µ). If δ → ∞ and ε → ∞, then both the M and E

classes disappear in the MSEIRS model leading to an SIRS model with

R0 = β/(γ + µ).

Global stability has been proved for some of these models. For the

SEIR model with constant population size, the global stability below the

threshold was proved in [116] and the global stability above the threshold

was proved using clever new methods [112]. Global stability of the endemic

equilibrium has been proved for the SEIRS model with short or long period

of immunity [114], and for the SEIR model with exponentially changing

population size under a mild restriction [113]. The global stabilities of

the SIR, SIRS, SEIS models with constant population sizes are proved by

standard phase plane methods [71, 75]. Epidemiology models with variable

population size are considered in my third tutorial lecture.

1.8. Threshold estimates using the two SIR models

The two basic SIR models in previous sections are very important as con-

ceptual models (similar to predator-prey and competing species models in

ecology). The SIR epidemic modeling yields the useful concept that the

replacement number R = σso is the threshold quantity which determines

when an epidemic occurs. It also yields formulas for the peak infective

fraction im and the final susceptible fraction s∞. The SIR endemic model

yields R0 = σ as the threshold quantity which determines when the dis-

ease remains endemic, the concept that the infective replacement number

σse is 1 at the endemic equilibrium, and the explicit dependence of the

infective fraction ie on the parameters. However, these simple SIR models

have obvious limitations. Although the assumption that the population is

uniform and homogeneously mixing is plausible since many children have

similar patterns of preschool and school attendance, mixing patterns can

depend on many factors including age (children usually have more adequate

contacts per day than adults). Moreover, different geographic and social-

economic groups have different contact rates. Despite their limitations,


42 H.W. Hethcote

the classic SIR models can be used to obtain some useful estimates and

comparisons.

From equation (1.5.4) for s∞ in the classic SIR epidemic model, the

approximation

σ ≈ln(so/s∞)

so − s∞

follows because io is negligibly small. By using data on the susceptible

fractions so and s∞ at the beginning and end of epidemics, this formula can

be used to estimate contact numbers for specific diseases [75]. Using blood

samples from freshmen at Yale University [56], the fractions susceptible to

rubella at the beginning and end of the freshman year were found to be

0.25 and 0.090, so the epidemic formula above gives σ ≈ 6.4. The fractions

so = 0.49 and s∞ = 0.425 for Epstein-Barr virus (related to mononucleosis)

lead to σ ≈ 2.2, and the fractions so = 0.911 and s∞ = 0.514 for influenza

(H3N2 type A “Hong Kong”) lead to σ ≈ 1.44. For the 1957 “Asian Flu”

(H2N2 type A strain of influenza) in Melbourne, Australia, the fractions

so = 1 and s∞ = 0.55 from [20] (p. 129) yield the contact number estimate

σ ≈ 1.33. Thus the easy theory for the basic SIR epidemic model yields the

formula above that can be used to estimate contact numbers from epidemic

data.

The basic SIR endemic model can also be used to estimate contact

numbers. If blood samples in a serosurvey are tested for antibodies to a

virus and it is assumed that the SIR model above holds in the population

with the disease at an endemic equilibrium, then the contact number can

be estimated from σ = 1/se, where se is the fraction of the samples that are

not seropositive, since se = 1 − ie − re. This approach is somewhat naive,

because the average seropositivity in a population decreases to zero as the

initial passive immunity declines and then increases as people age and are

exposed to infectives. Thus the ages of those sampled are critical in using

the estimate σ = 1/se.

For an age-structured SIR model with negative exponential survival

considered in my fourth tutorial lecture, one estimation formula for the

basic reproduction number is R0 = 1+L/A, where L is the average lifetime

1/µ and A is the average age of infection. This estimation formula can also

be derived heuristically from the basic SIR endemic model. The incidence

rate at the endemic equilibrium is βiese, so that βie is the incidence rate

constant, which with exponential waiting time implies that the average age

of infection (the mean waiting time in S) is A = 1/βie = 1/[µ(σ−1)]. Using



µ = 1/L, this leads to R0 = σ = 1 + L/A, since R0 = σ for this model.

1.9. Comparisons of some directly transmitted diseases

Data on average ages of infection and average lifetimes in developed coun-

tries have been used to estimate basic reproduction numbers R0 for some

viral diseases ([7], p. 70, [75]). From Table 4, we see that these estimates of

R0 are about 16 for measles, 11 for varicella (chickenpox), 8 for mumps, 7

for rubella, and 5 for poliomyelitis and smallpox. Because disease-acquired

immunity is only temporary for bacterial diseases such as pertussis (whoop-

ing cough), diphtheria, and scarlet fever, the formula R0 = σ = 1 + L/A

cannot be used to estimate R0 for these diseases (see [80] for estimates of

R0 and σ for pertussis).

Table 4. Estimates of basic reproduction numbers and fractions for herd immunity.

Min fraction Vacc. Min fraction

Disease Location A L R0 = σ r for herd efficacy vaccinated for= 1 + L/A immunity VE herd immunity

Measles England and 4.8 70 15.6 0.94 0.95 0.99Wales, 1956–59

USA, 1912–1928 5.3 60 12.3 0.92 0.95 0.97Nigeria, 1960–68 2.5 40 17.0 0.94 0.95 0.99

Chickenpox Maryland, USA, 6.8 70 11.3 0.91 0.90 1.01

(varicella) 1943

Mumps Maryland, USA, 9.9 70 8.1 0.88 0.95 0.93

1943

Rubella England and 11.6 70 7.0 0.86 0.95 0.91Wales, 1979

West Germany, 10.5 70 7.7 0.87 0.95 0.92

1972

Poliomyelitis USA, 1955 17.9 70 4.9 0.80 ?

Netherlands, 11.2 70 4.3 0.86 ?1960

Smallpox India 12.0 50 5.2 0.81 0.95 0.85

Herd immunity occurs for a disease if enough people have disease-

acquired or vaccination-acquired immunity, so that the introduction of one

infective into the population does not cause an invasion of the disease. Intu-


44 H.W. Hethcote

itively, if the contact number is σ, so that the typical infective has adequate

contacts with σ people during the infectious period, then the replacement

number σs must be less than one so that the disease does not spread. This

means that s must be less than 1/σ, so the immune fraction r must be

satisfy r > 1 − 1/σ = 1 − 1/R0. For example, if R0 = σ = 10, then the

immune fraction must satisfy r > 1 − 1/10 = 0.9, so that the replacement

number is less than one and the disease does not invade the population.

Table 4 contains data [4] on the average age of infection for various directly

transmitted diseases and average lifetimes. See [7] for more data. Table 4

also contains estimates of basic reproduction numbers R0 and the minimum

fractions that must be immune in order to achieve herd immunity. Using

the estimates above for R0, the minimum immune fractions for herd immu-

nity in these populations are about 0.94 for measles, 0.88 for mumps, 0.86

for rubella, and about 0.80 for poliomyelitis and smallpox.

Primary vaccination failure occurs when a vaccinated individuals does

not become immune. Secondary vaccination failure occurs when the im-

munity from an initially-successful vaccination wanes, so that the person

becomes susceptible to infection again at a later time. The fraction of

those vaccinated who initially become immune is called the vaccine efficacy

VE. The vaccine efficacy for many vaccines is about 0.95, so that primary

vaccination failure occurs in about 5% of those vaccinated. Secondary vac-

cination failure is rare for vaccines for most viral diseases in Table 2. In

order to calculate the minimum fraction that must be vaccinated to achieve

herd immunity, one divides the minimum fraction that must be immune by

the vaccine efficacy. These values are shown in the last column in Table 4.

The minimum fractions that must be vaccinated to achieve herd im-

munity in these populations are about 0.99 for measles, 0.93 for mumps,

0.91 for rubella, and about 0.85 for smallpox. Although these values give

only crude, ball park estimates for the vaccination level in a community

required for herd immunity, they are useful for comparing diseases. For

example, these numbers suggest that it should be easier to achieve herd

immunity for mumps, rubella, poliomyelitis, and smallpox than for measles

and chickenpox. Indeed, the theoretical vaccination levels of 0.99 for measles

and 1.01 for varicella (chickenpox) suggests that herd immunity would be

very hard to achieve for these diseases with a one dose program, because

the percentages not vaccinated would have to be below 1% for measles,

below 7% for mumps, and below 9% for rubella. Because vaccinating all

but 1% against measles would be difficult to achieve, a two dose program



for measles is used in many countries [34, 73, 74]. With a varicella (chick-

enpox) vaccination program whose goal is to give each child one dose, it

is impossible to achieve vaccination of 101% of the population. Moreover,

immunity from the varicella (chickenpox) vaccine wanes, so that children

vaccinated may become susceptible again about 20 years after vaccination

[143].

1.10. The effectiveness of vaccination programs

The conclusions from Table 4 are justified by the actual effectiveness of

vaccination programs in reducing, locally eliminating, and eradicating these

diseases (eradication means elimination throughout the world). The infor-

mation in this section verifies that smallpox has been eradicated worldwide

and polio should be eradicated worldwide within a few years, while the dis-

eases of rubella and measles still persist at low levels in the United States

and at higher levels in many other countries. Thus this section provides

historical context and verifies the disease comparisons obtained in Table 4

from the basic SIR endemic model.

1.10.1. Smallpox

Smallpox is believed to have appeared in the first agricultural settlements

around 6,000 BC. For centuries the process of variolation with material from

smallpox pustules was used in Africa, China, and India before arriving in

Europe and the Americas in the 18th century. Edward Jenner, an English

country doctor, observed over 25 years that milkmaids who had been in-

fected with cowpox did not get smallpox. In 1796 he started vaccinating

people with cowpox to protect them against smallpox [134]. This was the

world’s first vaccine (vacca is the Latin word for cow). Two years later, the

findings of the first vaccine trials were published, and by the early 1800s, the

smallpox vaccine was widely available. Smallpox vaccination was used in

many countries in the 19th century, but smallpox remained endemic. When

the World Health Organization (WHO) started a global smallpox eradica-

tion program in 1967, there were about 15 million cases per year, of which

2 million died and millions more were disfigured or blinded by the disease

[58]. The WHO strategy involved extensive vaccination programs, surveil-

lance for smallpox outbreaks, and containment of these outbreaks by local

vaccination programs. There are some interesting stories about the WHO

campaign including the persuasion of African chiefs to allow their tribes


46 H.W. Hethcote

to be vaccinated and monetary bounty systems for finding hidden small-

pox cases in India. Smallpox was slowly eliminated from many countries,

with the last case in the Americas in 1971. The last case worldwide was

in Somalia in 1977, so smallpox has been eradicated throughout the world

[14, 58, 134]. The WHO estimates that the elimination of worldwide small-

pox vaccination saves over two billion dollars per year. The smallpox virus

has been kept in USA and Russian government laboratories; the USA keeps

it so that vaccine could be produced if smallpox were ever used in biological

terrorism [145]. See [41] for a paper on smallpox attacks that compares the

effects of quarantine, isolation, vaccination, and behavior change.

1.10.2. Poliomyelitis

Most cases of poliomyelitis are asymptomatic, but a small fraction of cases

result in paralysis. In the 1950s in the United States, there were about

60,000 paralytic polio cases per year. In 1955 Jonas Salk developed an in-

jectable polio vaccine from an inactivated polio virus. This vaccine provides

protection for the person, but the person can still harbor live viruses in their

intestines and can pass them to others. In 1961 Albert Sabin developed an

oral polio vaccine from weakened strains of the polio virus. This vaccine

provokes a powerful immune response, so the person cannot harbor the

“wild-type” polio viruses, but a very small fraction (about one in 2 million)

of those receiving the oral vaccine develop paralytic polio [14, 134]. The Salk

vaccine interrupted polio transmission and the Sabin vaccine eliminated po-

lio epidemics in the United States, so there have been no indigenous cases

of naturally-occurring polio since 1979. In order to eliminate the few cases

of vaccine-related paralytic polio each year, the United States now recom-

mends the Salk injectable vaccine for the first four polio vaccinations, even

though it is more expensive [34]. In the Americas, the last case of paralytic

polio caused by the wild virus was in Peru in 1991. In 1988 WHO set a goal

of global polio eradication by the year 2000 [141]. Most countries are using

the live-attenuated Sabin oral vaccine, because it is inexpensive (8 cents

per dose) and can be easily administered into a mouth by an untrained

volunteer. The WHO strategy includes routine vaccination, National Im-

munization Days (during which many people in a country or region are

vaccinated in order to interrupt transmission), mopping-up vaccinations,

and surveillance for acute flaccid paralysis [90]. Polio has disappeared from

many countries in the past ten years, so that by 1999 it is concentrated

in the Eastern Mediterranean region, South Asia, West Africa and Cen-



tral Africa. It is likely that polio will be eradicated worldwide soon. WHO

estimates that eradicating polio will save about $1.5 billion each year in

immunization, treatment, and rehabilitation around the globe [29].

1.10.3. Measles

Measles is a serious disease of childhood that can lead to complications and

death. For example, measles caused about 7500 deaths in the United States

in 1920 and still causes about 1 million deaths worldwide each year [31, 32].

Measles vaccinations are given to children between 6 and 18 months of age,

but the optimal age of vaccination for measles seems to vary geographically

[74].

Consider the history of measles in the United States. In the prevaccine

era, every child had measles, so the incidences were approximately equal

to the sizes of the birth cohorts. After the measles vaccine was licensed in

1963 in the United States, the reported measles incidence dropped in a few

years to around 50,000 cases per year. In 1978 the US adopted a goal of

eliminating measles and vaccination coverage increased, so that there were

fewer than 5000 reported cases per year between 1981 and 1988. Pediatric

epidemiologists at meetings at Centers for Disease Control in Atlanta in

November 1985 and February 1988 decided to continue the one-dose pro-

gram for measles vaccinations instead of changing to a more expensive two-

dose program. But there were about 16,000, 28,000, and 17,000 reported

measles cases in the United States in 1989, 1990, and 1991, respectively;

there were also measles outbreaks in Mexico and Canada during these years

[91]. Because of this major measles epidemic, epidemiologists decided in

1989 that the one-dose vaccination program for measles, which had been

used for 26 years, should be replaced with a two-dose program with the

first measles vaccination at age 12-15 months and the second vaccination

at 4-6 years, just before children start school [34]. Reported measles cases

declined after 1991 until there were only 137, 100, and 86 reported cases

in 1997, 1998, and 1999, respectively. Each year some of the reported cases

are imported cases and these imported cases can trigger small outbreaks.

The proportion of cases not associated with importation has declined from

85% in 1995, 72% in 1996, 41% in 1997, to 29% in 1998. Analysis of the epi-

demiologic data for 1998 suggests that measles is no longer an indigenous

disease in the United States [31]. Measles vaccination coverage in 19-35

month old children was only 92% in 1998, but over 99% of children had at

least one dose of measles-containing vaccine by age 6 years. Because measles


48 H.W. Hethcote

is so easily transmitted and the worldwide measles vaccination coverage was

only 72% in 1998 [32, 134], this author does not believe that it is feasible to

eradicate measles worldwide using the currently available measles vaccines.

1.10.4. Rubella

Rubella (also called 3 day measles or German measles) is a mild disease

with few complications, but a rubella infection during the first trimester

of pregnancy can result in miscarriage, stillbirth, or infants with a pattern

of birth defects called Congenital Rubella Syndrome (CRS) [14]. Because

the goal of a rubella vaccination program is to prevent rubella infections

in pregnant women, special vaccination strategies such as vaccination of

12-14 year old girls are sometimes used [73, 76]. In the prevaccine era,

rubella epidemics and subsequent congenital rubella syndrome (CRS) cases

occurred about every 4 to 7 years in the United States. During a major

rubella epidemic in 1964, it is estimated that there were over 20,000 CRS

cases in the United States with a total lifetime cost of over $2 billion [73, 76].

Since the rubella vaccine was licensed in 1969, the incidences of rubella and

CRS in the United States have decreased substantially. Since many rubella

cases are subclinical and unreported, we consider only the incidence of

Congenital Rubella Syndrome (CRS). The yearly incidences of CRS in the

United States were between 22 and 67 in the 1970s, between 0 and 50 in

the 1980s, 11 in 1990, 47 in 1991, 11 in 1992, and then between 4 and 8

between 1993 and 1999 [28]. Although there have been some increases in

CRS cases associated with occasional rubella outbreaks, CRS has been at

a relatively low level in the United States in recent years. In recent rubella

outbreaks in the United States, most cases occurred among unvaccinated

persons aged at least 20 years and among persons who were foreign born,

primarily Hispanics (63% of reported cases in 1997) [30]. Although it does

not solve the problem of unvaccinated immigrants, the rubella vaccination

program for children has reduced the incidence of rubella and CRS in the

United States to very low levels. Worldwide eradication of rubella is not

feasible, because over two-thirds of the population in the world is not yet

routinely vaccinated for rubella. Indeed, the policies in China and India of

not vaccinating against rubella may be the best policies for those countries,

because most women of childbearing age in these countries already have

disease-acquired immunity.



1.10.5. Chickenpox (varicella)

The varicella zoster virus (VZV) is the agent for varicella, commonly known

as chickenpox. Chickenpox is usually a mild disease in children that lasts

about 4-7 days with a body rash of several hundred lesions. After a case of

chickenpox, the VZV becomes latent in the dorsal root ganglia, but VZV

can reactivate in the form of zoster, commonly known as shingles. Shingles

is a painful vesicular rash along one or more sensory root nerves that usu-

ally occurs when the immune system is less effective due to illness or aging

[14]. People with shingles are less infectious than those with chickenpox,

but they can transmit the VZV. Indeed, it was found that some isolated

Amazon tribes had no antibodies to diseases such as measles, mumps and

rubella, but they did have antibodies to VZV [16]. Thus it appears that

the persistence of VZV in these small isolated populations has occurred be-

cause VZV can be dormant in people for many years and then be spread in

the population by a case of shingles. Because of the transmission by those

with both chickenpox and shingles, the expression for R0 is more compli-

cated than for the MSEIR model [143]. A varicella vaccine was licensed

in the United States in 1995 and is now recommended for all young chil-

dren. But the vaccine-immunity wanes, so that vaccinated children can get

chickenpox as adults. Two possible dangers of this new varicella vaccina-

tion program are more chickenpox cases in adults, when the complication

rates are higher, and an increase in cases of shingles. An age-structured

epidemiologic-demographic model has been used with parameters estimated

from epidemiological data to evaluate the effects of varicella vaccination

programs [142]. Although the age distribution of varicella cases does shift

in the computer simulations, this shift does not seem to be a problem since

many of the adult cases occur after vaccine-induced immunity wanes, so

they are mild varicella cases with fewer complications. In the computer

simulations, shingles incidence increases in the first 30 years after initiation

of a varicella vaccination program, because people are more likely to get

shingles as adults when their immunity is not boosted by frequent expo-

sures, but after 30 years the shingles incidence starts to decrease as the

population includes more previously vaccinated people, who are less likely

to get shingles. Thus the simulations validate the second danger that the

new vaccination program could lead to more cases of shingles in the first

several decades [142].


50 H.W. Hethcote

1.10.6. Influenza

Type A influenza has three subtypes in humans (H1N1, H2N2, and H3N2)

that are associated with widespread epidemics and pandemics (i.e. world-

wide epidemics). Types B and C influenza tend to be associated with local

or regional epidemics. Influenza subtypes are classified by antigenic prop-

erties of the H and N surface glycoproteins, whose mutations lead to new

variants every few years [14]. An infection or vaccination for one variant

may give only partial immunity to another variant of the same subtype,

so that flu vaccines must be reformulated almost every year. Sometimes

the experts do not choose the correct variant for the vaccine. For example,

the A/California/7/2004(H3N2) variant was the dominant type A influenza

in the 2004-05 flu season in the United States [36]. However, the 2004-05

influenza vaccine still contained an A/Wyoming/3/2003(H3N2)-like virus,

which was the dominant type A influenza in the 2003-04 flu season in the

United States. An A/California/7/2004(H3N2)-like virus has been recom-

mended as the H3 component for the 2005-06 Northern Hemisphere flu

vaccine. If the influenza virus variant did not change, then it should be

easy to eradicate, because the contact number for flu has been estimated

above to be only about 1.4. But the frequent drift of the A subtypes to

new variants implies that flu vaccination programs cannot eradicate them

because the target is constantly moving.

Completely new A subtypes (antigenic shift) emerge occasionally from

unpredictable recombinations of human with swine or avian influenza anti-

gens. These new subtypes can lead to major pandemics. A new H1N1 sub-

type led to the 1918-19 pandemic that killed over one-half million people

in the United States and over 20 million people worldwide. Pandemics also

occurred in 1957 from the Asian Flu (an H2N2 subtype) and in 1968 from

the Hong Kong flu (an H3N2 subtype) [104]. When 18 confirmed human

cases with 6 deaths from an H5N1 chicken flu occurred in Hong Kong in

1997, there was great concern that this might lead to another antigenic

shift and pandemic. At the end of 1997, veterinary authorities slaughtered

all (1.6 million) chickens present in Hong Kong, and importation of chick-

ens from neighboring areas was stopped. Fortunately, the H5N1 virus has

so far not evolved into a form that is readily transmitted from person to

person [148, 155]. But outbreaks of avian influenza continues to occur in

many countries, particularly in Southeast Asia, so that the danger of the

occurrence of a new type A influenza that could spread in humans contin-

ues.



1.11. Other Epidemiology Models with Thresholds

Mathematical epidemiology has now evolved into a separate area of pop-

ulation dynamics that is parallel to mathematical ecology. Epidemiology

models are now used to combine complex data from various sources in or-

der to study equally complex outcomes. In these tutorial lectures we have

focused on the role of the basic reproduction number R0, which is defined

as the average number of people infected when a typical infective enters

an entirely susceptible population. We have illustrated the significance of

R0 by obtaining explicit expressions for R0 and proving threshold results

which imply that a disease can invade a completely susceptible population

if and only if R0 > 1. Using the SIR endemic model, the estimates of R0 for

various diseases show that some diseases are more easily spread than others,

so that they are more difficult to control or eradicate. We considered the

effects of these differences for six diseases (smallpox, polio, measles, rubella,

chickenpox, and influenza).

The results presented in these tutorial lectures provide a theoretical

background for reviewing some previous results. In this section we do not

attempt to cite all papers on infectious disease models with heterogeneity,

and spatial structure, but primarily cite sources that consider thresholds

and the basic reproduction number R0. The cited papers reflect the au-

thor’s interests, but additional references are given in these papers and

in the books and survey papers listed in the Introduction. We refer the

reader to other sources for information on stochastic epidemiology mod-

els [9, 11, 39, 43, 49, 62, 99, 133], discrete time models [2, 3], models involving

macroparasites [7, 43, 67], genetic heterogeneity [7, 67], plant disease models

[106, 153], and wildlife disease models [67]. Models with age structure are

surveyed in the written version of my fourth tutorial lecture.

Irregular and biennial oscillations of measles incidences have led to vari-

ous mathematical analyses including the following seven modeling explana-

tions, some of which involve age structure. Yorke and London [158] proposed

SEIR models with seasonal forcing in delay differential equations. Dietz [46]

proposed subharmonic resonance in a seasonally forced SEIR model using

ordinary differential equations. Schenzle [140] used computer simulations

to show that the measles outbreak patterns in England and Germany could

be explained by the primary school yearly calenders and entry ages. Ol-

son and Schaffer [135] proposed chaotic behavior in simple deterministic

SEIR models. Bolker and Grenfell [18] proposed realistic age structured

models with seasonal forcing and stochastic terms. Ferguson, Nokes, and


52 H.W. Hethcote

Anderson [59] proposed finely age-stratified models with stochastic fluc-

tuations that can shift the dynamics between biennial and triennial cycle

attractors. Earn, Rohani, Bolker, and Grenfell [53] proposed a simple, time-

forced SEIR model with slow variation in the average rate of recruitment

of new susceptibles.

In recent years the human immunodeficiency virus (HIV), which leads to

acquired immunodeficiency syndrome (AIDS), has emerged as an important

new infectious disease. Many models have been developed for HIV/AIDS.

May and Anderson [122] found R0 for some simple HIV transmission mod-

els. Bongaarts [19] and May et al. [124] used models with age structure

to examine the demographic effects of AIDS in African countries. The

book [24] by Castillo contains a review of HIV/AIDS modeling papers

including single-group models, multiple-group models, and epidemiologic-

demographic models. It also contains papers on AIDS models with HIV

class age, variable infectivity, distributions for the AIDS incubation period,

heterogeneity, and structured mixing. Busenberg and Castillo [21] found an

R0 expression for an HIV model with variable infectivity and continuous

chronological and HIV class age structure and proportionate mixing. Hy-

man, Li and Stanley [94] generalized these results on R0 to HIV models

with non-proportionate mixing and discrete or continuous risk.

