Of Viruses, Mosquitoes & Zombies

How Mathematical Models can Help Control Epidemics

Adnan KhanDepartment of MathematicsLahore University of Management Sciences

Of Viruses……..

• Literature Review– Hethcote, H. The Mathematics of Infectious

Diseases, SIAM Review 2000

– Imran A, Malik T, Rqfique H & Khan, A. A model of bi-mode transmission dynamics of hepatitis C with optimal control, Theory in Biosciences 2013

…….Mosquitoes

• Literature Review– Wearing, H. & Rohani P. Ecological and Immunological

Determinants of Dengue Epidemics, PNAS 2006

– Khan, A. Hassan M. & Imran M. Estimating the Basic Reproduction Number for Single-Strain Dengue Fever Epidemics, Journal of Biological Dynamics 2014

and…….Zombies• Literature Review

– P. Munz, I. Hudea, J. Imad and R.J. Smith? When zombies attack!: Mathematical modelling of an outbreak of zombie infection (Infectious Disease Modelling Research Progress 2009, in: J.M. Tchuenche and C. Chiyaka, eds, pp133-150).

– White Zombie - 1932– Revolt of the Zombies- 1936– Zombie of Mora Tau – 1957– El Muerto - 1961– Night of the Living Dead - 1968– Curse of the Living Dead - 1973– Day of the Dead – 1985– 28 Days Later – 2007

Zombie Attack!!

SimZombie.jar

https://code.google.com/p/simzombie/

Zombie Attack!!• There has been an outbreak of Zombies

• How to model this?

• How to suggest control measure?

• What about their efficacy?

Influenza Data Timeline from Google

googleflu.mov

T. Malik, A. Gumel, L. Thompson, T. Strome, S. Mahmud; "Google Flu Trends" and Emergency Department Triage Data Predicted the 2009 Pandemic H1N1 Waves in Winnipeg, Manitoba. Canadian Journal of Public Health. 102(4):294-97.

googleflu.mov

Influenza

• An Army base has a total staff of 8342. An individual who has