For many infectious diseases the transmission occurs in a diverse pop-

ulation, so the epidemiological model must divide the heterogeneous pop-

ulation into subpopulations or groups, in which the members have similar

characteristics. This division into groups can be based not only on mode of

transmission, contact patterns, latent period, infectious period, genetic sus-

ceptibility or resistance, and amount of vaccination or chemotherapy, but

also on social, cultural, economic, demographic or geographic factors. For

these models it is useful to find R0 from the threshold conditions for invasion

and endemicity, and to prove stability of the equilibria. For the SIS model

with n groups, the threshold was first found in terms of whether s(A) ≤ 0

or s(A) > 0, where s(A) is the largest real part of the eigenvalues of the

Jacobian matrix A at the disease-free equilibrium. The seminal paper [109]

of Lajmanovich and Yorke found this threshold condition and proved the

global stability of the disease-free and endemic equilibria using Liapunov

functions. This approach has been extended to SIR, SEIR, and SEIRS mod-

els with n groups [72, 149, 150]. For these models R0 can be shown to be

the spectral radius of a next generation matrix that is related to the Ja-

cobian matrix A [78, 83]. This next generation operator approach has also



been used for epidemiology models with a variety of features such as pro-

portionate mixing, preferred mixing, heterosexual transmission, host-vector

groups, multiple mixing groups, vaccination, and age structure [42, 43]. For

proportionate mixing models with multiple interacting groups, the basic

reproduction number R0 is the contact number σ, which is the weighted

average of the contact numbers in the groups [78, 83, 88]. The sexual trans-

mission of diseases often occurs in a very heterogeneous population, because

people with more sexual partners have more opportunities to be infected

and to infect others. The basic reproduction number R0 has been deter-

mined for many different models with heterogeneous mixing involving core,

social, and sexual mixing groups [88, 100, 102, 107, 108, 147]. It has been

shown that estimates of R0, under the false assumption that a heteroge-

neously mixing population is homogeneously mixing, are not greater than

the actual R0 for the heterogeneous population [1, 78]. Many models with

heterogeneity in the form of competing strains of infectious agents have

been considered for diseases such as influenza, dengue, and myxomatosis

[8, 25, 26, 27, 47, 52, 54, 55, 57, 123, 128].

HIV/AIDS is spread in a very heterogeneous population by heterosex-

ual intercourse, homosexual intercourse, and sharing of needles by inject-

ing drug users. Because of the great diversity and heterogeneity among

those at risk to HIV/AIDS, modeling this disease is a challenging task

[5, 7, 24, 79, 89, 92, 93, 97, 101]. For HIV/AIDS models with a continuous

distribution of sexual activity levels and with various preference mixing

functions, the proportionate mixing has been shown to be the only sep-

arable solution, and expressions for the basic reproduction number R0 in

the proportionate mixing case has been found [17, 21]. Expressions for R0

have also been found for HIV/AIDS models using groups of people based

on their sexual behavior, e.g. homosexual men, bisexual men, heterosex-

ual women, and heterosexual men, with further subdivisions based on their

numbers of sexual or needle-sharing partners. For staged progression mod-

els for HIV/AIDS with many infectious classes with different infectivities,

the basic reproduction number R0 is often the weighted average of the basic

reproduction numbers in the infectious classes, where the weights involve

the fraction of contacts (or partners) that result in an infection and the

probability of reaching that infectious stage [84, 95, 96, 103, 115].

There is clear evidence that infectious diseases spread geographically

and maps with isodate spread contours have been produced [7, 38, 126, 132].

Some estimated speeds of propagation are 30-60 kilometers per year for fox


54 H.W. Hethcote

rabies in Europe starting in 1939 [132], 18-24 miles per year for raccoon

rabies in the Eastern United States starting in 1977 [33], about 140 miles

per year for the plague in Europe in 1347-1350 [132], and worldwide in one

year for influenza in the 20th century [139]. Epidemiology models with spa-

tial structures have been used to describe spatial heterogeneity [7, 71, 83]

and the spatial spread of infectious diseases [23, 37, 43, 67, 132, 152]. There

seem to be two types of spatial epidemiology models [130, 152]. Diffusion

epidemiology models are formulated from non-spatial models by adding

diffusion terms corresponding to the random movements each day of sus-

ceptibles and infectives. Dispersal-kernel models are formulated by using

integral equations with kernels describing daily contacts of infectives with

their neighbors. For both types of spatial epidemiology models in infinite

domains, one often determines the thresholds (sometimes in terms of R0)

above which a traveling wave exists, finds the minimum speed of propaga-

tion and the asymptotic speed of propagation (which is usually shown to be

equal to the minimum speed), and determines the stability of the traveling

wave to perturbations [129, 137]. For spatial models in finite domains, sta-

tionary states and their stability have been investigated [23]. For stochastic

spatial models there is also a threshold condition, so that the disease dies

out below the threshold and approaches an endemic stationary distribution

above the threshold [51].

References

1. F. R. Adler, The effects of averaging on the basic reproduction ratio, Math.Biosci., 111 (1992), pp. 89-98.

2. L. J. S. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Math.Biosci., 124 (1994), pp. 83-105.

3. L. J. S. Allen and A. M. Burgin, Comparison of deterministic and stochasticSIS and SIR models in discrete-time, Math. Biosci., 163 (2000), pp. 1-33.

4. R. M. Anderson, ed., Population Dynamics of Infectious Diseases, Chapmanand Hall, London, 1982.

5. , The role of mathematical models in the study of HIV transmissionand the epidemiology of AIDS, J. AIDS, 1 (1988), pp. 241–256.

6. R. M. Anderson and R. M. May, eds., Population Biology of InfectiousDiseases, Springer-Verlag, Berlin, Heidelberg, New York, 1982.

7. , Infectious Diseases of Humans: Dynamics and Control, Oxford Uni-versity Press, Oxford, 1991.

8. V. Andreasen, J. Lin, and S. A. Levin, The dynamics of cocirculating in-fluenza strains conferring partial cross-immunity, J. Math. Biol., 35 (1997),pp. 825–842.



9. N. T. J. Bailey, The Mathematical Theory of Infectious Diseases, Hafner,New York, second ed., 1975.

10. , The Biomathematics of Malaria, Charles Griffin, London, 1982.11. M. Bartlett, Stochastic Population Models in Ecology and Epidemiology,

Methuen, London, 1960.12. N. Becker, The use of epidemic models, Biometrics, 35 (1978), pp. 295–305.13. , Analysis of Infectious Disease Data, Chapman and Hall, New York,

1989.14. A. S. Benenson, Control of Communicable Diseases in Man, American Pub-

lic Health Association, Washington, D.C., sixteenth ed., 1995.15. D. Bernoulli, Essai d’une nouvelle analyse de la mortalite causee par la

petite verole et des avantages de l’inoculation pour la prevenir, in Memoiresde Mathematiques et de Physique, Academie Royale des Sciences, Paris,1760, pp. 1–45.

16. F. L. Black, W. J. Hierholzer, F. d. P. Pinheiro, et al., Evidence for thepersistence of infectious disease agents in isolated human populations, Am.J. Epidemiol., 100 (1974), pp. 230–250.

17. S. P. Blythe and C. Castillo-Chavez, Like with like preference and sexualmixing models, Math. Biosci., 96 (1989), pp. 221–238.

18. B. M. Bolker and B. T. Grenfell, Chaos and biological complexity in measlesdynamics, Proc. Roy. Soc. London Ser. B, 251 (1993), pp. 75–81.

19. J. Bongaarts, A model of the spread of HIV infection and the demographicimpact of AIDS, Stat. Med., 8 (1989), pp. 103–120.

20. M. Burnett and D. O. White, Natural History of Infectious Disease, Cam-bridge University Press, Cambridge, fourth ed., 1974.

21. S. Busenberg and C. Castillo-Chavez, A general solution of the problemof mixing of subpopulations and its application to risk- and age-structuredmodels for the spread of AIDS, IMA J. Math. Appl. Med. Biol., 8 (1991),pp. 1–29.

22. S. Busenberg and K. L. Cooke, Vertically Transmitted Diseases, vol. 23 ofBiomathematics, Springer-Verlag, Berlin, 1993.

23. V. Capasso, Mathematical Structures of Epidemic Systems, vol. 97 of Lec-ture Notes in Biomathematics, Springer-Verlag, Berlin, 1993.

24. C. Castillo-Chavez, ed., Mathematical and Statistical Approaches to AIDSEpidemiology, vol. 83 of Lecture Notes in Biomathematics, Springer-Verlag,Berlin, 1989.

25. C. Castillo-Chavez, H. W. Hethcote, V. Andreasen, S. A. Levin, and W. M.Liu, Epidemiological models with age structure, proportionate mixing, andcross-immunity, J. Math. Biol., 27 (1989), pp. 233–258.

26. C. Castillo-Chavez, W. Huang, and J. Li, Competitive exclusion in gonor-rhea models and other sexually-transmitted diseases, SIAM J. Appl. Math.,56 (1996), pp. 494–508.

27. , Competitive exclusion and multiple strains in an SIS STD model,SIAM J. Appl. Math., 59 (1999), pp. 1790–1811.

28. Centers for Disease Control and Prevention, Summary of notifiable diseases,


56 H.W. Hethcote

United States, 1998, MMWR, 47 (1999), pp. 78–83.29. , Progress toward global poliomyelitis eradication, 1997–1998,

MMWR, 48 (1999), pp. 416–421.30. , Rubella outbreak—Westchester County, New York, 1997–1998,

MMWR, 48 (1999), pp. 560–563.31. , Epidemiology of measles—United States, 1998, MMWR, 48 (1999),

pp. 749–753.32. , Global measles control and regional elimination, 1998–99, MMWR,

48 (1999), pp. 1124–1130.33. , Update: Raccoon rabies epizootic—United States and Canada, 1999,

MMWR, 49 (2000), pp. 31–35.34. , Recommended childhood immunization schedule—United States,

2000, MMWR, 49 (2000), pp. 35–47.35. , Measles outbreak—Netherlands, April 1999–January 2000, MMWR,

49 (2000), pp. 299-303.36. , Update: Influenza Activity—United States and Worldwide, 2004-05

season, MMWR, 54 (2005), pp. 631-634.37. A. D. Cliff, Incorporating spatial components into models of epidemic

spread, in Epidemic Models: Their Structure and Relation to Data, D. Mol-lison, ed., Cambridge University Press, Cambridge, 1996, pp. 119–149.

38. A. D. Cliff and P. Haggett, Atlas of Disease Distributions: Analytic Ap-proaches to Epidemiological Data, Blackwell, London, 1988.

39. D. J. Daley and J. Gani, Epidemic Modelling: an Introduction, CambridgeUniversity Press, Cambridge, 1999.

40. M. C. M. De Jong, O. Diekmann, and J. A. P. Heesterbeek, How doestransmission depend on population size?, in Human Infectious Diseases,Epidemic Models, D. Mollison, ed., Cambridge University Press, Cambridge,1995, pp. 84–94.

41. S. Del Valle, H. W. Hethcote, J. M. Hyman, and C. Castillo-Chavez, Ef-fects of Behavioral Changes in a Smallpox Attack Model, MathematicalBiosciences 195 (2005) 228-251.

42. O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, On the definition andthe computation of the basic reproduction ratio R0 in models for infectiousdiseases in heterogeneous populations, J. Math. Biol., 28 (1990), pp. 365–382.

43. O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of In-fectious Diseases, Wiley, New York, 2000.

44. K. Dietz, Epidemics and rumours: A survey, J. Roy. Statist. Soc. Ser. A,130 (1967), pp. 505–528.

45. , Transmission and control of arbovirus diseases, in Epidemiology,K. L. Cooke, ed., SIAM, Philadelphia, 1975, pp. 104–121.

46. , The incidence of infectious diseases under the influence of seasonalfluctuations, in Mathematical Models in Medicine, J. Berger, W. Buhler,R. Repges, and P. Tautu, eds., vol. 11 of Lecture Notes in Biomathematics,Springer-Verlag, Berlin, 1976, pp. 1–15.



47. , Epidemiologic inferference of virus populations, J. Math. Biol., 8(1979), pp. 291–300.

48. , Density dependence in parasite transmission dynamics, Parasit. To-day, 4 (1988), pp. 91–97.

49. , The first epidemic model: A historical note on P. D. En’ko, Austral.J. Statist., 30A (1988), pp. 56-65.

50. K. Dietz and D. Schenzle, Mathematical models for infectious disease statis-tics, in A Celebration of Statistics, A. C. Atkinson and S. E. Feinberg, eds.,Springer-Verlag, New York, 1985, pp. 167–204.

51. R. Durrett, Stochastic spatial models, SIAM Rev., 41 (1999), pp. 677–718.52. G. Dwyer, S. A. Levin, and L. Buttel, A simulation model of the population

dynamics and evolution of myxomatosis, Ecological Monographs, 60 (1990),pp. 423–447.

53. D. J. D. Earn, P. Rohani, B. M. Bolker, and B. T. Grenfell, A simplemodel for complex dynamical transitions in epidemics, Science, 287 (2000),pp. 667–670.

54. L. Esteva and C. Vargas, Analysis of a dengue disease transmission model,Math. Biosci., 150 (1998), pp. 131–151.

55. , A model for dengue disease with variable human population, J. Math.Biol., 38 (1999), pp. 220–240.

56. A. S. Evans, Viral Infections of Humans, Plenum Medical Book Company,New York, second ed., 1982.

57. Z. Feng and J. X. Velasco-Hernandez, Competitive exclusion in a vector-hostmodel for the dengue fever, J. Math. Biol., 35 (1997), pp. 523–544.

58. F. Fenner, D. A. Henderson, I. Arita, Z. Jezek, and I. D. Ladnyi, Smallpoxand its Eradication, World Health Organization, Geneva, 1988.

59. N. M. Ferguson, D. J. Nokes, and R. M. Anderson, Dynamical complexity inage-structured models of the transmission of measles virus: Epidemiologicalimplications of high levels of vaccine uptake, Math. Biosci., 138 (1996),pp. 101–130.

60. N. M. Ferguson, C. A. Donnelly, R. M. Anderson, The Foot-and-MouthEpidemic in Great Britain: Pattern of Spread and Impact of Interventions,Science, 292 (2001) 1155-1160.

61. J. C. Frauenthal, Mathematical Modeling in Epidemiology, Springer-VerlagUniversitext, Berlin, 1980.

62. J. P. Gabriel, C. Lefever, and P. Picard, eds., Stochastic Processes in Epi-demic Theory, Springer-Verlag, Berlin, 1990.

63. L. Q. Gao and H. W. Hethcote, Disease transmission models with density-dependent demographics, J. Math. Biol., 30 (1992), pp. 717–731.

64. L. Gao, J. Mena-Lorca, and H. W. Hethcote, Four SEI endemic models withperiodicity and separatrices, Math. Biosci., 128 (1995), pp. 157–184.

65. , Variations on a theme of SEI endemic models, in Differential Equa-tions and Applications to Biology and to Industry, M. Martelli et al., eds.,World Scientific Publishing, Singapore, 1996, pp. 191–207.

66. L. Garrett, The Coming Plague, Penguin, New York, 1995.


58 H.W. Hethcote

67. B. T. Grenfell and A. P. Dobson, eds., Ecology of Infectious Diseases inNatural Populations, Cambridge University Press, Cambridge, 1995.

68. J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York,1969.

69. W. H. Hamer, Epidemic disease in England, Lancet, 1 (1906), pp. 733–739.70. H. W. Hethcote, Asymptotic behavior in a deterministic epidemic model,

Bull. Math. Biology 35 (1973) 607-614.71. H. W. Hethcote, Qualitative analyses of communicable disease models,

Math. Biosci., 28 (1976), pp. 335–356.72. , An immunization model for a heterogeneous population, Theoret.

Population Biol., 14 (1978), pp. 338–349.73. , Measles and rubella in the United States, Am. J. Epidemiol., 117

(1983), pp. 2–13.74. , Optimal ages of vaccination for measles, Math. Biosci., 89 (1988),

pp. 29–52.75. , Three basic epidemiological models, in Applied Mathematical Ecol-

ogy, L. Gross, T. G. Hallam, and S. A. Levin, eds., Springer-Verlag, Berlin,1989, pp. 119–144.

76. , Rubella, in Applied Mathematical Ecology, L. Gross, T. G. Hallam,and S. A. Levin, eds., Springer-Verlag, Berlin, 1989, pp. 212–234.

77. , A thousand and one epidemic models, in Frontiers in TheoreticalBiology, S. A. Levin, ed., vol. 100 of Lecture Notes in Biomathematics,Springer-Verlag, Berlin, 1994, pp. 504–515.

78. , Modeling heterogeneous mixing in infectious disease dynamics, inModels for Infectious Human Diseases, V. Isham and G. F. H. Medley, eds.,Cambridge University Press, Cambridge, 1996, pp. 215–238.

79. , Modeling AIDS prevention programs in a population of homosexualmen, in Modeling the AIDS Epidemic: Planning, Policy and Prediction,E. H. Kaplan and M. L. Brandeau, eds., Raven Press, New York, 1994,pp. 91–107.

80. H. W. Hethcote, The Mathematics of Infectious Diseases, Siam Review 42(2000) 599-653.

81. H. W. Hethcote and S. A. Levin, Periodicity in epidemiological models, inApplied Mathematical Ecology, L. Gross, T. G. Hallam, and S. A. Levin,eds., Springer-Verlag, Berlin, 1989, pp. 193–211.

82. H. W. Hethcote, H. W. Stech, and P. van den Driessche, Periodicity andstability in epidemic models: A survey, in Differential Equations and Appli-cations in Ecology, Epidemics and Population Problems, S. N. Busenbergand K. L. Cooke, eds., Academic Press, New York, 1981, pp. 65–82.

83. H. W. Hethcote and J. W. Van Ark, Epidemiological models with hetero-geneous populations: Proportionate mixing, parameter estimation and im-munization programs, Math. Biosci., 84 (1987), pp. 85–118.

84. , Modeling HIV Transmission and AIDS in the United States, vol. 95of Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 1992.

85. H. W. Hethcote and P. van den Driessche, Some epidemiological models



with nonlinear incidence, J. Math. Biol., 29 (1991), pp. 271–287.86. H. W. Hethcote and P. Waltman, Optimal vaccination schedules in a deter-

ministic epidemic model, Math. Biosci. 18 (1973) 365-382.87. H. W. Hethcote, W. Wang, and Y. Li, Species coexistence and periodcity

in host-host pathogen models, J. Math. Biol. (2005), to appear, now onlineat http://dx.doi.org/10.1007/s00285-005-0335-5

88. H. W. Hethcote and J. A. Yorke, Gonorrhea Transmission Dynamics andControl, vol. 56 of Lecture Notes in Biomathematics, Springer-Verlag,Berlin, 1984.

89. W. Huang, K. L. Cooke, and C. Castillo-Chavez, Stability and bifurcationfor a multiple-group model for the dynamics of HIV/AIDS transmission,SIAM J. Appl. Math., 52 (1992), pp. 835–854.

90. H. F. Hull, N. A. Ward, B. F. Hull, J. B. Milstein, and C. de Quatros,Paralytic poliomyelitis: Seasoned strategies, disappearing disease, Lancet,343 (1994), pp. 1331–1337.

91. S. S. Hutchins, L. E. Markowitz, W. L. Atkinson, E. Swint, and S. Hadler,Measles outbreaks in the United States, 1987 through 1990, Pediatr. Infect.Dis. J., 15 (1996), pp. 31–38.

92. J. M. Hyman and E. A. Stanley, Using mathematical models to understandthe AIDS epidemic, Math. Biosci., 90 (1988), pp. 415–473.

93. , The effect of social mixing patterns on the spread of AIDS, in Math-ematical Approaches to Problems in Resource Management and Epidemi-ology, C. Castillo-Chavez, S. A. Levin, and C. Shoemaker, eds., vol. 81 ofLecture Notes in Biomathematics, Springer-Verlag, New York, 1989.

94. J. M. Hyman, J. Li, and E. A. Stanley, Threshold conditions for the spreadof HIV infection in age-structured populations of homosexual men, J. Theor.Biol., 166 (1994), pp. 9–31.

95. , The differential infectivity and staged progression models for thetransmission of HIV, Math. Biosci., 155 (1999), pp. 77–109.

96. J. M. Hyman and J. Li, An intuitive formulation for the reproductive num-ber for the spread of diseases in heterogeneous populations. Math. Biosci.,to appear.

97. V. Isham, Mathematical modeling of the transmission dynamics of HIVinfection and AIDS: A review, J. Roy. Statist. Soc. Ser. A, 151 (1988),pp. 5–31.

98. V. Isham and G. Medley, eds., Models for Infectious Human Diseases, Cam-bridge University Press, Cambridge, 1996.

99. J. A. Jacquez and C. P. Simon, The stochastic SI model with recruitmentand deaths I: Comparison with the closed SIS model, Math. Biosci., 117(1993), pp. 77–125.

100. J. A. Jacquez, C. P. Simon, and J. S. Koopman, The reproduction numberin deterministic models of contagious diseases, Curr. Topics in Theor. Biol.,2 (1991), pp. 159–209.

101. J. A. Jacquez, J. S. Koopman, C. P. Simon, and I. M. Longini, Role of theprimary infection in epidemics of HIV infection in gay cohorts, J. AIDS, 7


60 H.W. Hethcote

(1994), pp. 1169–1184.102. J. A. Jacquez, C. P. Simon, and J. S. Koopman, Core groups and the R0’s for

subgroups in heterogeneous SIS models, in Epidemic Models: Their Struc-ture and Relation to Data, D. Mollison, ed., Cambridge University Press,Cambridge, 1996, pp. 279–301.

103. J. A. Jacquez, C. P. Simon, J. S. Koopman, L. Sattenspiel, and T. Perry,Modeling and analyzing HIV transmission: The effect of contact patterns,Math. Biosci., 92 (1988), pp. 119–199.

104. M. M. Kaplan and R. G. Webster, The epidemiology of influenza, ScientificAmerican, 237 (1977), pp. 88–105.

105. W. O. Kermack and A. G. McKendrick, Contributions to the mathematicaltheory of epidemics, part 1, Proc. Roy. Soc. London Ser. A, 115 (1927),pp. 700–721.

106. J. Kranz, ed., Epidemics of Plant Diseases: Mathematical Analysis and Mod-elling, Springer-Verlag, Berlin, 1990.

107. C. M. Kribs-Zaleta, Structured models for heterosexual disease transmis-sion, Math. Biosci., 160 (1999), pp. 83–108.

108. , Core recruitment effects in SIS models with constant total popula-tions, Math. Biosci., 160 (1999), pp. 109–158.

109. A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in anonhomogeneous population, Math. Biosci., 28 (1976), pp. 221–236.

110. H. A. Lauwerier, Mathematical Models of Epidemics, Mathematisch Cen-trum, Amsterdam, 1981.

111. R. Levins, T. Awerbuch, U. Brinkman, et al., The emergence of new diseases,American Scientist, 82 (1994), pp. 52–60.

112. M. Y. Li and J. S. Muldowney, Global stability for the SEIR model inepidemiology, Math. Biosci., 125 (1995), pp. 155–164.

113. M. Y. Li, J. R. Graef, L. Wang, and J. Karsai, Global dynamics of a SEIRmodel with varying population size, Math. Biosci., 160 (1999), pp. 191–213.

114. M. Y. Li, J. S. Muldowney, and P. van den Driessche, Global stability for theSEIRS models in epidemiology, Canadian Quart. Appl. Math., submitted.

115. X. Lin, H. W. Hethcote, and P. van den Driessche, An epidemiological modelfor HIV/AIDS with proportional recruitment, Math. Biosci., 118 (1993),pp. 181–195.

116. W. M. Liu, H. W. Hethcote, and S. A. Levin, Dynamical behavior ofepidemiological models with nonlinear incidence rates, J. Math. Biol., 25(1987), pp. 359–380.

117. W. M. Liu, S. A. Levin, and Y. Iwasa, Influence of nonlinear incidence ratesupon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986),pp. 187–204.

118. W. P. London and J. A. Yorke, Recurrent outbreaks of measles, chickenpoxand mumps I: Seasonal variation in contact rates, Am. J. Epidemiol., 98(1973), pp. 453–468.

119. I. P. Longini, E. Ackerman, L. R. Elveback, An optimization model forinfluenza A epidemics. Math. Biosci. 38 (1978) 141-157.



120. D. Ludwig and K. L. Cooke, eds., Epidemiology, SIMS Utah ConferenceProceedings, SIAM, Philadelphia, 1975.

121. P. Martens, How will climate change affect human health?, American Sci-entist, 87 (1999), pp. 534–541.

122. R. M. May and R. M. Anderson, Transmission dynamics of HIV infection,Nature, 326 (1987), pp. 137–142.

123. , Parasite-host coevolution, Parasitology, 100 (1990), pp. S89–S101.124. R. M. May, R. M. Anderson, and A. R. McLean, Possible demographic

consequences of HIV/AIDS, Math. Biosci., 90 (1988), pp. 475–506.125. A. G. McKendrick, Applications of mathematics to medical problems, Proc.

Edinburgh Math. Soc., 44 (1926), pp. 98–130.126. W. H. McNeill, Plagues and People, Blackwell, Oxford, 1976.127. J. Mena-Lorca and H. W. Hethcote, Dynamic models of infectious diseases

as regulators of population sizes, J. Math. Biol., 30 (1992), pp. 693–716.128. J. Mena-Lorca, J. X. Velasco-Hernandez, and C. Castillo-Chavez, Density-

dependent dynamics and superinfection in an epidemic model, IMA J. Math.Appl. Med. Biol., 16 (1999), pp. 307–317.

129. J. A. J. Metz and F. van den Bosch, Velocities of epidemic spread, in Epi-demic Models: Their Structure and Relation to Data, D. Mollison, ed., Cam-bridge University Press, Cambridge, 1996, pp. 150–186.

130. D. Mollison, Dependence of epidemic and population velocities on basicparameters, Math. Biosci., 107 (1991), pp. 255–287.

131. , ed., Epidemic Models: Their Structure and Relation to Data, Cam-bridge University Press, Cambridge, 1996.

132. J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989.133. I. Nasell, Hybrid Models of Tropical Infections, Springer-Verlag, Berlin,

1985.134. M. B. A. Oldstone, Viruses, Plagues, and History, Oxford University Press,

New York, 1998.135. L. F. Olson and W. M. Schaffer, Chaos versus noisy periodicity: Alternative

hypotheses for childhood epidemics, Science, 249 (1990), pp. 499–504.136. R Preston, The Hot Zone, Random House, New York, 1994.137. J. Radcliffe and L. Rass, Spatial Deterministic Epidemics. Mathematical

Surveys and Monographs, American Mathematical Society, Providence, toappear, 2000.

138. R. Ross, The Prevention of Malaria, Murray, London, second ed., 1911.139. L. A. Rvachev and I. M. Longini, A mathematical model for the global

spread of influenza, Math. Biosci., 75 (1985), pp. 3–22.140. D. Schenzle, An age-structured model of pre- and post-vaccination measles

transmission, IMA J. Math. Appl. Med. Biol., 1 (1984), pp. 169–191.141. L. Schlien, Hunting down the last of the poliovirus, Science, 279 (1998),

p. 168.142. M. C. Schuette and H. W. Hethcote, Modeling the effects of varicella vac-

cination programs on the incidence of chickenpox and shingles, Bull. Math.Biol., 61 (1999), pp. 1031–1064.


62 H.W. Hethcote

143. M. Schuette, Modeling the transmission of the varicella-zoster virus.Preprint, 2000.

144. M. E. Scott and G. Smith, eds., Parasitic and Infectious Diseases, AcademicPress, San Diego, 1994.

145. D. Shalala, Bioterrorism: How prepared are we?, Emerg. Infect. Dis., 5(1999). Available from URL:http://www.cdc.gov/ncidod/eid/vol5no4/shalala.htm.

146. R. Shilts, And The Band Played On, St. Martin’s Press, New York, 1987.147. C. P. Simon and J. A. Jacquez, Reproduction numbers and the stability of

equilibria of SI models for heterogeneous populations, SIAM J. Appl. Math.,52 (1992), pp. 541–576.

148. R. Snacken, A. P. Kendal, L. R. Haaheim, and J. M. Wood, The nextinfluenza pandemic: Lessons from Hong Kong, 1997, Emerg. Infect. Dis., 5(1999). Available from URL:http://www.cdc.gov/ncidod/eid/vol5no2/snacken.htm.

149. H. R. Thieme, Global asymptotic stability in epidemic models, in Equadiff82 Proceedings, H. W. Knobloch and K. Schmitt, eds., vol. 1017 of LectureNotes in Mathematics, Springer-Verlag, Berlin, 1983, pp. 609-615.

150. , Local stability in epidemic models for heterogeneous populations, inMathematics in Biology and Medicine, V. Capasso, E. Grosso, and S. L.Paveri-Fontana, eds., vol. 97 of Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 1985, pp. 185-189.

151. , Persistence under relaxed point-dissipativity (with an application toan endemic model), SIAM J. Math. Anal., 24 (1993), pp. 407–435.

152. F. van den Bosch, J. A. J. Metz, and O. Diekmann, The velocity of spatialpopulation expansion, J. Math. Biol., 28 (1990), pp. 529–565.

153. J. E. Vanderplank, Plant Diseases: Epidemics and Control, Academic Press,New York, 1963.

154. P. E. Waltman, Deterministic Threshold Models in the Theory of Epidemics,vol. 1 of Lecture Notes in Biomathematics, Springer-Verlag, Berlin, 1974.

155. R. G. Webster, Influenza: An emerging disease, Emerg. Infect. Dis., 4 (1998).Available from URL:http://www.cdc.gov/ncidod/eid/vol4no3/webster.htm.

156. K. Wickwire, Mathematical models for the control of pests and infectiousdiseases: A survey, Theoret. Population Biol., 11 (1977), pp. 182–238.

157. M. Woolhouse, M. Chase-Topping, D. Haydon, J. Friar, L. Matthews, G.Hughes, D. Shaw, J. Wilesmith, A. Donaldson, S. Cornel, M. Keeling, andBryan Grenfell, Foot-and-mouth disease under control in the UK. Nature411(2001) 258-259.

158. J. A. Yorke and W. P. London, Recurrent outbreaks of measles, chickenpox,and mumps II, Am. J. Epidemiol., 98 (1973), pp. 469–482.


EPIDEMIOLOGY MODELS WITH VARIABLE

POPULATION SIZE

Herbert W. Hethcote


University of Iowa

14 Maclean Hall



Basic epidemiology models often assume that births balance deaths, sothat the total population size is constant. But populations may be grow-ing or decreasing significantly due to differences in the natural birthand death rates, an excess disease-related mortality, or disease-relateddecreases in reproduction. In models with a variable total populationsize, the persistence of the infectious disease may slow the growth rateof a naturally growing population, lead to a lower equilibrium popula-tion size, or even reverse the population growth to a decay to extinction.Variable population size models of SIR and SEI type are formulated andanalyzed in order to demonstrate the effects of the disease on the pop-ulation size and the effects of the population structure on the diseasedynamics.

2.1 Introduction 66

2.2 Epidemiological Compartment Structures 68

2.3 Horizontal Incidences 69

2.4 Waiting Times in the E, I and R Compartments 70

2.5 Demographic Structures 71

2.6 Epidemiological-Demographic Interactions 73

2.7 SIRS model with recruitment-death and standard incidence 74

2.8 SIRS model with logistic demographics and standard incidence 75

2.9 SIRS model with exponential demographics and standard incidence 79

63


64 H.W. Hethcote

2.10 Discussion of the 3 previous SIRS models 81

2.11 Periodicity in SEI Endemic Models 83

2.11.1 The SEI Model for Fox Rabies with mass action incidence 84

2.11.2 An SEI Model with Standard Incidence 85

2.11.2.1 The equilibria in T 86

2.11.2.2 Asymptotic behavior 86

2.11.3 Discussion of the cause of periodicity in SEI models 87

References 88

2.1. Introduction

My third tutorial lecture is based on material in the following papers

[46, 15, 26, 16, 17]. Ecologists know that the sizes of plant and animal pop-

ulations are influenced and constrained by foraging, predation, competition

and limited resources. Infectious diseases can also affect population sizes.

Anderson and May [2] cite examples of the impacts of diseases on labora-

tory, domestic and wild populations of insects, birds and mammals. Indeed,

infectious diseases have also influenced human population sizes and histor-

ical events [44]. Even today parasitic diseases involving viruses, bacteria,

protozoans, helminths and arthropods combined with low nutritional status

are the primary reason for differences in the age-specific survival probabil-

ities in developing and developed countries. Measles still causes significant

mortality among malnourished children in developing countries. Recently,

human immunodeficiency virus (HIV) infections and the subsequent de-

velopment of acquired immunodeficiency syndrome (AIDS) has led to in-

creased death rates in various risk groups throughout the world. Thus it is

relevant to analyze models for infectious disease transmission with different

demographic structures in order to assess the influences of the disease on

the population dynamics.

Many epidemiological models are formulated so that the infectious dis-

ease spreads in a population which either is a fixed closed population or has

a fixed size with balancing inflows and outflows due to births and deaths

or migration. Results for the basic epidemiological models of these types

are given in my tutorial lectures I&II. Surveys of results for models with

constant size populations are given in Hethcote et al [29] and Hethcote and

Levin [27]. These models have been developed more extensively partly be-

cause they are easier to analyze than variable population size models and

partly because they are often realistic for human diseases.


Epidemiology Models with Variable Population Size 65

The assumption that the population is closed and fixed is often a rea-

sonable approximation when modeling epidemics where the disease spreads

quickly in the population and dies out within a short time (about one year).

The assumption that the population has fixed size with deaths balancing

births is often reasonable when modeling endemic diseases over many years

if the population size is constant or nearly constant and the disease rarely

causes deaths. However, if the human or animal population growth or de-

crease is significant or the disease causes enough deaths to influence the

population size, then it is not reasonable to assume that the population

size is constant. In these cases the models with a variable total population

size are suitable. These models are often more difficult to analyze math-

ematically because the population size is an additional variable which is

governed by a differential equation in a deterministic model.

There are many examples of situations where infectious diseases have

caused enough deaths so that the population size has not remained con-

stant or even approximately constant. Indeed, infectious diseases have often

had a big impact on population sizes and historical events [44]. The black

plague caused population decreases in excess of 25% and led to social, eco-

nomic and religious changes in Europe in the 14th century. Diseases such

as smallpox, diphtheria and measles brought by Europeans devastated na-

tive populations in the Americas. Even today diseases caused by viruses,

bacteria, protozoans, helminths, fungi, and arthropods combined with low

nutritional status cause significant early mortality in developing countries.

The longer life spans in developed countries seem to be primarily a result of

the decline of mortality due to communicable diseases. Infectious diseases

which have played a major role in the debilitation and regulation of human

populations include plague, measles, scarlet fever, diphtheria, tuberculosis,

smallpox, malaria, schistosomiasis, leishmaniasis, trypanosomiasis, filaria-

sis, onchocerciasis, hookworm, the gastroenteritises and the pneumonias.

See articles in [4, 22] for further discussions of the effects of diseases on host

populations.

Deterministic epidemic models with varying population size have been

formulated and analyzed mathematically by Anderson and May [2] [3], An-

derson et al. [1], Brauer [6] [7] [8], Bremmerman and Thieme [9], Busenberg

and van den Driessche[11], Busenberg and Hadeler [10], Derrick and van den

Driessche [14], Gao and Hethcote [15], Greenhalgh [18] [19] [20] [21], May

and Anderson [42] [43], Mena and Hethcote [46], Pugliese [48][49], Swart

[51], Zhou [55] and Zhou and Hethcote [56]. Some models of HIV/AIDS


66 H.W. Hethcote

with varying population size have been considered by Jacquez et al. [35],

Castillo-Chavez et al. [13], Hyman and Stanley [34], Lin [36][37], Anderson

et al. [5], Thieme and Castillo-Chavez [54], and Lin et al. [38].

For epidemiological models with varying population sizes, there are

many questions to be answered. Does the disease die out or persist in the

population? Is there an epidemic or does the disease remain endemic? How

does disease-related mortality affect the population size and the disease

persistence? Is disease-reduced reproduction an important factor? How

does the disease progress in the population as a function of time? Does

the disease incidence oscillate periodically forever or does it approach a

steady state? This tutorial lecture answers these questions for some models

of the dynamics of infectious diseases with disease-reduced reproduction

and disease-related deaths in populations with demographic dynamics.

2.2. Epidemiological Compartment Structures

The notation for the epidemiological classes in an infectious disease model

has now become somewhat standard. Let S be the number of individuals

who are susceptible, E be the number of individuals in the latent period, I

be the number of infectious individuals and R be the number of removed

individuals who have recovered with temporary or permanent immunity.

Using this notation eight possible compartmental models described by their

flow patterns are SI, SIS, SEI, SEIS, SIR, SIRS, SEIR, and SEIRS. For

example, in an SEIRS model, susceptibles become exposed in the latent

period (i.e. infected, but not yet infectious), then infectious, then removed

with temporary immunity and then susceptible again when the immunity

wears off. Sometimes a class P of passively immune newborns, a class A of

asymptomatic infectives, a class C of carriers, or a class V of vaccinated

persons is included in a model. For HIV/AIDS the models often include

classes corresponding to the stages of HIV infection leading to AIDS. Thus

there are at least 10 different models for directly transmitted diseases in a

single homogeneous population.

If there is an insect vector for the disease, then the compartment struc-

ture of the insect vector could conceivably have at least 10 different formu-

lations so that there could be over 100 different host-vector model formula-

tions. Similarly, there could be over 1000 models for a vector borne disease

with two distinct hosts. In heterogeneous populations where there are sub-

groups with different characteristics, then each subgroup could conceivably

have different epidemiological compartment structures so there could be



many possible models. For any model it is also possible to incorporate a

chronological age structure to get partial differential equations in age and

time.

The point of the paragraphs above is that there are lots of possibilities in

choosing a compartment structure. Even when this is chosen, there are still

many possible forms of the incidence, waiting times in the compartments,

demographic structures and epidemiological-demographic interactions. The

transfer diagram below indicates possibilities for an SEIRS model.

S E I R S

↓births or immigration ↓vertical incidence

horizontal incidence−−−−−−−−−−−−−−→

waiting time in E−−−−−−−−−−−−−→

waiting time in I−−−−−−−−−−−−→

waiting time in R−−−−−−−−−−−−−→

↓natural deathsor emigration ↓natural deaths

or emigration ↓natural deathsor emigration

or disease-related deaths

↓natural deathsor emigration

2.3. Horizontal Incidences

Horizontal incidence is the flow rate out of the susceptible class into the

exposed class or into the infectious class if the latent period is being ignored,

as in an SIRS model. If S(t) is the number of susceptibles at time t, I(t)

is the number of infectives, and N(t) is the total population size, then

s(t) = S(t)/N(t) and i(t) = I(t)/N(t) are the susceptible and infectious

fractions, respectively. Most forms of the incidence involve S, I and N so

we use f(S, I,N) for the incidence. If β is the average number of adequate

contacts (i.e. sufficient for transmission) of a person per unit time, then

βS/N is the average number of contacts with susceptibles per unit time of

one infective so the standard form of the incidence is

fstandard(S, I,N) = (βS/N)I = βs(Ni).

Since the form above depends on the infectious fraction i, it is also called

frequency-dependent incidence [33]. An alternate form of the incidence

based on the simple mass action law is

fmass-action(S, I,N) = ηSI = βNs(Ni)

where η is a mass action coefficient (which has no direct epidemiological

interpretation). For more information about the differences in models using

these two forms of the incidence, see Hethcote [24], Hethcote and Van Ark

[31], Mena and Hethcote [46], or Hethcote et al. [33]. Another possible

incidence has a density dependent contact rate β(N) so the form is


68 H.W. Hethcote

fdens-dep(S, I,N) = β(N)SI/N = β(N)s(Ni).

Population size dependent contact functions have been considered by

Brauer [7] [8], Pugliese [48] [49], Thieme [52] [53], and Zhou[55]. These pop-

ulation size dependent contact rates have also occurred in AIDS models by

Castillo-Chavez et. al. [13] and Thieme and Castillo-Chavez [54].

Various forms of nonlinear incidences have been considered. The satu-

ration incidence

f(S, I,N) = βI(1 − cI)S

was used by London and Yorke [41] in their modeling of measles. Another

saturation incidence

f(S, I,N) =βI

1 + cIS

was used by Capasso and Serio [12] in their modeling of cholera. Liu et

al. [40] used f(S, I,N) = βIpSq and Liu et al. [39] analyzed models with

nonlinear incidence of the form f(s, i,N) = βipsq. Hethcote and van den

Driessche [32] considered models with f(s, i,N) = βg(i)s and

f(s, i,N) =βip

1 + αiqs.

See Derrick and van den Driessche [14] for a discussion of which forms of

nonlinear incidence can not lead to periodic solutions. In any of the forms

of incidence above, the proportionality constant β or β could be a periodic

function of time t. See Hethcote and Levin [27] for a survey of periodicity

in epidemiological models. In a multigroup model the interactions between

groups are specified by a contact or mixing matrix [31]. In an n group

model there are n2 entries in the matrix so the number of possible contact

matrices is enormous [25].

2.4. Waiting Times in the E, I and R Compartments

The most common assumption is that the movements out of the E, I and

R compartments and into the next compartment are governed by terms

like εE, γI and δR in an ordinary differential equations model. It has been

shown [29] that these terms correspond to exponentially distributed waiting

times in the compartments. For example, the transfer rate εE corresponds



to P (t) = e−εt as the probability of still being in the exposed class t units

after entering this class and 1/ε as the mean waiting time.

Another possible assumption is that the probability of still being in the

compartment t units after entering is a nonincreasing, piecewise continuous

function P (t) with P (0) = 1 and P (∞) = 0. Then the probability of leaving

the compartment at time t is −P ′(t) so the mean waiting time in the

compartment is∫∞

0t(−P ′(t))dt =

∫∞

0P (t)dt. These distributed delays lead

to integral or integrodifferential or functional differential equations. If the

waiting time distribution is a step function given by

P (t) =

{

1 if 0 ≤ t ≤ τ

0 if τ ≤ t,

then the mean waiting time is τ and for t ≥ τ the model reduces to a

delay-differential equation [29].

Each waiting time can have a different distribution. Thus, in an SEIRS

model each of the three waiting time distributions can be chosen in three dif-

ferent ways so there are 27 different possibilities. In a male-female SEIRS

model for a sexually transmitted disease, there would be 36 possibilities.

Hence, there are many choices in choosing waiting time distributions in

models.

2.5. Demographic Structures

Many different demographic structures can be applied to an epidemic

model. A simple birth-death demographic structure is based on the dif-

ferential equation

dN

dt= bN − dN

where bN represents the births and dN represents the natural deaths. In an

SEIRS epidemic model without vaccination where the number of people

in the latent period is E, the newborns would be susceptible and natural

deaths would occur in each class so that a simple model would be:

dS/dt = bN − dS − f(S, I,N) + δR

dE/dt = f(S, I,N) − εE − dE

dI/dt = εE − γI − dI

dR/dt = γI − δR − dR

with N = S + E + I + R. So far no one has proved global stability of

the endemic equilibrium for this simple model with the standard incidence


70 H.W. Hethcote

and b = d [30]. Note that a similar model with all deaths occurring in the

removed class is ill-posed [24].

If there are no births, deaths or migration, then the above model with

b = d = 0 is suitable for describing an epidemic in a short time period less

than one year. If b = d 6= 0, then there is an inflow of newborn susceptibles,

but the population size remains constant. This inflow of new susceptibles

leads to an endemic or persistent equilibrium above the threshold [24]. If

r = b−d 6= 0, then the population would be naturally exponentially growing

or decaying in the absence of the infectious disease. The persistence of the

disease and disease-related deaths can affect the demographic behavior and

can even reverse exponential growth so there is a stable equilibrium or

exponential decay[10] [11] [15] [46].

Another possible demographic model is

dN

dt= A − dN

where A represents recruitment or immigration and dN represents natural

deaths. Without the disease the population size N approaches A/d. See

[2][43][46] for the formulation and analysis of models with this demographic

structure. Many of the models of HIV/AIDS referenced in Section 2.1 have

used this structure.

Another reasonably simple demographic structure is based on the logis-

tic equation:

dN

dt= r(1 − N/K)N

where the net growth rate is r = b − d and K is the carrying capacity of

the environment. Gao and Hethcote [15] have considered a logistic model

written as

dN

dt= (b − arN/K)N − [d + (1 − a)rN/K]N

with 0 ≤ a ≤ 1. Here the logistic term r(1 − N/K)N due to crowding or

limited resources is divided into the first term corresponding to a decreasing

birth rate and the second term corresponding to an increasing natural death

rate. A generalized logistic differential equation would have the form

dN

dt= [B(N) − D(N)]N

where B(N) is the density dependent birth rate and D(N) is the density

dependent death rate. Note that all of the above demographic structures



are special cases of this general formulation. Epidemic models with logistic

or generalized logistic demographic structure have been considered in [1],

[6], [7], [8], [9], [48] [56].

2.6. Epidemiological-Demographic Interactions

The infectious disease can affect the demographic process through disease-

related deaths and reduced fertility of exposed, infectious and removed in-

dividuals. The demographics can affect the epidemiological process through

vertical transmission. Since each of these aspects of an epidemic model can

be left out or included, there are at least 10 different possibilities. All of

these interactions are considered below.

If there is an increased death rate (above the natural death rate) due

to the infectious disease, then the excess deaths can be modeled by an

extra flow term αI out of the infectious class. In a birth-death demographic

process with birth rate B(N)N and death rate D(N)N, the disease-related

deaths can affect the population size. In this case the differential equation

for N becomes

dN

dt= [B(N) − D(N)]N − αI

where I is the number of infectives. Disease-related deaths and persistence

of the disease can change a naturally growing population into a stable or

decaying population. For examples see [2], [11], [15], [43], [46], [52], [53],

[55], [56].

As a consequence of the infectious disease, females in the exposed, in-

fectious and removed class may not have as many children as susceptible

females. A typical birth term which includes reduced fecundity or fertility

would be B(N)(S + ρ1E + ρ2I + ρ3R) where the last three terms with

0 ≤ ρi ≤ 1 are the reduced fertility terms.

Vertical incidence is the flow of newborns infected by their mothers (be-

fore, during or just after the birth process) into the infectious class. Vertical

incidence can be inserted as shown in the transfer diagram in Section 2.2;

note that an E category of vertically infected babies in a latent period might

also be appropriate. Since all vertically infected people are newborns, verti-

cal incidence is most naturally incorporated into an age structured model.

A typical vertical incidence term in a deterministic model would be the

product of the probability of transmission per birth, the birth rate and the

number of infected women.


72 H.W. Hethcote

2.7. SIRS model with recruitment-death and standard

incidence

This SIRS model uses the recruitment death demographics and the stan-

dard incidence [46]. The differential equations for this model are:

dS/dt = A − dS − βSI/N + ρR

dI/dt = βSI/N − (γ + α + d)I

dR/dt = γI − (ρ + d)R

dN/dt = A − dN − αI

(2.7.1)

where the equation for N(t) follows from N = S + I + R. This model is

mathematically and epidemiologicaJly well-posed. In the absence of disease,

the population size N(t) approaches A/d. Here the threshold is the contact

number σ, which is equal to the basic reproduction number R0, so that

R0 = σ = β/(γ + α + d),

which is the average number of adequate contacts of an infective during the

infectious period 1/(γ + α + d).

For this model the equilibria in SIR space are P0 = (A/d, 0, 0) and

P1 = (Se, Ie, Re) where

Se = Ne/σ

Ie = Ne(1 − 1/σ)/[1 + γ/(ρ + d)]

Re = γIe/(ρ + d)

Ne = A/{d + α(1 − 1/σ)/[1 + γ/(ρ + d)]}

P0 is the only equilibrium in the first octant of SIR space when R0 = σ ≤ 1.

For R0 = σ > 1, both equilibria are in the first octant. For this SIRS model

the asymptotic behaviors of solution paths are summarized in the Table

below, where LAS means locally asymptotically stable, GAS means globally

asymptotically stable, and NID means not in the domain.

P0 = (A/d, 0, 0) P1 = (Se, Ie, Re)

R0 = σ ≤ 1 GAS NID

R0 = σ > 1 saddle LAS

When the disease persists, the equilibrium population size is reduced

from A/d to Ne by the disease-related deaths. Another effect of the disease-

related death rate constant α is to decrease the average infectious period

slightly from 1/(γ + d) to 1/(γ + α + d). This means that the minimum



contact rate A so that R0 = σ > 1 and the disease does not die out must

be larger when there are disease-related deaths.

The proofs of the results for this SIRS model are given in [46]. All paths

approach the subset T in the first octant in SIR space where S + I + R ≤

A/d. For R0 = σ ≤ 1, the Liapunov function L = I can also be used here to

show that P0 is GAS. For R0 = σ > 1, the equilibrium P1 = (Se, Ie, Re) can

be shown to be locally asymptotically stable by linearization. We remark

that for α = 0, N(t) → A/d and in the S + I +R = A/d plane, the systems

above reduce to a constant population SIRS model so that global stability

results follow from those in my tutorial lectures I&II.

2.8. SIRS model with logistic demographics and standard

incidence

The epidemiological model formulated here has population dynamics cor-

responding to the logistic equation where the restricted growth is due to

density-dependence in both the birth and death rates [15]. The birth rate

decreases and the death rate increases as the population size increases to-

wards its carrying capacity. Vertical transmission from infected females to

their offspring before, during or just after birth is not included here. Thus

all newborns are susceptible. It is assumed that infection does not affect

fertility so that the birth rates are the same for women in all three epidemi-

ological states.

The transfer diagram for the numbers S, I and R of susceptibles, infec-

tives and removed individuals, respectively, is similar to that for the SEIRS

model.

The parameters in the model are:

a = convex combination constant

b = natural birth rate constant

d = natural death rate constant

r = b − d = net growth rate constant

K = carrying capacity of the environment

β = daily contact rate

α = disease-related death rate constant

γ = recovery rate constant

ρ = loss of immunity rate constant

We are following the convention of using Roman letters for demographic

parameters and Greek letters for epidemiological parameters. We assume

that b, d, K, β, and γ are positive, a is in the interval [0, 1], and α and ρ are


74 H.W. Hethcote

nonnegative. The autonomous differential equations for this SIRS model

are:

dS/dt = [b − arN/K]N − βSI/N − [d + (1 − a)rN/K]S + ρR

dI/dt = βSI/N − [γ + α + d + (1 − a)rN/K]I

dR/dt = γI − [ρ + d + (1 − a)rN/K]R

dN/dt = r[1 − N/K]N − αI

where one of the equations is redundant since N = S + I + R. In the

absence of disease the differential equation for N is the logistic equation;

the second term in this differential equation for N corresponds to disease-

related deaths. For 0 < a < 1 the birth rate decreases and the death rate

increases as N increases to its carrying capacity K; these are consistent with

the limited resources associated with density- dependence. The birth rate is

density independent when a = 0 and the death rate is density independent

when a = 1.

We consider the case where r > 0 so the logistic equation really does

describe restricted growth. The birth rate in the model does not make sense

if it is negative, so we consider the positively invariant subset of the first

octant in SIR space where N < bK/ar. Since dN/dt < 0 for N > K, all

solution paths in the subset above approach, enter or stay in the subset

where N = S + I + R < K. Solution paths with N0 > K which do not

enter the region N < K in finite time must have their omega limit sets on

the N = K plane. Thus for r > 0 it suffices to analyze solution paths and

omega limits sets in the subset of the first octant where N < K. If r = 0

and there is no disease, then the population size remains constant. Thus if

r = 0, we would consider the system in the entire positively- invariant first

octant in SIR space. If r < 0 and there is no disease, then the population

size decreases to zero if N0 < K and increases to infinity if N0 > K. In this

case we would consider only the subspace of the first octant where N < K

when r < 0.

It is convenient to reformulate the model above in terms of the frac-

tions i = I/N and rm = R/N of the population, which are infectious

and removed, respectively. The susceptible fraction s = S/N satisfies

s = 1 − i − rm. The last three differential equations in the system above

become

di/dt = [β − [γ + α + b) − (β − α)i − βrm + arN/K]i

drm/dt = γi − (ρ + b)rm + arNrm/K + αrm

dN/dt = r[1 − N/K − αi]N

(2.8.1)



In irmN space the positively invariant subset corresponding to the sub-

set S + I + R < K in the first octant of SIR space is

D = {(i, rm, N) |i ≥ 0, rm ≥ 0, i + rm ≤ 1, 0 ≤ N ≤ K }.

The continuity of the right side of (2.8.1) and its derivatives implies that

unique solutions exist on maximal time intervals. Since for r > 0 solution

paths approach, enter or stay in D, these paths are always bounded and

continuable so they exist for all positive time ([23], pp. 18–27). Thus the

initial value problem for system (2.8.1) is mathematically well-posed and is

epidemiologically reasonable since the fractions i and rm remain between 0

and 1. It is also well-posed in the first octant when r = 0 and in the subset

where N < K if r < 0.

In the absence of disease the population size approaches its carrying

capacity K if r > 0 and N0 > 0. If disease is initially present, then the

population size can go to zero, approach an equilibrium size below the

carrying capacity or approach the carrying capacity, and the disease can

die out or persist (remain endemic) depending on the values of several

threshold quantities. The contact number is the average number of adequate

contacts of an infective during the infectious period. In a totally susceptible

population the contact number is the average number of new infections

(secondary cases) produced per infective, so it is sometimes called the basic

reproduction number. Here the contact rate is β, and the average infectious

period when the population is at its carrying capacity is 1/(γ + a + d +

(1−a)r) where r = b−d, so that the basic reproduction number or contact

number is

R0 = σ = β/(γ + α + b − ar). (2.8.2)

A closely-related threshold quantity is the modified contact number θ

given by

θ = β/(γ + α + b).

Note that R0 = σ and θ coincide when a = 0. When r > 0 and the disease

persists, the net growth threshold

φ =rβ

α[β − (γ + α + d)](1 +

γ

ρ + d)

determines whether the population size decreases to zero or approaches a

constant size. The net growth threshold φ primarily reflects the relative


76 H.W. Hethcote

effects of the disease-related death rate constant α and the growth rate

constant r.

The system (2.8.1) can have up to four equilibria in the subregion D

of irmN space. They are found by setting the right sides of (2.8.1) equal

to zero. System (2.8.1) always has the equilibria P1 = (0, 0, 0) and P2 =

(0, 0,K) corresponding to fadeout of the disease with the population size

at zero or at the carrying capacity K.

For N = 0 the right sides of (2.8.1) are equal to zero if

i = [β − (γ + α + b) − βrm]/(β − α)

i = (ρ + b)rm/(γ + αrm)

Analysis of the graphs of these equations shows that they intersect and

yield a distinct equilibrium P3 = (i3, rm3, 0) in D iff the modified contact

number θ satisfies θ > 1. Note that P3 → P1, as θ → 1+.

For nonzero i and N values at equilibrium, use rN/K = r − αi in the

equilibrium equations for i and rm to obtain

i = [β − (γ + α + b − ar) − βrm]/[β − (1 − a)α]

i = (ρ + b − ar)rm/[γ + (1 − a)αrm]

The graphs of these equations intersect at a nonzero point in D iff the

contact number σ satisfies R0 = σ > 1. These equations imply that rm4 is

the positive root of the quadratic equation,

β(1 − a)αr2m + [(1 − a)α(γ + α − ρ − β) + β(γ + gr + b − ar)]rm

+ γ(γ + ga + b − ar − β) = 0

To obtain an equation for N4, we find equations for s4, i4 and rm4 in

terms of N4 and set their sum to N4. The resulting quadratic equation

for χ = rN4/K has a positive solution if the left side is negative at χ = 0

which is equivalent to φ > 1. See [15] for details. Thus P4 = (i4, rm4, N4)

is a distinct equilibrium in D iff R0 = σ > 1 and φ > 1. Note that P4 → P2

as σ → 1+. Also note that P4 → P3 as φ → 1+ for R0 = σ > 1. Note that

rm3 in the equilibrium P3 is the larger root of the quadratic equation above

with a = 0. In the special case when a = 1 so that the birth rate is density

dependent and the death rate is density independent, the coordinates of

the equilibrium P4 can be found explicitly [15].

The local and global stabilities of the equilibrium points given in the

table below are proved in [15], where LAS means locally asymptotically

stable, GAS means globally asymptotically stable, NID means not in the



domain, UN means unstable node, NAS means numerical solutions suggest

asymptotic stability.

P1 = (0, 0, 0) P2 = (0, 0,K) P3 = (i3, rm3, 0) P4 = (i4, rm4, N4)

θ ≤ 1 σ ≤ 1 saddle GAS NID NID

σ > 1 saddle saddle NID LAS

θ > 1 φ < 1 UN saddle LAS NID

φ = 1 UN saddle NAS NID

φ > 1 UN saddle saddle LAS

2.9. SIRS model with exponential demographics and

standard incidence

Here we consider a natural birth process in which the birth rate is propor-

tional to the population size [46]. Thus the population dynamics without

disease is given by dn/dt = (b−d)N , so that there is exponential growth or

decline of the population except in the special case of a constant size pop-

ulation when the birth and death rate constants b and d are equal. Using

the usual parameters the differential equations for the model are:

dS/dt = bN − dS − βSI/N + ρR

dI/dt = βSI/N − (γ + α + d)I

dR/dt = γI − (ρ + d)R

dN/dt = (b − d)N − αI

where one of the equations is redundant since N = S + I + R. This model

is well-posed in the first octant of SIR space.

The net growth rate constant in a disease-free population is r = b − d.

In the absence of disease, the population size N(t) declines exponentially

to 0 if r < 0, remains constant if r = 0 and grows exponentially if r > 0. If

disease is present, the population still declines to zero if r < 0. For r > 0

the population can go to zero, remain finite or grow exponentially and the

disease can die out or persist depending on the values of several threshold

quantities.

The contact number, which is equal to the basic reproduction number,

R0 = σ = β/(γ + α + d)

is the average number of adequate contacts of an infective during the mean

infectious period 1/(γ + α + d). A closely related threshold quantity is the

modified contact number


78 H.W. Hethcote

θ = β/(γ + α + b),

where the death rate constant d in R0 = σ is replaced by the birth rate

constant b. When r > 0 and the disease persists, the net growth threshold

φ =rσ

α(σ − 1)(1 +

γ

ρ + d)

determines whether the populations size N(t) decreases exponentially, re-

mains finite or grows exponentially. The net growth threshold φ primarily

reflects the combined effects of the disease-related death rate constant α

and the net growth rate constant r.

The results for this model with r > 0 are summarized in the Table

below. These results and the results with r ≤ 0 are proved in [46] using the

differential equations for the fractions (s, i, rm) in the susceptible, infectious,

and removed classes. Although this Table may look complicated, it is not.

The modified contact number θ always determines whether the disease dies

out (the fraction i → 0 if θ ≤ 1) or persists (the fraction i → ie if θ > 1).

The population size behavior is usually determined by the net growth rate

constant r = b − d except for the three cases where the disease persistence

affects the population size behavior. When r = 0, the population would

normally remain constant, but for α > 0 and θ = σ > 1, the disease-

related deaths cause the population to become extinct. In the second and

third cases, r > 0 so the population would normally grow exponentially,

but for θ > 1, the disease persists and the disease-related deaths cause the

population to remain finite when φ = 1 and to become extinct when φ < 1.

Note that φ ≤ 1 is equivalent to

α(1 − 1/σ)/[1 + γ/(ρ + d)] ≥ r,

which displays more clearly the concept that the disease-related deaths

defined by α overcome the intrinsic net growth rate r of the population and

prevent the population from growing exponentially.



global attractor asymptotic asymptotic

in plane behavior behavior

s + i + r = 1 of N(t) in SIR space

θ ≤ 1 R0 = σ < 1 (1, 0, 0) N → ∞ (∞, 0, 0) is GAS

R0 = σ = 1 (1, 0, 0) N → ∞ line (∞, Ie, Re) is GAS

R0 = σ > 1 (1, 0, 0) N → ∞ (∞,∞,∞) is GAS

θ > 1 φ < 1 (se, ie, rme) N → 0 (0, 0, 0) is GAS

φ = 1 (se, ie, rme) N → Ne line (Se, Ie, Re) is GAS

φ > 1 (se, ie, rme) N → ∞ (∞,∞,∞) is GAS

2.10. Discussion of the 3 previous SIRS models

The epidemiological and demographic processes in a dynamic model affect

each other. Mena-Lorca and Hethcote [46] studied epidemiological models

with two demographic processes: recruitment with the death rate propor-

tional to the population size, and both the birth and death rates propor-

tional to the population size. Using the same notation and terminology, Gao

and Hethcote [15] considered infectious disease models with density- depen-

dent restricted growth corresponding to the logistic equation. These models

with logistic growth are really an entire range of models with 0 < a < 1

where, roughly speaking, the fraction a of the density dependence is al-

located to reducing the birth rate and the fraction 1 − a is allocated to

increasing the death rate. The demographic and epidemiological aspects of

these models affect each other by altering the expected asymptotic behav-

iors.

For the SIRS model with recruitment with the death rate proportional

to the population size in Section 2.7, the behavior is like that of the basic

SIR endemic model. Below the threshold (R0 = σ ≤ 1), the disease dies

out and the population size approaches its usual limit A/d. Above the

threshold (R0 = σ > 1), the disease remains endemic and the population

size approaches a new value below its usual limit A/d.

The population size in a logistic demographic model usually approaches

the carrying capacity K, but in Section 2.8 we found that when the disease

persists, the disease-related deaths either lower the asymptotic population

size or cause the population to approach extinction. The competing effects

of the positive growth rate constant r and the disease-related death rate

constant α are measured by the net growth threshold φ. If θ > 1 and

φ < 1, then the disease-related deaths overcome the intrinsic growth rate


80 H.W. Hethcote

corresponding to positive r and cause the population size to decrease to

zero. In this case the equilibrium P3 = (i3, rm3, 0) corresponds to popula-

tion extinction due to disease-related deaths and the persistence of the dis-

ease. If φ > 1, the intrinsic growth rate corresponding to r is large enough

so that the population persists, but the disease-related deaths cause the

asymptotic population size N4 to be lower than the carrying capacity K.

Hence the equilibrium P4 = (i4, rm4, N4) corresponds to a decreased equi-

librium population size due to disease-related deaths and the persistence of

the disease. In the two cases above, the infectious disease dynamics clearly

affect the population size dynamics.

The population size dynamics also affects the infectious disease dynam-

ics. In the model in Section 2.9 with exponential growth demographics, the

modified contact number θ always determines whether the disease dies out

(θ ≤ 1) or remains endemic (θ > 1). This is also generally true for the mod-

els in Section 2.8 with logistic growth demographics, but for R0 = σ > 1,

the disease remains endemic in one case even though the modified contact

number θ satisfies θ ≤ 1. In this case the logistic demographic structure

causes the disease to persist even though it would normally have died out

with the exponential growth demographic structure in Section 2.9. Thus

the demographic aspects affect the epidemiological aspects. Note that the

epidemiological thresholds are also affected by the logistic demographics

since the contact number R0 = σ, the modified contact number θ and the

net growth threshold φ involve the demographic parameters a, b, d and r.

The net growth threshold φ plays a similar role in Sections 2.8 and 2.9.

In these models with r > 0 and θ > 1, the population size would normally

approach the carrying capacity or grow exponentially, but in both models

the disease-related deaths cause the population to become extinct if φ < 1.

For φ > 1 the disease-related deaths cause the population size to grow at a

slower rate in the exponential demographic model and to approach a size

below the carrying capacity in the logistic demographic models in Section

2.8.

Based on numerical calculations with a variety of parameter sets and

the global stability results in the special cases when a = 0 or α = 0 in

[15], we conjecture that the local stability results in the table in Section

2.8 are actually global. Proofs of these global stability results would be of

some interest, but our results here have already revealed the main concepts

about density-dependent demographics in infectious disease models. The

SIRS model reduces to an SIR model when the immunity loss rate constant



ρ is zero and both the SIRS and SIS models reduce to an SI model when

the recovery rate constant γ is zero. All of the results in Section 2.8 hold

for these special cases provided b is positive so that there is some inflow

into the susceptible class.

2.11. Periodicity in SEI Endemic Models

This remaining sections are based on the results in two papers of Gao,

Mena, and Hethcote [16, 17]; see these papers for more details. Identify-

ing the mechanisms which can lead to periodic solutions in epidemiological

models is important, so that all possibilities can be considered when pe-

riodicity is observed in epidemiological data. It is not surprising that sea-

sonal (periodic) contact rates can cause periodic solutions in models for

the spread of infectious diseases, but periodic solutions can also occur in

models without seasonal contact rates. For example, delays in the removed

class R in an SIRS model can cause periodic solutions for some parameter

values [28] [29]. Some epidemiological models with nonlinear incidence have

been shown to have periodic solutions [40] [39]. Some of the mechanisms

in epidemiological models which lead to periodic solutions are described in

two surveys [29] [27].

A model for fox rabies was formulated by Anderson et.al. [1]; they found

numerically that stable limit cycles (i.e. periodic solutions) occurred for

some parameter values. Their model is of SEI type, since susceptible foxes

become exposed and then infectious. There is no removed or recovered

class R, since the foxes die from the rabies infection. Without the disease

the fox population has logistic growth, but exposed and infectious foxes

do not have any surviving offspring. The reader is referred to Anderson

et.al. [1] for more details on this model and for parameter estimates. Other

authors have verified numerically and analytically using Hopf bifurcation

that periodic solutions do occur in this SEI model [45] [51] [49].

Here we focus on the mathematical question of identifying those features

of the SEI model of Anderson et.al. [1] which are necessary for periodic so-

lutions. Thus we examine a variety of models of SEI type and also consider

related SI and SIS models, in order to determine whether these models also

have periodic solutions or have stable endemic equilibria. We say that a dis-

ease persists or remains endemic in a population if the infectious fraction of

the population is bounded away from zero. For example, the persistence of

a disease combined with disease-related deaths and disease-reduced repro-

duction can drive a population to extinction with the number of infectives


82 H.W. Hethcote

going to zero, but the disease is still said to be endemic because the infective

fraction is bounded away from zero.

2.11.1. The SEI Model for Fox Rabies with mass action

incidence

Anderson et.al. [1] formulated an SEI model for fox rabies in Europe. They

used different letters for their parameters, but in our notation, their model

has no reproduction by exposed and infective individuals, uses the simple

mass action incidence ηSI, and has a logistic death demographic structure

so that B(N) = b and D(N) = d + rN/K. The transfer diagram for this

SEI model is

↓bS

SηSI

−−−−−−−−−−−−−→ EεE

−−−−−−−−−−−−−−−−−→ I

↓(d+rN/K)S ↓(d+rN/K)E ↓[α+d+rN/K]I

They obtained a basic reproduction number R0 = ηKε/[(α + b)(ε + b)]

as the threshold quantity which determines whether the disease dies out

(R0 ≤ 1) or remains endemic (R0 > 1). Then they reinterpret this as a

threshold in the carrying capacity density K, so the disease remains endemic

if their carrying capacity density K exceeds KT = (a + b)(ε + b)/ηε. Here

the latent period is 1/ε. In the two dimensional parameter space with the

latent period 1/ε as abscissa and the density K as the ordinate, but all

other parameter values fixed, they determined (from numerical solutions of

the differential equations) a region where there is limit cycle behavior (i.e.

periodic solutions). They also calculated the period of the oscillations which

range from 2 to 6 years. They explained intuitively that periodic solutions

arose because rabies acts ”as a time-delayed density dependent regulator

of fox population growth”.

Here we examine similar epidemiological models in order to determine

which features of this SEI model are essential in order to obtain periodic

solutions. Global stability results of Zhou and Hethcote [56] on the SIS

model with the simple mass action incidence ηSI and logistic demographics,

which is the direct analog of the fox rabies SEI model, show that it does

not have periodic solutions. Hence the SI model which is a special case also

does not have periodic solutions. Thus, an SEI model with an exposed class

E is necessary in order to obtain periodic solutions.



2.11.2. An SEI Model with Standard Incidence

Here we consider an SEI model which is similar to the SEI model for fox

rabies, but it uses the standard incidence and the reproduction by the

susceptible, exposed and infective individuals is the same. This model

never has periodic solutions. Thus changing from the simple mass action

incidence ηSI to the standard incidence βSI/N eliminates the possibility

of periodic solutions in SEI models with uniform reproduction of offspring

by all individuals.

The transfer diagram for the SEI model considered here is

↓bN

SβSI/N

−−−−−−−−−−−−−→ EεE

−−−−−−−−−−−−−−−−−→ I

↓(d+rN/K)S ↓(d+rN/K)E ↓[α+d+rN/K]I

Assume that the parameters b, d, r = b − d, K, β and ε are positive and

that α is nonnegative.

The differential equations for this SEI model are

dS/dt = bN − βSI/N − (d + rN/K)S,

dE/dt = βSI/N − εE − (d + rN/K)E,

dI/dt = εE − [α + d + rN/K]I,

dN/dt = r(1 − N/K)N − αI,

(2.11.1)

where one of the equations is redundant since N = S + E + I. For conve-

nience in the analysis, we change to the differential equations in terms of

the fractions s and i in the susceptible and infectious classes. The system

(2.11.1) becomes

ds/dt = b(1 − s) + (α − β)si,

di/dt = ε(1 − s) − (ε + α + b − αi)i,

dN/dt = r(1 − N/K)N − αiN.

(2.11.2)

This model is mathematically and epidemiologically well-posed. The fea-

sible region of this system is also the positively invariant region U given

by

U = {(s, i,N)| 0 ≤ s, i, s + i ≤ 1, 0 ≤ N ≤ K}, (2.11.3)

The first two equations in (2.11.2) do not involve the population size N, so

that the solution behavior of these equations can be analyzed separately in

the triangle T = {(s, i)| 0 ≤ s, i, s + i ≤ 1}.


84 H.W. Hethcote

The threshold quantity which determines the asymptotic behavior of

solutions of (2.11.2) is the contact number (which is also called the basic

reproduction number R0) given by

R0 = σ =

(

β

α + b

)(

ε

ε + b

)

. (2.11.4)

Although it is similar to the basic reproduction number for the simple mass

action incidence SEI model, here the contact number is the product of the

contact rate β, the mean infectious period 1/(α+b) at the carrying capacity

K, and the fraction ε/(ε + b) of the exposed individuals who live through

the exposed class when N = K.

2.11.2.1. The equilibria in T

The S and I equations in system (2.11.2) are analyzed in the triangle T

in the SI plane and then the behavior of N is determined. The system

(2.11.2) always has the disease-free equilibrium P1 = (1, 0). From the first

equilibrium equation, we find s = A/(i+A) with the same A = b/(β−α) >

0. Substituting this into the second equilibrium equation yields i = 0 or

g(i) = αi2 + i[αA − (ε + α + b)] + [ε − (ε + α + b)A] = 0. (2.11.5)

The roots of this quadratic equation are

i± =[

(ε + α + b − αA) ±√

(ε + α + b + αA)2 − 4αε]

/2α. (2.11.6)

Now i+ ≥ 1 at A = 0 and di+/dA > 0, so the positive root i+ > 1 is

outside the region T for A > 0. However, i− ≤ 1 at A = 0 and di−/dA < 0,

so the root i− is in the interval (0,1) iff g(0) = ε − (ε + α + b)A > 0,

which is equivalent to σ > 1. Thus there is an endemic equilibrium with

ie = i− if the contact number given by (2.11.4) satisfies σ > 1. In summary,

the disease-free equilibrium P1 = (1, 0) is always in T and for σ > 1 the

endemic equilibrium P2 = (se, ie) is also in T.

2.11.2.2. Asymptotic behavior

This SEI model has the typical behavior for an epidemiological model;

namely, the contact number σ determines whether the disease dies out

(σ ≤ 1) or remains endemic (σ > 1). More precise statements of these

results are given in the theorems below.



Theorem 2.11.1: If σ ≤ 1 in the system (2.11.2), then T is an asymptotic

stability region for the disease-free equilibrium P1 = (1, 0) and N → K as

t → ∞.

Proof. For σ ≤ 1, P1 = (1, 0) is the only equilibrium in the positively

invariant region T and it is locally asymptotically stable. There are no

periodic solutions in T, since there are no interior equilibria in T. by the

Poincare-Bendixson theory [47], all paths in T must approach P1. Applying

a Lemma in [55] to the equation for N in (2.11.2), we find that N → K as

t → ∞. ¤

Theorem 2.11.2: If σ > 1 in the system (2.11.2), then T except the

equilibrium P1 = (1, 0) is an asymptotic stability region for the endemic

equilibrium P2 = (se, ie). If r−αie > 0, then N → [1−αie/r]K as t → ∞;

if r − αie ≤ 0, then N → 0.

Proof. For σ > 1, P1 = (1, 0) is a saddle with the stable manifold outside

T and a branch of the unstable manifold in T. The endemic equilibrium

P2 = (se, ie) is locally asymptotically stable. By Dulacs criterion [50] with

multiplier 1/(1 − i), there are no periodic solutions in T. By the Poincare-

Bendixson theory, paths in T must approach an equilibrium. Thus all paths

in T except P1 approach P2. The asymptotic behavior of N is found by using

a Lemma in [55] on the equation for N in (2.11.2). ¤

2.11.3. Discussion of the cause of periodicity in SEI models

In the previous subsection we showed that an SEI model with the stan-

dard incidence and uniform reproduction does not have periodic solutions.

The analogous SEI model with simple mass action and uniform reproduc-

tion does have periodic solutions [16]. When Gao, Mena and Hethcote [16]

analyzed an SEI model with standard incidence and disease-reduced repro-

duction, they found unusual behavior with a separatrix that divides the

relevant region into attractive subregions for two radically different equilib-

ria, but they did not find periodic solutions. Thus two SEI models with the

simple mass action incidence ηSI were found to have periodic solutions for

some parameter values, but two SEI models with the standard mass action

incidence βSI/N did not have periodic solutions. Thus we conclude that it

is the simple mass action incidence ηSI which causes periodicity in these

SEI models with logistic demographic structure.


86 H.W. Hethcote

In a second paper Gao, Mena, and Hethcote [17] continued to look

for periodic solutions in SEI models by examining six models with two

different demographic structures. Four of these models have exponential

demographic structure and either uniform or disease-reduced reproduction.

They found that the two SEI models with the mass action incidence ηSI

can have periodic solutions, but the two SEI models with the standard in-

cidence βSI/N do not have periodic solutions. The two SEI models with

recruitment-death demographic structure and incidences ηSI or βSI/N

do not have periodic solutions. Models with the recruitment-death demo-

graphic structure appear to be more stable because the inflow is constant

and does not depend on the population size.

Thus the four SEI models with incidence ηSI and either logistic or ex-

ponential demographics can have periodic solutions, but the four analogous

SEI models with the standard incidence βSI/N do not. Whether the re-

production is uniform or disease-reduced does not influence the occurrence

of periodicity. These combined results provide overwhelming evidence that

the feature which leads to periodicity in SEI models is the mass action in-

cidence form ηSI. This conclusion implies that periodic solutions can also

occur in the more general SEIS, SEIR, and SEIRS models with the inci-

dence ηSI. This insight emphasizes that the choice of either ηSI or βSI/N

as the incidence in an endemic model for a specific disease is absolutely

crucial, since one can lead to periodic solutions and the other does not.

References

1. Anderson, R.M., Jackson, H.C., May, R.M. and Smith, A.D.M. Populationdynamics of fox rabies in Europe. Nature 289 (1981) 765-777.

2. Anderson, R.M. and May, R.M. Population biology of infectious diseases I.Nature 180 (1979) 361-367.

3. Anderson, R.M. and May, R.M. The population dynamics of microparasitesand their invertebrate hosts. Phil. Trans. Roy. Soc. London B 291 (1981)451-524.

4. Anderson, R.M. and May, R.M., eds. Population Biology of Infectious Dis-eases. Springer-Verlag, Berlin, Heidelberg, New York, 1982.

5. Anderson, R.M., Medley, G.P., May, R.M. and Johnson, A.M. A preliminarystudy of the transmission dynamics of the human immunodeficiency virus,the causative agent of AIDS. IMA J. Math. Appl. Med. Biol. 3 (1986) 229-263.

6. Brauer, F. Epidemic models in populations of varying size. In: Mathemat-ical Approaches to Problems in Resource Management and Epidemiology.Castillo-Chavez, C.C., Levin, S.A. and Shoemaker, C., eds. Lecture Notes



in Biomathematics 81, Springer-Verlag, Berlin, Heidelberg, New York, 1989,109-123.

7. Brauer, F. Models for the spread of universally fatal diseases. J. Math. Biol.28 (1990) 451-462.

8. Brauer, F. Models for universally fatal diseases, II. In: Differential EquationsModels in Biology, Epidemiology and Ecology. Busenberg, S. and Martelli,M., eds. Lecture Notes in Biomathematics 92, Springer-Verlag, Berlin, Hei-delberg, New York, 1991, 57-69.

9. Bremermann, H.J. and Thieme, H.R. A competitive exclusion principle forpathogen virulence. J. Math. Biol. 27 (1989) 179-190.

10. Busenberg, S.N. and Hadeler, K.P. Demography and epidemics. Math. Biosci.101 (1990) 41-62.

11. Busenberg, S.N. and van den Driessche, P. Analysis of a disease transmissionmodel in a population with varying size. J. Math. Biol. 28 (1990) 257-270.

12. Capasso, V. and Serio, G. A generalization of the Kermack-McKendrick de-terministic epidemic model. Math. Biosci. 42 (1978) 41-61.

13. Castillo-Chavez, C., Cooke, K., Huang, W. and Levin, S.A. On the role oflong incubation periods in the dynamics of AIDS I: Single Population Models.J. Math. Biol. 27 (1989) 373-398.

14. Derrick, W.R. and van den Driessche, P. A disease transmission model in anonconstant population. J. Math. Biol. 31 (1992) 495-512.

15. Gao, L.Q. and Hethcote, H.W. Disease transmission models with density-dependent demographics. J. Math. Biol. 30 (1992) 717-731.

16. Gao, L.Q., Mena-Lorca, J., and Hethcote, H.W. Four SEI endemic modelswith periodicity and separatrices, Math. Biosci. 128 (1995) 157-184.

17. Gao, L.Q., Mena-Lorca, J., and Hethcote, H.W. Variations on a theme ofSEI endemic models, In Differential Equations and Applications to Biologyand to Industry, M. Martelli et. al. (eds.), World Scientific, Singapore, 1996,191-207.

18. Greenhalgh, D. An epidemic model with a density-dependent death rate.IMA J. Math. Appl. Med. Biol. 7 (1990) 1-26.

19. Greenhalgh, D. Vaccination in density dependent epidemic models. Bull.Math. Biol. 54 (1992) 733-758.

20. Greenhalgh, D. Some threshold and stability results for epidemic models witha density dependent death rate. Theor. Pop. Biol. 42 (1992) 130-151.

21. Greenhalgh, D. Some results for an SEIR epidemic model with density de-pendence in the death rate. IMA J. Math. Appl. Med. Biol. 9 (1992) 67-106.

22. Grenfell, B. T. and Dobson, A. P., eds., Ecology of Infectious Diseases inNatural Populations, Cambridge University Press, Cambridge, 1995.

23. Hale, J.K. Ordinary Differential Equations, Wiley-Interscience, New York,1969.

24. Hethcote, H.W. Qualitative analyses of communicable disease models. Math.Biosci. 28 (1976) 335-356.

25. Hethcote, H.W. Modeling heterogeneous mixing in infectious disease dynam-ics. In Models for Infectious Human Diseases, V. Isham and G.F.H. Medley,


88 H.W. Hethcote

(eds.), Cambridge University Press, Cambridge, 1996, 215-238.26. Hethcote, H.W. A thousand and one epidemic models, In Frontiers in

Mathematical Biology, S. Levin, ed., Lecture Notes in Biomathematics 100,Springer, Berlin, 1994, 504-515.

27. Hethcote, H.W. and Levin, S.A. Periodicity in epidemiological models. In:Applied Mathematical Ecology, Gross, L., Hallam, T.G. and Levin, S.A.,eds. Springer-Verlag, Berlin, Heidelberg, New York, 1989, 193-211.

28. Hethcote, H.W., Stech, H.W. and van den Driessche, P. Nonlinear oscillationsin epidemic models, SIAM J. Appl. Math. 40 (1981) 1-9.

29. Hethcote, H.W., Stech, H.W. and van den Driessche, P. Periodicity and sta-bility in epidemic models: A survey. In: Differential Equations and Applica-tions in Ecology, Epidemics and Population Problems, Busenberg, S..N. andCooke, K.L., eds. Academic Press, New York, 1981, 65-82.

30. Hethcote, H.W. and Thieme, H.R. Stability of the endemic equilibrium inepidemic models with subpopulations. Math. Biosci. 75 (1985) 205-227.

31. Hethcote, H.W. and Van Ark, J.W. Epidemiological models with heteroge-neous populations: Proportionate mixing, parameter estimation and immu-nization programs. Math Biosci. 84 (1987) 85-118.

32. Hethcote, H.W. and van den Driessche. Some epidemiological models withnonlinear incidence. J. Math. Biol. 29 (1991) 271-287.

33. Hethcote, H.W., Wang, W., and Li, Y. Species Coexistence and Periodicityin Host-Host-Pathogen Models, J Math Biology, in press

34. Hyman, J.M. and Stanley, E.A. Using mathematical models to understandthe AIDS epidemic. Math. Biosci. 90 (1988) 415-473.

35. Jacquez, J.A., Simon, C.P., Koopman, J., Sattenspiel, L. and Perry, T. Mod-eling and analyzing HIV transmission: The effect of contact patterns. Math.Biosci. 92 (1988) 119-199.

36. Lin, X. On the uniqueness of endemic equilibria of an HIV/AIDS transmissionmodel for a heterogeneous population. J. Math. Biol. 29 (1991) 779-790.

37. Lin, X. Qualitative analysis of an HIV transmission model. Math. Biosci. 104(1991) 111-134.

38. Lin, X., Hethcote, H.W. and van den Driessche, P. An epidemiological modelfor HIV/AIDS with proportional recruitment. Math. Biosci. 118 (1993) 181-195.

39. Liu, W.-M., Hethcote, H.W. and Levin, S.A. Dynamical behavior of epi-demiological models with nonlinear incidence rates. J. Math. Biol. 25 (1987)359-380.

40. Liu, W.-M., Levin, S.A. and Iwasa, Y. Influence of nonlinear incidence ratesupon the behavior of SIRS epidemiological models. J. Math. Biol. 23 (1986)187-204.

41. London, W.P. and Yorke, J.A. Recurrent outbreaks of measles, chickenpoxand mumps I: Seasonal variation in contact rates. Am. J. Epidemiol. 98 (1973)453-468.

42. May, R.M. and Anderson, R.M. Regulation and stability of host-parasite pop-ulation interactions II: Destabilizing processes. J. Animal Ecology 47 (1978)



248-267.43. May, R.M. and Anderson, R.M. Population biology of infectious diseases II.

Nature 280 (1979) 455-461.44. McNeill, W.H. Plagues and People. Blackwell, Oxford, 1976.45. Mena-Lorca, J. Periodicity and stability in epidemiological models with

disease-related deaths, Ph.D. Dissertation: University of Iowa, 1988.46. Mena-Lorca, J. and Hethcote, H.W. Dynamic models of infectious diseases

as regulators of population sizes, J. Math. Biol. 30:693-716 (1992).47. Miller, R.K. and Michel, A.N. Ordinary Differential Equations, Academic

Press, New York, 1982.48. Pugliese, A. Population models for diseases with no recovery. J. Math. Biol.

28 (1990) 65-82.49. Pugliese, A. An SEI epidemic model with varying population size. In: Differ-

ential Equation Models in Biology, Epidemiology and Ecology. Busenberg, S.and Martelli, M., eds. Lecture Notes in Biomathematics 92, Springer-Verlag,Berlin, Heidelberg, New York, 1991, 121-138.

50. Strogatz, S,H. Nonlinear Dynamics and Chaos, Addison-Wesley, Reading,Massachusetts, 1994.

51. Swart, J.H. Hopf bifurcation and stable limit cycle behavior in the spreadof infectious disease, with special application to fox rabies. Math. Biosci. 95(1989) 199-207.

52. Thieme, H.R. Persistence under relaxed point-dissipativity, SIAM J MathAnal 24 (1993) 407-435.

53. Thieme, H.R. Epidemic and demographic interaction in the spread of poten-tially fatal diseases in growing populations, Math. Biosci. 111 (1992) 99-130.

54. Thieme, H.R. and Castillo-Chavez, C. On the role of variable infectivity in thedynamics of human immunodeficiency virus. In: Mathematical and StatisticalApproaches to AIDS Epidemiology. Castillo-Chavez, C., ed. Lecture Notesin Biomathematics 83, Springer-Verlag, Berlin, Heidelberg, New York, 1989,157-176.

55. Zhou, J. An epidemiological model with population size dependent incidence.Rocky Mt. J. Math. 24 (1992), 429-445.

56. Zhou, J. and Hethcote, H.W. Population size dependent incidence in modelsfor diseases without immunity. J. Math. Biology 32 (1994) 809-834.


90 H.W. Hethcote


AGE-STRUCTURED EPIDEMIOLOGY MODELS AND

EXPRESSIONS FOR R0

Herbert W. Hethcote


University of Iowa

14 Maclean Hall



Demographic models with either continuous age or age groups are de-veloped and then extended to MSEIR and SEIR endemic models for thespread of infectious diseases in populations. Expressions for the basicreproduction number R0 are obtained and threshold theorems are ob-tained. Values of R0 and the contact number σ are estimated for measlesin Niger and pertussis in the United States.

3.1 Introduction 94

3.2 Three threshold quantities: R0, σ, and R 95

3.3 Two Demographic Models 96

3.3.1 The demographic model with continuous age 96

3.3.2 The demographic model with age groups 99

3.4 The MSEIR Model with Continuous Age Structure 100

3.4.1 Formulation of the MSEIR model 102

3.4.2 The basic reproduction number R0 and stability 104

3.4.3 Expressions for the average age of infection A 107

3.4.4 Expressions for R0 and A with negative exponential survival 108

3.4.5 The MSEIR model with vaccination at age Av 111

3.4.6 Expressions for R0 and A for a step survival function 113

3.5 The SEIR Model with Age Groups 114

3.5.1 Formulation of the SEIR model with age groups 115

91


92 H.W. Hethcote

3.5.2 The basic reproduction number R0 and stability 116

3.5.3 Expressions for the average age of infection A 119

3.6 Application to Measles in Niger 120

3.7 Application to Pertussis in the United States 123

3.8 Discussion 126

References 128

3.1. Introduction

This is my fourth tutorial lecture. In my tutorial lectures I&II we found

that the threshold for many epidemiology models is the basic reproduction

number R0, which determines when an infection can invade and persist

in a new host population. Here we extend these results to epidemiology

models with age structure based on either continuous age or age groups.

The definitions and notation here are the same as in my tutorial lectures

I&II, as shown in Table 5. For example, the variables M, S, E, I, and R

are used for the passively immune, susceptible, exposed (latent), infectious,

and removed epidemiological classes, respectively. This lecture is based on

the last part of the paper [36].

Realistic infectious disease models include both time t and age a as inde-

pendent variables, because age is often an important factor in the transmis-

sion process. For example, age groups mix heterogeneously, the recovered

fraction usually increases with age, risks from an infection may be related

to age, vaccination programs often focus on specific ages, and epidemiologic

data is often age specific. First demographic models with either continu-

ous age or age groups are formulated and analyzed. These two demographic

models demonstrate the role of the population reproduction numbers in de-

termining when the population grows asymptotically exponentially. Then

the MSEIR with continuous age structure is formulated and analyzed. Gen-

eral expressions for the basic reproduction number R0 and the average age

of infection A are obtained. Special expressions for these quantities are

found in the cases when the survival function of the population is a neg-

ative exponential and a step function. In addition the endemic threshold

and the average age of infection are obtained when vaccination occurs at

age Av. Then the SEIR model with age groups is formulated and analyzed.

The expressions for the basic reproduction number R0 and the average age

of infection A are analogous to those obtained for the MSEIR model with

continuous age structure.


Age-Structured Epidemiology Models 93

The theoretical expressions for the basic reproduction number R0 are

used to obtain estimates of the basic reproduction number R0 and the av-

erage age of infection A for measles in Niger, Africa. These estimates are

affected by the very rapid 3.3% growth of the population in Niger. Es-

timates of the basic reproduction number R0 and the contact number σ

are obtained for pertussis (whooping cough) in the United States. Because

pertussis infectives with lower infectivity occur in previously infected peo-

ple, the contact number σ at the endemic steady state is less than the basic

reproduction number R0.

3.2. Three threshold quantities: R0, σ, and R

In my tutorial lectures I&II, we defined the basic reproduction number

R0 as the average number of secondary infections that occur when one

infective is introduced into a completely susceptible host population [16].

The contact number σ is defined as the average number of adequate contacts

of a typical infective during the infectious period [28, 39]. An adequate

contact is one that is sufficient for transmission, if the individual contacted

by the susceptible is an infective. The replacement number R is defined to be

the average number of secondary infections produced by a typical infective

during the entire period of infectiousness [28]. We noted that these three

quantities R0, σ, and R are all equal at the beginning of the spread of an

infectious disease when the entire population (except the infective invader)

is susceptible.


M passively-immune infants

S susceptibles

E exposed people in the latent period

I infectives

R recovered people with immunity

m, s, e, i, r fractions of the population in the classes above

β contact rate

1/δ average period of passive immunity

1/ε average latent period

1/γ average infectious period


σ contact number

R replacement number

Although R0 is only defined at the time of invasion into a completely


94 H.W. Hethcote

susceptible population, σ and R are defined at all times. For most models,

the contact number σ remains constant as the infection spreads, so it is

always equal to the basic reproduction number R0. In these models σ and

R0 can be used interchangeably and invasion theorems can be stated in

terms of either quantity. But for the pertussis models considered here, the

contact number σ becomes less than the basic reproduction number R0 after

the invasion, because new classes of infectives with lower infectivity appear

when the disease has entered the population. The replacement number R is

the actual number of secondary cases from a typical infective, so that after

the infection has invaded a population and everyone is no longer susceptible,

R is always less than the basic reproduction number R0. Also after the

invasion, the susceptible fraction is less than one, so that not all adequate

contacts result in a new case. Thus the replacement number R is always

less than the contact number σ after the invasion. Combining these results

we observe that

R0 ≥ σ ≥ R,

with equality of the three quantities at the time of invasion. Note that

R0 = σ for most models, and σ > R after the invasion for all models.

3.3. Two Demographic Models

Before formulating the age-structured epidemiological models, we present

the underlying demographic models, which describe the changing size and

age structure of a population over time. These demographic models are a

standard partial differential equations model with continuous age and an

analogous ordinary differential equations model with age groups.

3.3.1. The demographic model with continuous age

The demographic model consists of an initial-boundary value problem with

a partial differential equation for age dependent population growth [40]. Let

U(a, t) be the age distribution of the total population, so that the number

of individuals at time t in the age interval [a1, a2] is the integral of U(a, t)

from a1 to a2. The partial differential equation for the population growth

is∂U

∂a+

∂U

∂t= −d(a)U, (3.3.1)

where d(a) is the age specific death rate. Note that the partial derivative

combination occurs because the derivative of U(a(t), t) with respect to t is



∂U∂a

dadt + ∂U

∂t , and dadt = 1. Let f(a) be the fertility per person of age a, so

that the births at time t are given by

B(t) = U(0, t) =

∫ ∞

0

f(a)U(a, t)da. (3.3.2)

The initial age distribution is given by U(a, 0) = U0(a) with U0(0) = B(0).

This model was used by Lotka [47] in 1922 for population modeling, by

McKendrick [48] in 1926 in conjunction with epidemic models, and by

von Foerster [60] for cell proliferation, so it is sometimes called the Lotka-

McKendrick model or the McKendrick-von Foerster model.

We briefly sketch the proof ideas for analyzing the asymptotic behavior

of U(a, t) when d(a) and f(a) are reasonably smooth [40, 41]. Solving along

characteristics with slope 1, we find U(a, t) = B(t−a)e−R

a

0d(v)dv for t ≥ a,

and U(a, t) = u0(a − t)e−R

a

a−td(v)dv for t < a. If the integral in (3.3.2) is

subdivided at a = t, then substitution of the expressions for U(a, t) on the

intervals yields

B(t) = U(0, t)

=

∫ t

0

f(a)B(t − a)e−R

a

0d(v)dvda +

∫ ∞

t

f(a)U0(a)e−R

a

a−td(v)dvda.

This equation with a kernel K(a) in the first integral and g(t) for the second

integral becomes the renewal equation B(t) =∫ t

0K(a)B(t−a)da+g(t). To

analyze this convolution integral equation for B(t), take Laplace transforms

and evaluate the contour integral form of the inverse Laplace transform

by a residue series. As t → ∞, the residue for the extreme right pole

dominates, which leads to U(a, t) → eqtA(a) as t → ∞. Thus the population

age distribution approaches the steady state A(a), and the population size

approaches exponential growth or decay of the form eqt.

To learn more about the asymptotic age distribution A(a), assume a sep-

aration of variables form given by U(a, t) = T (t)A(a). Substituting this into

the partial differential equation (3.3.1) and solving the separated differential

equations yields U(a, t) = T (0)eqtA(0)e−D(a)−qa, where D(a) =∫ a

0d(v)dv.

Substituting this expression for U(a, t) into the birth equation (3.3.2), we

obtain the Lotka characteristic equation given by

1 =

∫ ∞

0

f(a) exp[−D(a) − qa]da. (3.3.3)

If the population reproduction number given by

Rpop =

∫ ∞

0

f(a) exp[−D(a)]da (3.3.4)


96 H.W. Hethcote

is less than, equal to, or greater than 1, then the q solution of (3.3.3) is

negative, zero, or positive, respectively, so that the population is decaying,

constant, or growing, respectively.

In order to simplify the demographic aspects of the epidemiological mod-

els, so there is no dependence on the initial population age distribution, we

assume that the age distribution in the epidemiology models has reached

a steady state age distribution with the total population size at time 0

normalized to 1, so that

U(a, t) = ρeqte−D(a)−qa, with ρ = 1

/∫ ∞

0

e−D(a)−qada . (3.3.5)

In this case the birth equation (3.3.2) is equivalent to the characteristic

equation (3.3.3).

If the age specific death rate d(a) is constant, then (3.3.5) is U(a, t) =

eqt(d + q)e−(d+q)a. Intuitively, when q > 0, the age distribution is (d +

q)e−(d+q)a, because the increasing inflow of newborns gives a constantly in-

creasing young population, so that the age distribution decreases with age

faster than de−da, corresponding to q = 0. Note that the negative exponen-

tial age structure may be a reasonable approximation in some developing

countries, but it is generally not realistic in developed countries, where a

better approximation would be that everyone lives until a fixed age L such

as 75 years and then dies. In this case, d(a) is zero until age L and infinite

after age L, so that D(a) is zero until age L and is infinite after age L. These

two approximate survival functions given by the step function and the neg-

ative exponential are called Type I and Type II mortality, respectively, by

Anderson and May [4]. Of course, the best approximation for any country

is found by using death rate information for that country to estimate d(a).

This approach is used in the models with age groups in Sections 3.6 and

3.7.

The factor w(a) = e−D(a) gives the fraction of a birth cohort surviving

until age a, so it is called the survival function. The rate of death is −w′(a),

so that the expected age a of dying is E[a] =∫∞

0a[−w′(a)]da =

∫∞

0wda.

When the death rate coefficient d(a) is constant, then w(a) = e−da and the

mean lifetime L is 1/d. For a step survival function, the mean lifetime is

the fixed lifetime L.



3.3.2. The demographic model with age groups

This demographic model with age groups has been developed from the

initial-boundary value problem in the previous section for use in age struc-

tured epidemiologic models for pertussis [34]. It consists of a system of n

ordinary differential equations for the sizes of the n age groups defined by

the age intervals [ai−1, ai] where 0 = a0 < a1 < a2 < · · · < an−1 < an = ∞.

A maximum age is not assumed, so the last age interval [an−1,∞) corre-

sponds to all people over age an−1. For a ∈ [ai−1, ai], assume that the death

rates and fertilities are constant with d (a) = di and f(a) = fi. We also as-

sume that the population has reached an equilibrium age distribution with

exponential growth in the form U(a, t) = eqtA(a) given by (3.3.5), so that

the number of individuals in the age bracket [ai−1, ai] is given by

Ni(t) =

∫ ai

ai−1

U(a, t)da = eqt

∫ ai

ai−1

A(a)da = eqtPi, (3.3.6)

where Pi is the size of the ith age group at time 0.

Substituting U(a, t) = eqtA(a) into (3.3.1) yields the ordinary differen-

tial equation dA/da = −[d(a) + q]A, which can be solved on the interval

[ai−1, ai] to obtain

A(a) = A(ai−1) exp[−(di + q)(a − ai−1)]. (3.3.7)

Integrate this A(a) over the interval [ai−1, ai] to get

Pi = A(ai−1){1 − exp[−(di + q)(ai − ai−1)]}/(di + q). (3.3.8)

For i = 1, 2, · · · , n−1, it is convenient to define the constants ci by A(ai) =

ciPi. Use this definition of the constants ci with (3.3.7) and (3.3.8) to obtain

ci =A(ai)

Pi=

di + q

exp[(di + q)(ai − ai−1)] − 1. (3.3.9)

Integration of (3.3.1) on the intervals [ai−1, ai] and (3.3.6) yields

dN1/dt =

n∑

j=1

fjNj − (c1 + d1)N1,

dNi/dt = ci−1Ni−1 − (ci + di)Ni, 2 ≤ i ≤ n − 1,

dNn/dt = cn−1Nn−1 − dnNn.

(3.3.10)

Thus the constants ci are the transfer rate constants between the successive

age groups.


98 H.W. Hethcote

Equations (3.3.7) and (3.3.8) imply A(ai) − A(ai−1) = −[di + q]Pi.

Substituting A(ai) = ciPi leads to Pi = ci−1Pi−1/(ci + di + q) for i ≥ 2.

Iterative use of this equation leads to the following equation for Pi in terms

of P1:

Pi =ci−1 · · · c1P1

(ci + di + q) · · · (c2 + d2 + q). (3.3.11)

The birth equation A(0) =∑n

i=1 fiPi, A(0) = (c1 + d1 + q)P1, and (3.3.11)

lead to the age-group form of the Lotka characteristic equation (3.3.3) given

by

1 =

f1 + f2c1

(c2 + d2 + q)+ · · · + fn

cn−1 · · · c1

(cn + dn + q) · · · (c2 + d2 + q)

(c1 + d1 + q).

(3.3.12)

For this demographic model with n age groups, the population reproduction

number is given by

Rpop = f11

(c1 + d1)+ f2

c1

(c2 + d2)(c1 + d1)

+ · · · + fncn−1 · · · c1

(cn + dn + q) · · · (c1 + d1). (3.3.13)

If the fertility constants fi and the death rate constants di for the age

groups are known, then the equation (3.3.12) with each ci given by (3.3.9)

can be solved for the exponential growth rate constant q. If the population

reproduction number Rpop is less than, equal to, or greater than 1, then

the q solution of (3.3.12) is negative, zero, or positive, respectively, so that

the population is decaying, constant, or growing, respectively. As in the

continuous demographic model, it is assumed that the population starts

at a steady state age distribution with total size 1 at time 0, so that the

group sizes Pi remain fixed and add up to 1. See [34] for more details on

the derivation of this demographic model for age groups.

3.4. The MSEIR Model with Continuous Age Structure

For many endemic models the basic reproduction number can be deter-

mined analytically by either of two methods. One method is to find the

threshold condition above which a positive (endemic) equilibrium exists for

the model and to interpret this threshold condition as R0 > 1. The second

method is to do a local stability analysis of the disease-free equilibrium and



to interpret the threshold condition at which this equilibrium switches from

asymptotic stability to instability as R0 > 1. As shown in tutorial lectures

I&II, both of these methods give the same R0 for the basic SIR endemic

model, because the two equilibria exchange stability with each other in the

sense that as the contact rate increases, the unstable, nontrivial equilibrium

with a negative coordinate moves from outside the feasible region through

the disease-free equilibrium at R0 = 1 and into the feasible region, where

it becomes a positive, stable endemic equilibrium. Similar methods work

to obtain the basic reproduction number for age-structured epidemiological

models; both are demonstrated for an SIR model with continuous age de-

pendence in [13]. Here we use the appearance of an endemic steady state age

distribution to identify expressions for the basic reproduction number R0,

and then show that the disease-free steady state is globally asymptotically

stable if and only if R0 ≤ 1.

M

?

births

-δM

?deaths

S

?

births

-λS

?deaths

E -εE

?deaths

I -γI

?deaths

R

?deaths

Fig. 9. Transfer diagram for the MSEIR model with the passively-immune class M, thesusceptible class S, the exposed class E, the infective class I, and the recovered class R.

This age-structured MSEIR model uses the transfer diagram of Figure

9 and the notation in Tables 5 and 6. The age distributions of the numbers

in the classes are denoted by M(a, t), S(a, t), E(a, t), I(a, t), and R(a, t),

where a is age and t is time, so that, for example, the number of susceptible

individuals at time t in the age interval [a1, a2] is the integral of S(a, t) from

a1 to a2. Because information on age-related fertilities and death rates are

available for most countries and because mixing is generally heterogeneous,

epidemiology models with age groups are now used frequently when ana-

lyzing specific diseases. However, special cases with homogeneous mixing

and asymptotic age distributions that are a negative exponential or a step

function are considered later. These special cases of the continuous MSEIR

model are often used as approximate models. For example, the negative

exponential age distribution is used for measles in Niger in Section 3.6.


100 H.W. Hethcote


f(a), fi fertilities for continuous age, age groups

d(a), di death rate coefficients for continuous age, age groups

L average lifetime

Rpop population reproduction number

q population growth rate constant

U(a, t) distribution of the total population for continuous age

A(a) steady state age distribution for continuous age

N1(t), · · · , Nn(t) distribution of total population at time t for age groups

P1, · · · , Pn steady state age distribution for age groups

ci rate constant for transfer from ith age class

λ(a, t), λi force of infections on susceptibles of age a, in age group i

b(a)b(a) contact rate between people of ages a and a

bibj contact rate between people in age groups i and j


A average age of infection

3.4.1. Formulation of the MSEIR model

The rate constants δ, ε, and γ are the transfer rates out of the M , E, and

I classes. Here it is assumed that the contact rate between people of age

a and age a is separable in the form b(a)b(a), so that the force of infec-

tion λ is the integral over all ages of the contact rate times the infectious

fraction I(a, t)/∫∞

0U(a, t)da at time t. The division by the total popula-

tion size∫∞

0U(a, t)da makes the contact rate λ(a, t) independent of the

population size, so the contact number is independent of the population

size [15, 29, 32, 49]. One example of separable mixing is proportionate mix-

ing, in which the contacts of a person of age are distributed over those of

other ages in proportion to the activity levels of the other ages [33, 52]. If

l(a) is the average number of people contacted by a person of age a per

unit time, u(a) is the steady state age distribution for the population, and

D =∫∞

0l(a)u(a)da is the total number of contacts per unit time of all

people, then b(a) = l(a)/D1/2 and b(a) = l(a)/D1/2. Another example of

separable mixing is age-independent mixing given by b(a) = 1 and b(a) = β.

The system of partial integro-differential equations for the age distribu-

tions are:



∂M/∂a + ∂M/∂t = −(δ + d(a))M,

∂S/∂a + ∂S/∂t = δM − (λ(a, t) + d(a))S,

with λ(a, t) =

∫ ∞

0

b(a)b(a)I(a, t)da

/∫ ∞

0

U(a, t)da ,

∂E/∂a + ∂E/∂t = λ(a, t)S − (ε + d(a))E,

∂I/∂a + ∂I/∂t = εE − (γ + d(a))I,

∂R/∂a + ∂R/∂t = γI − d(a)R.

(3.4.1)

Note that M + S + E + I + R = U(a, t). As in the MSEIR model without

age structure, infants born to mothers in the classes M, E, I, and R have

passive immunity. Thus the boundary conditions at age 0 are

M(0, t) =

∫ ∞

0

f(a)[M + E + I + R]da,

S(0, t) =

∫ ∞

0

f(a)Sda,

while the other distributions at age 0 are zero. Initial age distributions at

time 0 complete the initial-boundary value problem for this MSEIR model.

For each age a the fractional age distributions of the population in

the epidemiological classes at time t are m(a, t) = M(a, t)/U(a, t), s(a, t) =

S(a, t)/U(a, t), etc., where U(a, t) is given by (3.3.5) in the previous section.

Because the numerators and denominator contain the asymptotic growth

factor eqt, these fractional distributions do not grow exponentially. The

partial differential equations for m, s, e, i, and r found from (3.4.1) are

∂m/∂a + ∂m/∂t = −δm,

∂s/∂a + ∂s/∂t = δm − λ(a, t)s,

with λ(a, t) = b(a)

∫ ∞

0

b(a)i(a, t)ρe−D(a)−qada,

∂e/∂a + ∂e/∂t = λ(a, t)s − εe,

∂i/∂a + ∂i/∂t = εe − γi,

∂r/∂a + ∂r/∂t = γi,

(3.4.2)


102 H.W. Hethcote

and the boundary conditions at age 0 are zero except for

m(0, t) =

∫ ∞

0

f(a)[1 − s(a, t)]e−D(a)−qada,

s(0, t) =

∫ ∞

0

f(a)s(a, t)e−D(a)−qada,

(3.4.3)

where m(0, t) + s(0, t) = 1 by (3.3.3).

For this endemic MSEIR model, the steady state age distributions m(a),

s(a), e(a), i(a), and r(a) add up to 1 and satisfy the ordinary differential

equations corresponding to the equations (3.4.2) with the time derivatives

set equal to zero. The steady state solutions m(a), s(a), e(a), and i(a) are

m(a) = (1 − s0)e−δa,

s(a) = e−Λ(a)

[

s0 + δ(1 − s0)

∫ a

0

e−δx+Λ(x)dx

]

,

e(a) = e−εa

∫ a

0

λ(y)eεy−Λ(y)

[

s0 + δ(1 − s0)

∫ y

0

e−δx+Λ(x)dx

]

dy,

i(a) = e−γa

∫ a

0

εe(γ−ε)z

×

∫ z

0

λ(y)eεy−Λ(y)

[

s0 + δ(1 − s0)

∫ y

0

e−δx+Λ(x)dx

]

dydz,

(3.4.4)

where Λ(a) =∫ a

0λ(α)dα with λ = kb(a) for some constant k. At the

disease-free steady state, k is zero, s = 1, and m = e = i = r = 0. The

endemic steady state corresponds to k being a positive constant.

3.4.2. The basic reproduction number R0 and stability

We now use the solutions of the MSEIR model to examine the basic repro-

duction number R0. Substituting the steady state solution i(a) in (3.4.4)

into the expression for λ in (3.4.2) yields

λ(a) = b(a)

∫ ∞

0

b(a)ρe−D(a)−qa−γa

∫ a

0

εe(γ−ε)z

×

∫ z

0

λ(y)eεy−Λ(y)

[

s0 + δ(1 − s0)

∫ y

0

e−δx+Λ(x)dx

]

dydzda. (3.4.5)

Using the definition of s0 and (3.4.4), we find that

s0 = s0Fλ + δ(1 − s0)F∗, (3.4.6)



where Fλ =∫∞

0f(a)e−Λ(a)−D(a)−qada and

F∗ =

∫ ∞

0

f(a)e−Λ(a)−D(a)−qa

∫ a

0

e−δx+Λ(x)dxda. (3.4.7)

Substituting the solution s0 in (3.4.6) into (3.4.5) and cancelling λ(a) =

kb(a) yields

1 =

∫ ∞

0


∫ a

0

εe(γ−ε)z

∫ z

0

b(y)eεy

[

δF∗e−k

R

y

0b(α)dα

+ δ(1 − Fλ)

∫ y

0

e−δx−kR

y

xb(α)dαdx

]/

(δF∗ + 1 − Fλ)dydzda. (3.4.8)

The right side of this equation can be shown to be a decreasing function of

k, so that (3.4.8) has a positive solution k corresponding to a positive force

of infection λ(a) = kb(a) if and only if R0 > 1, where the basic reproduction

number R0 below is found by setting k = 0 in the right side of equation

(3.4.8):

R0 =

∫ ∞

0


∫ a

0

εe(γ−ε)z

∫ z

0

b(y)eεydydzda. (3.4.9)

Note that R0 > 1 implies that (3.4.8) has a positive solution k, which gives

a positive force of infection λ(a) = kb(a) and Λ(a) = k∫ a

0b(α)dα defining

the endemic steady state solution (3.4.4). This expression (3.4.9) for the

basic reproduction number in the MSEIR model seems to be new.

Determining the local stability of the disease-free steady state (at which

λ = kb(a) = 0 and s = 1) by linearization is possible following the method

in [13], but we can construct a Liapunov function to show the global stability

of the disease-free steady state when R0 ≤ 1. The feasible set for (3.4.2)

consists of nonnegative fractions that add to 1. Consider the Liapunov

function

V =

∫ ∞

0

[α(a)e(a, t) + β(a)i(a, t)]da,

where the positive, bounded functions α(a) and β(a) are to be determined.

The formal Liapunov derivative is

V =

∫ ∞

0

{α(a)[λs − εe − ∂e/∂a] + β(a)[εe − γi − ∂i/∂a]}da,

=

∫ ∞

0

{λsα(a) + e[α′(a) − εα(a) + εβ(a)] + [β′(a) − γβ(a)]i}da.


104 H.W. Hethcote

Choose α(a) so that the coefficient of the e term is zero. Then

V =

∫ ∞

0

sb(a)εeεa

∫ ∞

a

e−εzβ(z)dzda

∫ ∞

0

b(a)iρe−D(a)−qada

+

∫ ∞

0

[β′ − γβ]ida.

Choose β(y) so that the last integral is the negative of the next to last

integral. Then

V =

[∫ ∞

0

sb(a)εeεa

∫ ∞

a

e(γ−ε)z

∫ ∞

z

b(x)ρe−D(x)−qx−γxdxdzda − 1

]

×

∫ ∞

0

b(a)i(a, t)ρe−D(a)−qada.

Now s ≤ 1 and the triple integral in the first factor in V above with s = 1

is equal to R0 in (3.4.9) after changing the order of integration. Thus

V ≤ (R0 − 1)

∫ ∞

0

b(a)i(a, t)ρe−D(a)−qada ≤ 0 if R0 ≤ 1.

Hence solutions of (3.4.2) move downward through the level sets of V as

long as they do not stall on the set where V = 0. The set with V = 0 is

the boundary of the feasible region with i = 0, but di(a(t), t)/dt = εe on

this boundary, so that i moves off this boundary unless e = 0. If e = i = 0

so there are no exposed or infectious people, then (3.4.1) implies that there

would be no removed people or infants with passive immunity after several

generations, so everyone would be susceptible. Thus the disease-free steady

state is the only positively invariant subset of the set with V = 0. If there

is a finite maximum age (so that all forward paths have compact closure),

then either Corollary 2.3 in [50] or Corollary 18.5 in [1] (Liapunov-Lasalle

theorems for semiflows) implies that all paths in the feasible region approach

the disease-free steady state.

If R0 > 1, then we have V > 0 for points sufficiently close to the disease-

free steady state with s close to 1 and i > 0 for some age, so that the

disease-free steady state is unstable. This implies that the system (3.4.2) is

uniformly persistent when R0 > 1, as for the ordinary differential equation

models in the basic SIR and MSEIR endemic models in my tutorial lectures

I&II, but the assumption of a finite maximum age seems to be necessary to

satisfy the condition in Theorem 4.6 in [57] that there is a compact set that

attracts all solutions. Although the endemic steady state would usually be

stable, this may not be true in unusual cases. For example, the endemic



steady state can be unstable in the age-structured SIR model when b(a)

is decreasing and b(a) is constant [7] and when b(a) is concentrated at a

certain age while b(a) is constant [56]. Some types of mixing cannot be

written in the separable form b(a)b(a). For example, in preferred mixing,

certain age groups are more likely to mix with their own age group [33]. For

more general mixing, the endemic steady state might not be unique, but

some conditions that guarantee existence, uniqueness, and local stability

have been given [14, 43].

Because the basic reproduction number for the MSEIR model does not

depend on δ or on whether recovered people have no, temporary, or perma-

nent immunity, the expression (3.4.9) for R0 also works for the MSEIRS,

SEIR, SEIRS, and SEIS models, but the equations (3.4.8) for k would be

different. For example, in the SEIR model all newborns are susceptible, so

s0 = 1 and the equation for k is (3.4.8) with Fλ = 1. For the SIR model with

no passively immune or latent classes, an analysis similar to that above for

the MSEIR model leads to an equation for the force of infection constant

k given by

1 =

∫ ∞

0


∫ a

0

b(y)eγy−kR

y

0b(α)dαdyda. (3.4.10)

and a basic reproduction number given by

R0 =

∫ ∞

0


∫ a

0

b(y)eγydyda. (3.4.11)

This expression is similar to previous R0 expressions for SIR models with

constant population size [13, 18]. The expression (3.4.11) for R0 can also be

used for SIRS and SIS models, but the equations for the positive k when

R0 > 1 would be different. Proofs of stability and persistence for the models

in this paragraph are similar to those for the MSEIR model.

3.4.3. Expressions for the average age of infection A

We now find an expression for the average age of infection for the MSEIR

model at the endemic steady state age distribution. Although the steady

state age distribution of the population is ρe−D(a)−qa, the age distribution

for a specific birth cohort is e−D(a)/∫∞

0e−D(a)da. Thus the rate that in-

dividuals in a birth cohort leave the susceptible class due to an infection

is λ(a)s(a)e−D(a)/∫∞

0e−D(a)da, where s(a) is given in (3.4.4). Hence the


106 H.W. Hethcote

expected age A for leaving the susceptible class is

A = E[a]

=

∫∞

0aλ(a)e−D(a)[δF∗e

−Λ(a) + δ(1 − Fλ)∫ a

0e−δx−Λ(a)+Λ(x)dx]da

∫∞

0λ(a)e−D(a)[δF∗e−Λ(a) + δ(1 − Fλ)

∫ a

0e−δx−Λ(a)+Λ(x)dx]da

.

(3.4.12)

This expression assumes that the force of infection λ(a) = kb(a) at the

endemic steady state age distribution has already been determined, so that

Λ(a), Fλ, and F∗ are known. For the SEIR and SIR models, s(a) = e−Λ(a),

so that the expression for the average age of infection is

A = E[a] =

∫∞

0aλ(a)e−Λ(a)−D(a)da

∫∞

0λ(a)e−Λ(a)−D(a)da

. (3.4.13)

3.4.4. Expressions for R0 and A with negative exponential

survival

When the death rate coefficient d(a) is independent of the age a, the age

distribution (3.3.5) becomes U(a, t) = eqt(d + q)e−(d+q)a. Also the waiting

times in M , E, and I have negative exponential distributions, so that, after

adjusting for changes in the population size, the average period of passive

immunity, the average latent period, and the average infectious period are

1/(δ + d + q), 1/(ε + d + q), and 1/(γ + d + q), respectively. Here it is also

assumed that the contact rate is independent of the ages of the infectives

and susceptibles, so we let b(a) = 1 and b(a) = β. In this case (3.4.9)

defining the basic reproduction number becomes

R0 = βε/[(γ + d + q)(ε + d + q)], (3.4.14)

which has the same interpretation as R0 in the MSEIR model without age

structure.

With the assumptions above, λ is a constant and the equation (3.4.5)

for λ becomes

1 =(d + q)R0

λ + d + q

[

s0 +δ(1 − s0)

δ + d + q

]

, (3.4.15)

If s is the integral average of the susceptible steady state age distribution

s(a)(d + q)e−(d+q)a over all ages, then using the endemic steady state solu-

tion s(a) given in (3.4.4), we find that R0s = 1 is equivalent to the equation

(3.4.15). Thus the infective replacement number R0s is one at the endemic



equilibrium for this model. However, this is generally not true, so it is not

valid to use R0 = 1/s to derive an expression for the basic reproduction

number.

Using the definition of s0 and the solutions (3.4.4), we find that

s0 =δ − λs0

δ − λFλ −

δ(1 − s0)

δ − λFδ, (3.4.16)

where Fλ =∫∞

0f(a)e−(λ+d+q)ada and Fδ =

∫∞

0f(a)e−(δ+d+q)ada. Note

that F∗ in (3.4.7) is equal to (Fλ −Fδ)/(δ−λ), so that (3.4.6) is equivalent

to (3.4.16). Here the equations (3.4.15) and (3.4.16) are two simultaneous

equations in the unknowns R0, s0, and λ. One can solve (3.4.16) for s0 to

obtain

s0 = δ(Fλ − Fδ)/[δ(1 − Fδ) − λ(1 − Fλ)]. (3.4.17)

The right side of (3.4.17) is a decreasing function of λ with Fλ = 1 and

s0 = 1 at λ = 0. Substituting (3.4.17) into (3.4.15) yields the equation

corresponding to equation (3.4.8) given by

1 =

R0(d + q)δ

[

Fλ − Fδ +(δ − λ)(1 − Fλ)

δ + d + q

]

(λ + d + q)[δ(1 − Fδ) − λ(1 − Fλ)], (3.4.18)

which relates R0 and λ. Because the right side of (3.4.18) is a decreasing

function of λ that goes from R0 at λ = 0 to zero as λ → ∞, the equation

(3.4.18) has a positive solution λ if and only if R0 > 1. If d(a) = d and

f(a) = b = d + q, then equation (3.4.18) reduces to a simple equation for

λ in an ordinary differential equations MSEIR model ([36], p. 620). When

R0 ≤ 1, solutions of (3.4.2) approach the disease-free steady state (3.4.4)

with λ = 0, and for fixed R0 > 1, we expect solutions to approach the

endemic steady state (3.4.4) with the constant λ determined by solving

either (3.4.18) or the combination of (3.4.15) and (3.4.16).

We now find an expression for the average age of infection for this

MSEIR model. Here the steady state age distribution of the population

is (d + q)e−(d+q)a, and the age distribution for a specific birth cohort is

de−da. Thus the rate that individuals in a birth cohort leave the suscepti-

ble class due to an infection is λs(a)de−da, where s(a) is given in (3.4.4).

Here the equation for the expected age A for leaving the susceptible class


108 H.W. Hethcote

is

A = E[a] =λd∫∞

0a[c1e

−(λ+d)a + c2e−(δ+d)a]da

λd∫∞

0[c1e−(λ+d)a + c2e−(δ+d)a]da

=

δ − λs0

(λ + d)2−

δ(1 − s0)

(δ + d)2

δ − λs0

(λ + d)−

δ(1 − s0)

(δ + d)

.

(3.4.19)

It is useful to consider limiting cases of the model and the corresponding

limiting equations for R0 and A. If δ → ∞, the M class disappears, so that

the MSEIR model becomes an SEIR model with s0 = 1, and the equations

above reduce to λ = (d + q)(R0 − 1) and A = 1/(λ + d), where R0 is still

given by (3.4.14). These same equations also hold for the SIR model, but

R0 = β/(γ+d+q) for this model. For the SEIR and SIR models it is possible

to solve explicitly for R0 in terms of the average lifetime L = 1/d and the

average age of infection A to obtain R0 = (q + 1/A)/(q + 1/L). When

the population has constant size with q = 0, the R0 expression reduces

to R0 = L/A which is the usual formula for the SEIR and SIR models

[34]. By not including the death factor e−da when considering the rate

of leaving the susceptible class, one obtains the widely-cited approximate

formula R0 = 1 + L/A for the SEIR and SIR models [16]. But the death

factor really should be included, since we want to calculate the average age

for those who survive long enough to become infected.

As another limiting case, consider the MSEIR model for a very virulent

disease in which almost every mother has been infected. In the limiting

situation every newborn infant has passive immunity, so that m0 → 1 and

s0 → 0. In this case λ = (d + q)[R0δ/(δ + d + q) − 1] and A = 1/(δ +

d) + 1/(λ + d). Note that the formula for λ is for an endemic steady state

for a virulent disease, so it does not imply that R0δ/(δ + d + q) > 1 is

the threshold condition for existence of a positive endemic steady state age

distribution; compare with [4] (p. 81). The formula for A is plausible since

it is the sum of the average period p = 1/(δ + d) of passive immunity and

the average age of attack 1/(λ + d) from the SEIR model. Thus for a very

virulent disease, adding a passively immune class to a model increases the

average age of attack by the mean period of passive immunity. Solving for

R0 in terms of the average period p of passive immunity and the average

lifetime L = 1/d, we obtain

R0 =[q + 1/(A − p)](1 + pq)

(q + 1/L)(1 − p/L). (3.4.20)

For a constant population size with q = 0, we have R0 = L/[(A − p)(1 −



p/L)]. For q = 0 and p ¿ L, we obtain the approximation R0 ≈ (L +

p)/(A − p). For this MSEIR model with constant size, it seems naively

that one could just subtract off the average period p of passive immunity

from the average age A of infection and the average lifetime L to obtain

the approximation R0 ≈ (L − p)/(A − p) used in [4] (p. 79, p. 658), [31],

but our careful analysis here shows that this naive formula does not work.

Of course, when q = 0 and p = 0, the expression (3.4.20) reduces to the

previous expression R0 = L/A for the SEIR model with constant population

size.

3.4.5. The MSEIR model with vaccination at age Av

Now we modify the age structured MSEIR endemic model above with con-

stant coefficients to include vaccination at age Av. The results using this

approximate model for measles in Niger are compared with the correspond-

ing results for the MSEIR model with age groups in Section 3.6. Let g be the

fraction of the population vaccinated successfully at age Av (i.e. the frac-

tion of the population which has permanent immunity after vaccination). In

epidemiological terminology, g is the product of the fraction vaccinated and

the vaccine efficacy. This vaccination at age Av causes a jump discontinuity

in the susceptible age distribution given by s(Av + 0) = (1 − g)s(Av − 0),

where s(Av − 0) is the limit from the left and s(Av + 0) is the limit from

the right.

With this jump condition, the ordinary differential equations corre-

sponding to (3.4.2) without time derivatives, but with constant d and λ,

are solved first on the interval [0, Av] and then on the interval [Av,∞). The

details are omitted, but substituting the steady state solutions i(a) on these

intervals into the expression for λ yields

1 =R0(d + q)

λ + d + q

[

s0 +δ(1 − s0)

δ + d + q− g[c1e

−(λ+d+q)Av + c2e−(δ+d+q)Av ]

]

,

(3.4.21)

where c1 = (δ−λs0)/(δ−λ) and c2 = −δ(1−s0)/(δ−λ). Note that equation

(3.4.21) reduces to (3.4.15) when g = 0. The analog here of (3.4.16) is

s0 = c1Fλ + c2Fδ − g[

c1 + c2e(λ−δ)Av

]

FAv, (3.4.22)

where FAv=∫∞

Avf(a)e−(λ+d+q)ada, and Fλ and Fδ are given in the previous

subsection. Given g, Av, and the values for the parameters β, γ, ε, δ, d,

and q, the equations (3.4.21) and (3.4.22) are two simultaneous equations


110 H.W. Hethcote

in the unknowns R0, s0, and λ. It is possible to solve (3.4.22) for s0 and

then substitute into (3.4.21), but we do not present the resulting, rather

complicated expression, which relates R0 and λ. For an SEIR and SIR

models, s0 = 1, so that (3.4.21) reduces to

1 =R0(d + q)

λ + d + q

[

1 − ge−(λ+d+q)Av

]

. (3.4.23)

For fixed parameters and R0 > 1, it is interesting to find how large

the successfully vaccinated fraction g must be in order to achieve herd

immunity. Recall that a population has herd immunity if a large enough

fraction is immune, so that the disease would not spread if an outside

infective were introduced into the population. To determine this threshold

we consider the situation when the disease is at a very low level with λ nearly

zero, so that almost no one is infected. Thus the initial passively immune

fraction m0 is very small and the initial susceptible fraction s0 is nearly 1.

In the limit as s0 → 1, equation (3.4.21) for the MSEIR model reduces to

λ = (d + q)(R0[1 − ge−(λ+d+q)Av ] − 1), which has a positive solution λ if

and only if ge−(d+q)Av < 1− 1/R0. If the successfully-vaccinated fraction g

at age Av is large enough so that

ge−(d+q)Av ≥ 1 − 1/R0, (3.4.24)

then the population has herd immunity and the disease cannot spread in

this population. It may seem surprising that this condition is the same for

the SEIR and the MSEIR models, but for very low disease levels, almost

no newborn children have passive immunity, so that the passively immune

class M has no influence on the threshold condition. A similar criterion for

herd immunity with vaccination at two ages in a constant population is

given in [30].

If the condition (3.4.24) is satisfied, then we expect solutions of (3.4.2)

to approach the steady state age distribution with λ = 0, s(a) = 1, and all

other distributions equal to zero, so that the disease disappears. Intuitively,

there are so many immunes that the average infective cannot replace itself

with at least one new infective during the infectious period and, conse-

quently, the disease dies out. If the inequality above is not satisfied and

there are some infecteds initially, then we expect the susceptible fraction

to approach the stable age distribution given by the jump solution with a

positive, constant λ that satisfies (3.4.21) and (3.4.22).



For an MSEIR model an expression for the average age of infection is

A =1

λ + d−

gAv[c1e−(λ+d)Av + c2e

−(δ+d)Av ] + c2δ−λ

(δ+d)2

c1[1 − ge−(λ+d)Av ] + c2[λ+dδ+d − ge−(δ+d)Av ]

.

The analogous expression for an SEIR or SIR model has c2 = 0. The neg-

ative signs in the expression for A makes it seem as if A is a decreasing

function of the successfully vaccinated fraction g, but this is not true since

the force of infection λ is a decreasing function of g.

3.4.6. Expressions for R0 and A for a step survival

function

For the demographic model in which everyone survives until age L and then

dies, d(a) is zero until age L and infinite after age L, so that D(a) is zero

until age L and is infinite after age L. It is assumed that the population

is constant, so q = 0 and ρ = 1/L in (3.3.5). Mixing is homogeneous,

so b(a) = 1 and b(a) = β. For the MSEIR and SEIR models the basic

reproduction number found from (3.4.9) is

R0 =β

γ

[

1 +γ

ε − γ

1 − e−εL

εL−

ε

(ε − γ)

1 − e−γL

γL

]

. (3.4.25)

An epidemiological interpretation is that the right side of (3.4.25) except

for the contact rate β is the average infectious period. For the SEIR model

the equation (3.4.8) for the constant λ at the endemic steady state age

distribution becomes

1 = βε

[

1 − e−λL

(γ − λ)(ε − λ)λL+

1 − e−εL

(ε − λ)(ε − γ)εL−

1 − e−γL

(γ − λ)(ε − γ)γL

]

.

(3.4.26)

For the MSEIR model the integrals in the equation (3.4.8) do not simplify

very much, so this equation for the constant λ is not presented. The basic

reproduction numbers for the MSEIRS and SEIRS models are also given by

(3.4.25), but the equations for the constant λ at the endemic steady state

age distributions would be different for these models.

For the analogous SIR model R0 found from (3.4.11) is given by

R0 =β

γ

[

1 −1 − e−γL

γL

]

, (3.4.27)


112 H.W. Hethcote

and the equation (3.4.10) for the constant λ at the endemic steady state

age distribution is

1 =β

γ − λ

[

1 − e−λL

λL−

1 − e−γL

γL

]

. (3.4.28)

These expressions can also be found heuristically by letting ε → ∞ in

(3.4.25) and (3.4.26), so that the exposed class E disappears.

For the SEIR and SIR models equation (3.4.13) for the average age of

attack becomes

A =1

λ−

Le−λL

1 − e−λL. (3.4.29)

The analogous equation for the MSEIR model does not simplify as much.

For the SEIR and SIR models, the average susceptible fraction is

s =

∫ L

0

e−λa

Lda =

1 − e−λL

λL. (3.4.30)

It is easy to see from (3.4.25) and (3.4.27) that R0s 6= 1 for the SEIR

and SIR models, so using R0 = 1/s gives incorrect expressions for R0.

Expressions similar to those in this section can be found for a nonconstant

population with ρ = q/(1 − e−qL), but they are not presented here.

Typically the lifetime L is larger than the average age of attack A ≈ 1/λ,

and both are much larger than the average latent period 1/ε and the average

infectious period 1/γ. Thus for typical directly transmitted diseases, λL is

larger than 5 and γL, εL, γ/λ, and ε/λ are larger than 50. Hence R0 ≈ β/γ

from (3.4.25) and (3.4.27), 1 = β(1−e−λL)/γλL from (3.4.26) and (3.4.28),

and A ≈ 1/λ from (3.4.29). Thus R0 ≈ λL/(1 − e−λL) ≈ λL ≈ L/A,

and R0s ≈ 1. Hence many of the formulas for Type I mortality in the

Anderson and May book ([4], Ch. 4, App. A) are either correct or reasonable

approximations.

3.5. The SEIR Model with Age Groups

Here we develop an expression for the basic reproduction number R0 in

an SEIR model with n separate age groups. This SEIR model is similar to

the MSEIR model shown in Figure 9, but there is no class M for passively

immune infants. In Sections 3.6 and 3.7 we estimate the basic reproduction

number in models with age groups for measles in Niger and pertussis in the

United States.



3.5.1. Formulation of the SEIR model with age groups

The SEIR model uses the same notation as the MSEIR model. The initial-

boundary value problem for this model is given below.

∂S/∂a + ∂S/∂t = −λ(a, t)S − d(a)S,

λ(a, t) =

∫ ∞

0

b(a)b(a)I(a, t)da

/∫ ∞

0

U(a, t)da ,

∂E/∂a + ∂E/∂t = λ(a, t)S − εI − d(a)E,

∂I/∂a + ∂I/∂t = εI − γI − d(a)I,

∂R/∂a + ∂R/∂t = γI − d(a)R.

(3.5.1)

The initial conditions are the values of the age distributions at time 0.

The boundary values at age 0 are all zero except for the births given by

S(0, t) =∫∞

0f(a)U(a, t)da.

The population is partitioned into n age groups as in the demographic

model with age groups. The subscripts i denote the parts of the epidemio-

logic classes in the ith age interval [ai−1, ai], so that Si(t) =∫ ai

ai−1S(a, t)da,

etc. Assume that the transfer rate coefficients on the age intervals are εi

and γi. Also assume that the separable contact rate is constant for the

interactions between age groups, so that b(a) = bi for a ∈ [ai−1, ai] and

b(a) = bj for a ∈ [aj−1, aj ]. By integrating the partial differential equations

(3.5.1) on the age intervals [ai−1, ai], using∑n

j=1 fiPi = (c1 + d1 + q)P1,

S(ai, t) = ciSi, E(ai, t) = ciEi, etc. as in the demographic model, and

using the boundary conditions, we obtain an initial value problem for 4n

ordinary differential equations for the sizes of the epidemiological classes in

the ith age group. The total in the four epidemiologic classes for the ith

age group is the size Ni(t) = eqtPi of the ith group, which is growing ex-

ponentially, but the age distribution P1, P2,..., Pn remains at a steady state

and∑n

i=1 Pi = 1.

Because the numbers are all growing exponentially by eqt, the fractions

of the population in the epidemiologic classes are of more interest than the

numbers in these epidemiologic classes. These fractions are given by si(t) =

Si(t)/eqt, etc., so that the fractions si, ei, ii,and ri add up to the age group

size Pi. The derivatives of these fractions satisfy s′i(t) = S′i(t)/e

qt − qsi,


114 H.W. Hethcote

etc., so that the differential equations for these fractions are

ds1/dt = (c1 + d1 + q)P1 − [λ1 + c1 + d1 + q]s1,

dsi/dt = ci−1si−1 − [λi + ci + di + q]si, i ≥ 2,

λi = bi

n∑

j=1

bjij ,

de1/dt = λ1s1 − [ε1 + c1 + d1 + q]e1,

dei/dt = λisi + ci−1ei−1 − [εi + ci + di + q]ei, i ≥ 2,

di1/dt = ε1e1 − [γ1 + c1 + d1 + q]i1,

dii/dt = εiei + ci−1ii−1 − [γi + ci + di + q]ii, i ≥ 2,

dr1/dt = γ1i1 − [c1 + d1 + q]r1,

dri/dt = γiii + ci−1ri−1 − [ci + di + q]ri, i ≥ 2.

(3.5.2)

3.5.2. The basic reproduction number R0 and stability

Here we follow the same procedure used in the continuous model to find

an expression for the basic reproduction number R0. Note that the steady

state age distribution for the differential equations (3.5.2) is the equilibrium

with

s1 = c1P1/λ1, si = ci−1si−1/λi, for i ≥ 2,

e1 = λ1s1/ε1, ei = (λisi + ci−1ei−1)/εi, for i ≥ 2,

i1 = ε1e1/γ1, ii = (εiei + ci−1ii−1)/γi, for i ≥ 2.

(3.5.3)

where we use λi for λi+ci +di +q, εi for εi +ci +di+q, γi for γi+ci +di +q,

and c1 for c1 + d1 + q. Substituting si−1 successively, we find that si =

Ci−1/[λi · · · λ1] for i ≥ 2, where Ci−1 for ci−1 · · · c1c1P1. Next we substitute

the si−1 and ei−1 successively into the ei quotient in (3.5.3) to obtain

e1 = λ1c1P1/(ε1λ1), and

ei =λiCi−1

εiλi · · · λ1

+λi−1Ci−1

εiεi−1λi−1 · · · λ1

+λi−2Ci−1

εiεi−1εi−2λi−2 · · · λ1

+· · ·+λ1Ci−1

εi · · · ε1λ1

for i ≥ 2. When the expressions for ei and ii−1 are substituted into the

expression for ii in (3.5.3), we obtain i1 = ε1λ1c1P1/(γ1ε1λ1), and for



i ≥ 2,

iiCi−1

=εi

γi

(

λi

εiλi · · · λ1

+λi−1

εiεi−1λi−1 · · · λ1

+ · · · +λ1

εi · · · ε1λ1

)

+εi−1

γiγi−1

(

λi−1

εi−1λi−1 · · · λ1

+λi−2

εi−1εi−2λi−2 · · · λ1

+ · · · +λ1

εi−1 · · · ε1λ1

)

+ · · · +ε2

γi · · · γ2

(

λ2

ε2λ2λ1

+λ1

ε2ε1λ1

)

+ε1

γi · · · γ1

(

λ1

ε1λ1

)

. (3.5.4)

From (3.5.2), we observe that λi = kbi, where k is a constant given by

k =∑n

j=1 bjij . Now the expressions for ii and λi = kbi can be substituted

into this last summation to obtain

1 =

n∑

j=1

bjCj−1

[

εj

γj

(

bj

εj bj · · · b1

+bj−1

εj εj−1bj−1 · · · b1

+ · · · +b1

εj · · · ε1b1

)

+εj−1

γj γj−1

(

bj−1

εj−1bj−1 · · · b1

+bj−2

εj−1εj−2bj−2 · · · b1

+ · · · +b1

εj−1 · · · ε1b1

)

+ · · · +ε2

γj · · · γ2

(

b2

ε2b2b1

+b1

ε2ε1b1

)

+ε1

γj · · · γ1

(

b1

ε1b1

)]

, (3.5.5)

where bj = bjk + ci + di + q and C0 = c1P1.

The right side of (3.5.5) is a decreasing function of k, so that it has

a solution for a positive k if and only if R0 > 1, where R0 is the basic

reproduction number below defined by setting k = 0 in (3.5.5).

R0 =

n∑

j=1

bjCj−1

[

εj

γj

(

bj

εj cj · · · c1+

bj−1

εj εj−1cj−1 · · · c1+ · · · +

b1

εj · · · ε1c1

)

+εj−1

γj γj−1

(

bj−1

εj−1cj−1 · · · c1+

bj−2

εj−1εj−2cj−2 · · · c1+ · · · +

b1

εj−1 · · · ε1c1

)

+ · · · +ε2

γj · · · γ2

(

b2

ε2c2c1+

b1

ε2ε1c1

)

+ε1

γj · · · γ1

(

b1

ε1c1

)]

, (3.5.6)

where ci = ci+di+q. The expression (3.5.6) for R0 is the discrete age group

analog of the triple integral expression (3.4.9) of R0 for the SEIR model

with continuous age. As for the continuous age model, the expression (3.5.6)

for R0 is also valid for the analogous MSEIR, MSEIRS, SEIRS, and SEIS

models with age groups, but the equations involving the force of infection

constant k would be different from (3.5.5) for these other models. The


116 H.W. Hethcote

equation for k for the MSEIR model could be found by tedious calculations

following the method used above.

If R0 > 1 for the SEIR model, then equation (3.5.5) can be solved for

a positive k to get the forces of infection λi = kbi, which give the unique

endemic equilibrium in the age groups from (3.5.3). Determining the sta-

bility of the disease-free equilibrium (at which everyone is susceptible) by

linearization is intractable except for small n, but we can construct a Lia-

punov function to prove the global stability of the disease-free equilibrium

when R0 ≤ 1 by taking a linear combination of the exposed and infectious

fractions. Here the feasible region is the subset of the nonnegative orthant in

the 4n dimensional space with the class fractions in the ith group summing

to Pi. Let V =∑

(αiei + βiii), where the coefficients are to be determined.

In the Liapunov derivative V , choose the αi coefficients so that the ei terms

cancel out by letting αn = βnεn/εn and αj−1 = (βj−1εj−1 + cj−1αj)/εj−1

for αn−1, · · · , α1. Then

V =∑

αibisi

∑

bjij − (β1γ1 − β2c1)i1

− · · · − (βn−1γn−1 − βncn−1)in−1 − βnγnin.

Now choose the βi so that the coefficients of the ij in the last n terms are

−bj by letting βn = bn/γn and βj−1 = (bj−1 + βjcj−1)/γj−1 for βn−1, · · · ,

β1. Using si ≤ Pi, we obtain V ≤ (R0 − 1)∑

bjij ≤ 0 if R0 ≤ 1. The set

where V = 0 is the boundary of the feasible region with ij = 0 for every

j, but dij/dt = εjej on this boundary, so that ij moves off this boundary

unless ej = 0. When ej = ij = 0, drj/dt = −cjrj , so that rj → 0. Thus

the disease-free equilibrium is the only positively invariant subset of the set

with V = 0, so that all paths in the feasible region approach the disease-free

equilibrium by the Liapunov-Lasalle theorem ([25], p. 296). Thus if R0 ≤ 1,

then the disease-free equilibrium is asymptotically stable in the feasible re-

gion. If R0 > 1, then we have V > 0 for points sufficiently close to the

disease-free equilibrium with si close to Pi and ij > 0 for some j, so that

the disease-free equilibrium is unstable. The system (3.5.2) can be defined

to be uniformly persistent if lim inf t→∞ ij(t) ≥ c for some c > 0 for all j

and all initial points such that ej(0) + ij(0) > 0 for some j. The instabil-

ity of the disease-free equilibrium and Theorem 4.5 in [57] imply that the

system (3.5.2) is uniformly persistent if R0 > 1. The endemic equilibrium

(3.5.3) corresponding to positive k would usually be asymptotically stable

in specific applications, but as for the continuous age model, it could be

unstable for unusual or asymmetric choices of bi and bi.



Using the same methods for an SIR model, the equation for the k in the

forces of infection λi = kbi is

1 =

n∑

j=1

bjCj−1

[

bj

γj bj · · · b1

+bj−1

γj γj−1bj−1 · · · b1

+ · · · +b2

γj · · · γ2b2b1

+b1

γj · · · γ1b1

]

, (3.5.7)

and the equation for the basic reproduction number is

R0 =n∑

j=1

bjCj−1

[

bj

γj cj · · · c1+

bj−1

γj γj−1cj−1 · · · c1

+ · · · +b2

γj · · · γ2c2c1+

b1

γj · · · γ1c1

]

. (3.5.8)

These equations can be also derived heuristically from those for the SEIR

model by letting εi → ∞ for every i. The R0 formula (3.5.8) also works

for the SIRS and SIS models with age groups, but the equations for k

would be different. Proofs of stability and persistence for the models in this

paragraph are similar to those for the SEIR model.

3.5.3. Expressions for the average age of infection A

From Section 3.4.3 we know that the average age of infection A is given by

A = E[a] =

∫∞

0aλ(a)s(a)e−D(a)da

∫∞

0λ(a)s(a)e−D(a)da

=

∑ni=1

∫ ai

ai−1aλ(a)s(a)e−D(a)da

∑ni=1

∫ ai

ai−1λ(a)s(a)e−D(a)da

.

In each integral above over the interval [ai−1, ai] of length ∆i, we have

the endemic equilibrium values s(a) = si, λ(a) = λi = kbi, and e−D(a) =

πi−1e−di(a−ai−1), where πi−1 =

∏i−1j=1e

−dj∆j . The integrals over the inter-

vals can be evaluated to obtain the following expression for the average

age of infection at the endemic equilibrium for the MSEIR, SEIR, and SIR

model with age groups.

A =

∑ni=1 bisiπi−1[1 + diai−1 − (1 + diai)e

−di∆i ]/d2i

∑ni=1 bisiπi−1[1 − e−di∆i ]/di

. (3.5.9)


118 H.W. Hethcote

3.6. Application to Measles in Niger

A deterministic compartmental mathematical model has been developed for

the study of the effects of heterogeneous mixing and vaccination distribu-

tion on disease transmission in Africa [44]. This study focuses on vaccination

against measles in the city of Naimey, Niger, in sub-Saharan Africa. The

rapidly growing population consists of a majority group with low trans-

mission rates and a minority group of seasonal urban migrants with higher

transmission rates. Demographic and measles epidemiological parameters

are estimated from data on Niger.

Here we consider the MSEIR model with 16 age groups for a homoge-

neously mixing, unvaccinated population in Niger [44]. The fertility rates

and the death rates in the 16 age groups are obtained from Niger census

data. Using the Lotka equation (3.3.12) for the demographic model with

age groups, the value of q corresponds to a growth of 3.36% per year. This

is consistent with the estimate from 1988 census data of 3.3% growth per

year. From measles data, it is estimated that the average period of passive

immunity 1/δ is 6 months, the average latent period 1/ε is 14 days and

the average infectious period 1/γ is 7 days. From data on a 1995 measles

outbreak in Niamey, the force of infection λ is estimated to be the con-

stant 0.762 per year [44]. A computer calculation using the demographic

and epidemiological parameter values in the formula (3.5.6) for the basic

reproduction number yields R0 = 18.83. The average age of infection at the

endemic equilibrium found from (3.5.9) is A = 2.4 years.

We now consider two methods for finding approximations to R, R0, and

A. The first method finds approximate values during the computer simu-

lations of the MSEIR measles model. Recall that the replacement number

R is the actual number of new cases per infective during the infectious pe-

riod. At an equilibrium in the basic SIR endemic model, R would be the

incidence divided by the quotient of the prevalence and the duration. For

this MSEIR model with age groups, R can be approximated by computing

the sum over all age groups of the daily incidence divided by the sum over

all age groups of the quotients of the prevalences and the product of the

average infectious periods times the fractions surviving the latent periods,



so that

R ∼=

16∑

j=1

λjsjPj

∑16j=1 ijPj

/[(

1

γj + dj + q

)(

εj

εj + dj + q

)] .

At the prevaccination endemic equilibrium, this approximation is computed

to be R ∼= 0.99988, which is consistent with the concept that the average

replacement number is equal to one at the endemic equilibrium.

For this MSEIR model there is only one class of infectives, so that the

basic reproduction number R0 is equal to the contact number σ at the pre-

vaccination endemic equilibrium. This contact number σ is approximated

by computing the sum of the daily incidences when all contacts are assumed

to be with susceptibles divided by the sum over all age groups of the quo-

tients of the prevalences and the product of the average infectious periods

times the fractions surviving the latent periods. When all of the sj in the

numerator in the expression for the replacement number R are replaced by

1, then we obtain the expression for the contact number σ given by

R0 = σ ∼=

16∑

j=1

λjPj

∑16j=1 ijPj

/[(

1

γj + dj + q

)(

εj

εj + dj + q

)] .

At the prevaccination endemic equilibrium, this yields R0∼= 18.85, which

is very close to the formula value of 18.83. Note that the expressions for

R0 and R given here are slightly different from those given in [36]; based on

calculations for several pertussis models, the expressions given here seem

to the correct ones.

The average age of infection can be crudely approximated in the measles

computer simulations by the quotient of the sum of the average age in

each age group times the incidence in that age group and the sum of the

incidences. Hence

A ∼=

∑16j=1[

aj−1+aj

2 ]λjsjPj∑16

j=1 λjsjPj

.

This approach gives A ∼= 2.2 years, which is slightly less than the formula

value of 2.4 years.


120 H.W. Hethcote

The second approximation method is to use the formulas for the MSEIR

endemic model in my tutorial lectures I&II, which has uniform constant

mortality and a negative exponential age distribution. This model is plau-

sible because the age distribution of the Niger population is closely approx-

imated by a negative exponential [44]. From census data the death rate for

the population is 22 per thousand per year. Using this d value and the fer-

tilities in the Lotka characteristic equation for discrete age groups (3.3.12),

we solve iteratively to obtain q = 0.02326 per year. This q value corre-

sponds to a population growth rate of 2.3% per year, which is less than the

recent census value of 3.3% growth per year, but this difference may occur

because our model is a simplification of the actual demographics. The value

d + q = 0.045 per year is consistent with the Niger population surviving

fraction as a function of age, which is very close to the exponential e−0.045a

for age a in years.

Recall that the replacement number R is 1 at the endemic equilibrium

for this model. Using the values of d+q, δ, and λ, the equations (3.4.15) and

(3.4.16) can be solved iteratively to obtain a basic reproduction number

of R0 = 17.4 and an susceptible fraction at age 0 of s0 = 1.6 × 10−6.

Thus in this population nearly every mother is infected with measles before

childbearing age, so almost every newborn child has passive immunity. In

the limit as s0 → 0, equation (3.4.15) becomes

R0 = [1 + λ/(d + q)][1 + (d + q)/δ],

which also leads to R0 = 17.4. This value is a reasonable approximation to

the value of R0 = 18.83 estimated above in the MSEIR model with 16 age

groups. The average age of infection of A = 1.8 years can be found from

either (3.4.12) or the approximation A = 1/(δ + d) + 1/(λ + d). This value

is less than the value of A = 2.4 years estimated above using the MSEIR

model with 16 age groups; this difference may be due to the high infant

mortality that occurs in the model with age groups. Using the estimated

parameter values and a vaccination age of Av = 0.75 years (9 months) in the

herd immunity condition (3.4.24), we find that to achieve herd immunity

the successfully vaccinated fraction g at age 9 months must satisfy g ≥ 0.98.

A measles vaccine efficacy of 0.95 implies that the fraction vaccinated would

have to be 1.03, which is impossible to achieve with a program that has at

most one vaccination per person. This result is confirmed by the measles

computer simulations for Niger, in which herd immunity is not achieved

when all children are vaccinated at age 9 months.



3.7. Application to Pertussis in the United States

Previous estimates ([4], p. 70) of 10 to 18 for R0 for pertussis (whoop-

ing cough) are based on the formula R0 = 1 + L/A, which is derived in

my tutorial lectures I&II for SEIR or SIR models of a disease that con-

fers permanent immunity in a uniform, homogeneously mixing population.

However, these estimates of R0 are not realistic, because pertussis gives

only temporary immunity and spreads by heterogeneous mixing. In the age

structured epidemiologic models developed specifically for pertussis [34, 35],

there are 32 age groups. Using fertilities and death rates from United States

census information for 1990, the value of q in (3.3.12) corresponds to 0.065%

growth per year, which is nearly zero. Thus the age distribution in the per-

tussis models is assumed to have become stable with a constant population

size. More details and graphs of the actual and theoretical age distributions

are given in [34].

Immunity to pertussis is temporary, because the agent Bordetella per-

tussis is bacterial, in contrast to the viral agents for measles, mumps and

rubella. As the time after the most recent pertussis infection increases,

the relative immunity of a person decreases. When people become infected

again, the severity of their symptoms and, consequently, their transmission

effectiveness (i.e. their infectivity) depends on their level of immunity at

the time of infection. Thus people with lower immunity have more symp-

toms and higher infectivity. Of course, infected people who were previously

fully susceptible are generally the most effective transmitters. In the age-

structured pertussis models [34, 35], the epidemiological classes include a

susceptible class S, an infective class I, a class R4 of those removed people

with very high immunity, classes R3, R2, and R1 for those with decreasing

immunity. In the two pertussis models, there are 3 or 4 levels of infectivity

and 32 age groups, so that not all infectives are equally effective in creat-

ing new infectives [35]. Infectives in those age groups that mix more with

other age groups are more effective transmitters than those in age groups

that mix less. Thus it might seem necessary in considering R0 to define a

“typical infective” by using some type of average over all infectivities and

age groups, so that R0 would be the average number of secondary cases

produced when a “typical infective” is introduced into a completely suscep-

tible population. In the next paragraph, we explain why averaging over age

groups is necessary, but averaging over classes with different infectivities is

not appropriate.

The occurrence of the first infection in a fully susceptible population


122 H.W. Hethcote

seems to be an unpredictable process, because it depends on random intro-

ductions of infectious outsiders into the host population. The probability

that a first infection occurs in the host population depends on the infectiv-

ity of the outside invader, on how the invader (with a mixing activity level

based on its age group) mixes in the host population, and the length of

time that the invader is in the population. It is clear that outside invaders

from high infectivity classes and high mixing activity age groups are more

likely to create a first new infection in a host population, especially if they

are in the population for their entire infectious period. We believe that the

definition of R0 should not depend on the circumstances under which an

outsider creates a first case, but on whether or not an infection with a first

case can persist in a fully susceptible population.

After the first infection in the host population, the infected people in

the next generations could be less effective transmitters, so that the in-

fection would die out. Thus the definition of R0 should be based on the

circumstances under which a disease with a first case would really invade a

fully-susceptible host population more extensively. In order for an infection

to survive the first 10 or 20 generations, so that it really does invade and

persist in the new host population, the number of secondary cases produced

by infectious members of the host population must exceed one. Thus R0

should be the number of secondary cases produced by averaging over all age

groups of the infectives that have not been previously infected. Because all

of the cases in the first generations of an invasion occur in fully susceptible

people, only infectives who were previously fully susceptible are relevant.

Thus R0 is calculated for the SIR4 part of the pertussis models and it is

not necessary to average over the classes with various infectivity levels. Al-

though the SIR model formula (3.5.8) for R0 works for the pertussis models,

the formula (3.5.7) for the constant k in the forces of infection λi = kbi at

the endemic equilibrium does not work, because the pertussis models have

temporary immunity and classes with different infectivities.

The fertilities fj , death rate constants dj , and transfer rate constants cj

are determined in the demographic model. The average infectious period is

21 days, so that the rate constant γ is 1/21. The form of separable mixing

used in the pertussis model is proportionate mixing, which has activity

levels lj in each of the 32 age groups. The activity levels lj are found from

the forces of infection λj and the infective fractions ij , as explained in

Appendix C of [34]. Then bj = bj = lj/D1/2, where D =

∑32j=1 ljPj is

the total number of people contacted per unit time. Using the SIR model



formula (3.5.8) for R0 in the pertussis computer simulation programs with

the baseline parameter sets, the values of the basic reproduction number R0

are 5.4 for the pertussis model in [34, 35] and 3.7 for the second pertussis

model in [35]. In the first model each pertussis booster moves the individual

back up one vaccinated or removed class, but for those in the second model

who have had a sequence of at least 4 pertussis vaccinations or have had

a previous pertussis infection, a pertussis booster raises their immunity

back up to the highest level. Thus the second model incorporates a more

optimistic view of the effectiveness of pertussis booster vaccinations. Note

that the R0 values here of 5.4 and 3.7 are much lower than the estimates

of R0 between 10 and 18 cited above.

Neither of the two methods used to find approximations of R0 for

measles in Niger work for the pertussis models. The replacement number R

at the pertussis endemic equilibrium depends on the fractions infected in

all of the 3 or 4 infective classes. For example, in the first pertussis model

R ∼=

∑32j=1 λj(sj + r1j + r2j)Pj

∑32j=1(ij + imj + iwj)Pj/[1/(γj + dj)]

,

where ij , imj , and iwj are the infective prevalences in the full-, mild-, and

weak-disease classes I, Im, and Iw. In the computer simulations for both

pertussis models, R is 1 at the endemic equilibrium. If the expression for R

is modified by changing the factor in parentheses in the numerator to one,

which corresponds to assuming that all contacts are with susceptibles, then

we obtain the contact number

σ ∼=

∑32j=1 λjPj

∑32j=1(ij + imj + iwj)Pj/[1/(γj + dj)]

,

which gives the average number of cases due to all infectives. At the endemic

equilibrium in the pertussis simulations, σ = 3.0 using the first model and

σ = 1.8 using the second model. Thus the basic reproduction number R0

is not equal to the contact number σ at the endemic equilibrium, because

the forces of infection λj in the approximation of σ are due to the contacts

of the infectives in the I, Im, and Iw classes instead of just the contacts

of those in the I class. Thus it is not possible to use the estimate of the

contact number σ during the computer simulations as an approximation

for R0 in the pertussis models. Since the age distribution of the population

in the United States is poorly approximated by a negative exponential and

the force of infection is not constant, the second method used for measles


124 H.W. Hethcote

in Niger also does not work to approximate R0 for pertussis in the United

States.

The ultimate goal of a pertussis vaccination program is to vaccinate

enough people to get the replacement number less than 1, so that pertussis

fades away and herd immunity is achieved. Because the mixing for per-

tussis is not homogeneous and the immunity is not permanent, we cannot

use the simple criterion for herd immunity that the fraction with vaccine-

induced or infection-induced immunity is greater than 1 − 1/R0. Indeed,

the low numerical R0 values of 5.4 and 3.7 for a disease like pertussis with

waning immunity do not indicate that herd immunity for pertussis is easy

to achieve. None of the vaccination strategies, including those that give

booster vaccinations every five years, have achieved herd immunity in the

pertussis computer simulations [34, 35].

3.8. Discussion

Age-structured epidemiology models with either continuous age or age

groups are essential for the incorporation of age-related mixing behavior,

fertility rates, and death rates, for the estimation of R0 from age specific

data, and for the comparison of vaccination strategies with age-specific risk

groups and age-dependent vaccination rates. Indeed, some of the early epi-

demiology models incorporated continuous age structure [8, 46]. Modern

mathematical analysis of age structured models appears to have started

with Hoppensteadt [40], who formulated epidemiology models with both

continuous chronological age and infection class age (time since infection),

showed that they were well posed, and found threshold conditions for en-

demicity. Expressions for R0 for models with both chronological and in-

fection age were obtained by Dietz and Schenzle [18]. In age-structured

epidemiology models, proportionate and preferred mixing parameters can

be estimated from age-specific force of infection data [33]. Mathematical

aspects such as existence and uniqueness of solutions, steady states, stabil-

ity, and thresholds have now been analyzed for many epidemiology models

with age structure; more references are cited in the following papers. These

SIS and SIR models with continuous age structure have included vertical

transmission [9, 10, 19], age dependent disease transmission [5, 16, 24, 56],

infection class age [55, 61], cross immunity [13], intercohort transmission

[11, 12, 14, 42, 43], short infectious period [6, 7], and optimal vaccination pat-

terns [21, 22, 27, 45, 51, 53].

Age structured models have been used in the epidemiology modeling of



many diseases [4]. Dietz [16, 17], Hethcote [30], Anderson and May [2, 3], and

Rouderfer, Becker, and Hethcote [52] used continuous age structured models

for the evaluation of measles and rubella vaccination strategies. Tudor [58]

found threshold conditions for a measles model with age groups. Hethcote

[31] considered optimal ages of vaccination for measles on three continents.

Halloran, Cochi, Lieu, Wharton, and Fehrs [26], Ferguson, Anderson, and

Garnett [20], and Schuette and Hethcote [54] used age structured models

to study the effects of varicella (chickenpox) vaccination programs. Grenfell

and Anderson [23] and Hethcote [34, 35, 37, 59] have used age structured

models in evaluating pertussis (whooping cough) vaccination programs.

Many epidemiology models now used to study infectious diseases involve

age structures, because fertilities, death rates, and contact rates all depend

on the ages of the individuals. Thus the basic reproduction number R0

must be found for these epidemiologic-demographic models. For MSEIR,

MSEIRS, SEIR, SEIRS, and SEIS models, expressions for R0 are given by

(3.4.9) and (3.5.6) when the demographic structures are continuous age and

age groups, respectively. Analogous expressions for R0 for the SIR, SIRS,

and SIS models are given by (3.4.11) and (3.5.8). These expressions for R0

are found by examining when there is a positive (endemic) equilibrium in

the feasible region, and then it is verified that the disease persists if and

only if R0 > 1.

To illustrate the application of the theoretical formulas for R0 in models

with age groups, two applications have been included in this paper. Based

on demographic and epidemiologic estimates for measles in Niger, Africa,

the value of the basic reproduction number found from (3.5.6) is R0 = 18.8.

The interesting aspect of this measles application is that R0 is found for a

very rapidly growing population. In contrast, the current fertility and death

data in the United States suggests that the population is approaching a

stable age distribution with constant total size.

Using previously developed models for pertussis (whooping cough) in

which the immunity is temporary [34, 35], the basic reproduction numbers

are estimated to be R0 = 5.4 and R0 = 3.7 for two pertussis models. It is

interesting that these basic reproduction numbers are found using the R0

expression derived for an SIR model, even though pertussis immunity is

temporary.

Recall that the contact number σ is the average number of adequate

contacts of a typical infective during the infectious period. The interesting

aspect of the pertussis calculations is that new types of infectives with lower


126 H.W. Hethcote

infectivity occur after the invasion, because infected people who previously

had pertussis have lower infectivity when reinfected. Thus typical infectives

after the invasion include those who have lower infectivities than the infec-

tives who had been fully susceptible. Although the contact number σ is

equal to R0 when pertussis first invades the population, the new broader

collection of typical infectives implies that σ < R0 after the invasion. Us-

ing numerical approximations during the computer simulations, the contact

numbers at the endemic equilibrium are estimated to be σ = 3 for the first

age group pertussis model and σ = 1.8 for the second pertussis model. This

phenomenon that σ < R0 at the endemic equilibrium also holds for three

relatively simple pertussis models based on ordinary differential equations

[38]. For the pertussis model with four removed groups in [38], the three

infective classes with decreasing infectivity are I, Im, and Iw, where the

infective classes Im and Iw are non-empty as soon as pertussis has invaded.

For this model the contact number σ satisfies

σ = R0[I + ρmIm + ρwIw]/[I + Im + Iw] < R0,

because the relative infectivities ρm and ρw are less than one. As pointed out

in Section 3.2 the basic reproduction number R0, the contact number σ, and

the replacement number R are all equal at the time when the disease invades

the population. For nearly all models R0 = σ > R after the invasion, but

for the pertussis models, R0 > σ > R after the invasion. Thus the pertussis

models have led to an entirely new way of thinking about the differences

between the contact number σ and the basic reproduction number R0.

References

1. H. Amann, Ordinary Differential Equations, Walter de Gruyter, Berlin,1990.

2. R. M. Anderson and R. M. May, Vaccination against rubella and measles:Quantitative investigations of different policies, J. Hyg. Camb., 90 (1983),pp. 259–325.

3. , Age related changes in the rate of disease transmission: Implicationfor the design of vaccination programs, J. Hyg. Camb., 94 (1985), pp. 365–436.

4. , Infectious Diseases of Humans: Dynamics and Control, Oxford Uni-versity Press, Oxford, 1991.

5. V. Andreasen, Disease regulation of age-structured host populations, The-oret. Population Biol., 36 (1989), pp. 214–239.

6. , The effect of age-dependent host mortality on the dynamics of anendemic disease, Math. Biosci., 114 (1993), pp. 29–58.



7. , Instability in an SIR-model with age-dependent susceptibility, inMathematical Population Dynamics, Vol 1, O. Arino, D. Axelrod, M. Kim-mel, and M. Langlais, eds., Wuerz Publishing, Ltd., Winnipeg, 1995, pp. 3–14.

8. D. Bernoulli, Essai d’une nouvelle analyse de la mortalite causee par lapetite verole et des avantages de l’inoculation pour la prevenir, in Memoiresde Mathematiques et de Physique, Academie Royale des Sciences, Paris,1760, pp. 1–45.

9. S. Busenberg and K. L. Cooke, Vertically Transmitted Diseases, vol. 23 ofBiomathematics, Springer-Verlag, Berlin, 1993.

10. S. N. Busenberg, K. L. Cooke, and M. Iannelli, Endemic thresholds andstability in a class of age-structured epidemics, SIAM J. Appl. Math., 48(1988), pp. 1379–1395.

11. S. N. Busenberg and K. P. Hadeler, Demography and epidemics, Math.Biosci., 101 (1990), pp. 41–62.

12. S. N. Busenberg, M. Iannelli, and H. R. Thieme, Global behavior of an age-structured epidemic model, SIAM J. Math. Anal., 22 (1991), pp. 1065–1080.

13. C. Castillo-Chavez, H. W. Hethcote, V. Andreasen, S. A. Levin, and W. M.Liu, Epidemiological models with age structure, proportionate mixing, andcross-immunity, J. Math. Biol., 27 (1989), pp. 233–258.

14. Y. Cha, M. Iannelli, and F. A. Milner, Existence and uniqueness of endemicstates for age-structured S-I-R epidemic model, Math. Biosci., 150 (1998),pp. 177–190.

15. M. C. M. De Jong, O. Diekmann, and J. A. P. Heesterbeek, How doestransmission depend on population size?, in Human Infectious Diseases,Epidemic Models, D. Mollison, ed., Cambridge University Press, Cambridge,1995, pp. 84–94.

16. K. Dietz, Transmission and control of arbovirus diseases, in Epidemiology,K. L. Cooke, ed., SIAM, Philadelphia, 1975, pp. 104–121.

17. , The evaluation of rubella vaccination strategies, in The MathematicalTheory of the Dynamics of Populations, Vol. 2, R. W. Hiorns and D. Cooke,eds., Academic Press, London, 1981, pp. 81–97.

18. K. Dietz and D. Schenzle, Proportionate mixing models for age-dependentinfection transmission, J. Math. Biol., 22 (1985), pp. 117–120.

19. M. El-Doma, Analysis of nonlinear integro-differential equations arising inage-dependent epidemic models, Nonlinear Anal., 11 (1987), pp. 913–937.

20. N. M. Ferguson, R. M. Anderson, and G. P. Garnett, Mass vaccination tocontrol chickenpox: The influence of zoster, Rev. Med. Virology, 6 (1996),pp. 151–161.

21. D. Greenhalgh, Analytical threshold and stability results on age-structuredepidemic models with vaccination, Theoret. Population Biol., 33 (1988),pp. 266–290.

22. , Vaccination campaigns for common childhood diseases, Math.Biosci., 100 (1990), pp. 201–240.

23. B. T. Grenfell and R. M. Anderson, Pertussis in England and Wales: An


128 H.W. Hethcote

investigation of transmission dynamics and control by mass vaccination,Proc. Roy. Soc. London Ser. B, 236 (1989), pp. 213–252.

24. G. Gripenberg, On a nonlinear integral equation modelling an epidemic inan age-structured population, J. Reine Angew. Math., 341 (1983), pp. 147–158.

25. J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York,1969.

26. M. E. Halloran, S. L. Cochi, T. A. Lieu, M. Wharton, and L. Fehrs, Theo-retical epidemiologic and morbidity effects of routine varicella immunizationof preschool children in the United States, Am. J. Epidemiol., 140 (1994),pp. 81–104.

27. M. E. Halloran, L. Watelet, and C. J. Struchiner, Epidemiological effects ofvaccines with complex direct effects in an age-structured population, Math.Biosci., 121 (1994), pp. 193–225.

28. H. W. Hethcote, Qualitative analyses of communicable disease models,Math. Biosci., 28 (1976), pp. 335–356.

29. , An immunization model for a heterogeneous population, Theoret.Population Biol., 14 (1978), pp. 338–349.

30. , Measles and rubella in the United States, Am. J. Epidemiol., 117(1983), pp. 2–13.

31. , Optimal ages of vaccination for measles, Math. Biosci., 89 (1988),pp. 29–52.

32. , A thousand and one epidemic models, in Frontiers in TheoreticalBiology, S. A. Levin, ed., vol. 100 of Lecture Notes in Biomathematics,Springer-Verlag, Berlin, 1994, pp. 504–515.

33. , Modeling heterogeneous mixing in infectious disease dynamics, inModels for Infectious Human Diseases, V. Isham and G. F. H. Medley, eds.,Cambridge University Press, Cambridge, 1996, pp. 215–238.

34. , An age-structured model for pertussis transmission, Math. Biosci.,145 (1997), pp. 89–136.

35. , Simulations of pertussis epidemiology in the United States: Effectsof adult booster vaccinations, Math. Biosci., 158 (1999), pp. 47–73.

36. H. W. Hethcote, The Mathematics of Infectious Diseases, Siam Review 42(2000) 599-653.

37. H. W. Hethcote, P. Horby, and P. McIntyre, Using computer simulationsto compare pertussis vaccination strategies in Australia, Vaccine 22 (2004)2181-2191.

38. H. W. Hethcote, Y. Li, and Z. J. Jing, Hopf bifurcation in models for per-tussis epidemiology, Math. Comput. Modelling, 30 (1999), pp. 29–45.

39. H. W. Hethcote and J. W. Van Ark, Epidemiological models with hetero-geneous populations: Proportionate mixing, parameter estimation and im-munization programs, Math. Biosci., 84 (1987), pp. 85–118.

40. F. Hoppensteadt, Mathematical Theories of Populations: Demographics,Genetics and Epidemics, SIAM, Philadelphia, 1975.

41. M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics,



vol. 7 of C.N.R. Applied Mathematics Monograph, Giardini Editori, Pisa,1994.

42. M. Iannelli, F. A. Milner, and A. Pugliese, Analytic and numerical results forthe age-structured S-I-S model with mixed inter-intracohort transmission,SIAM J. Math. Anal., 23 (1992), pp. 662–688.

43. H. Inaba, Threshold and stability results for an age-structured epidemicmodel, J. Math. Biol., 28 (1990), pp. 411–434.

44. A. V. Kaninda, S. Richardson, and H. W. Hethcote, Influence of hetero-geneous mixing on measles transmission in an African context. Preprint,2000.

45. W. Katzmann and K. Dietz, Evaluation of age-specific vaccination strate-gies, Theoret. Population Biol., 25 (1984), pp. 125–137.

46. W. O. Kermack and A. G. McKendrick, Contributions to the mathematicaltheory of epidemics, part 1, Proc. Roy. Soc. London Ser. A, 115 (1927),pp. 700–721.

47. A. J. Lotka, The stability of the normal age distribution, Proc. Nat. Acad.Sci. U.S.A., 8 (1922), pp. 339–345.

48. A. G. McKendrick, Applications of mathematics to medical problems, Proc.Edinburgh Math. Soc., 44 (1926), pp. 98–130.

49. J. Mena-Lorca and H. W. Hethcote, Dynamic models of infectious diseasesas regulators of population sizes, J. Math. Biol., 30 (1992), pp. 693–716.

50. K. Mischaikow, H. Smith and H. R. Thieme, Asymptotically autonomoussemiflows: chain recurrence and Lyapunov functions, Trans. A. M. S., 53(1993), pp. 1447-1479.

51. J. Muller, Optimal vaccination patterns in age-structured populations,SIAM J. Appl. Math., 59 (1998), pp. 222–241.

52. V. Rouderfer, N. Becker, and H. W. Hethcote, Waning immunity and itseffects on vaccination schedules, Math. Biosci., 124 (1994), pp. 59–82.

53. V. Rouderfer and N. Becker, Assessment of two-dose vaccination schedules:Availability for vaccination and catch-up, Math. Biosci., 129 (1995), pp. 41–66.

54. M. C. Schuette and H. W. Hethcote, Modeling the effects of varicella vac-cination programs on the incidence of chickenpox and shingles, Bull. Math.Biol., 61 (1999), pp. 1031–1064.

55. H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integralequations and asymptotic speeds for the spread of populations, J. ReineAngew. Math., 306 (1979), pp. 94–121.

56. , Stability change of the endemic equilibrium in age-structured mod-els for the spread of S-I-R type infectious diseases, in Differential Equa-tions Models in Biology, Epidemiology, and Ecology, S. Busenberg and M.Martelli, eds., vol. 92 of Lecture Notes in Biomathematics, Springer-Verlag,Berlin, 1991, pp. 139-158.

57. , Persistence under relaxed point-dissipativity (with an application toan endemic model), SIAM J. Math. Anal., 24 (1993), pp. 407–435.

58. D. W. Tudor, An age-dependent epidemic model with application to


130 H.W. Hethcote

measles, Math. Biosci., 73 (1985), pp. 131–147.59. A. Van Rie and H. W. Hethcote, Adolescent and adult pertussis vaccination:

Computer simulations of five new strategies, Vaccine 22 (2004) 3154-3165.60. H. von Foerster, The Kinetics of Cellular Proliferation, Grune and Stratton,

New York, 1959.61. G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics,

Marcel Dekker, New York, 1985.

August 10, 2005 Master Review Vol. 9in x 6in { (for ... · August 10, 2005 Master Review Vol. 9in x...

Documents

Transcript of August 10, 2005 Master Review Vol. 9in x 6in { (for ... · August 10, 2005 Master Review Vol. 9in x...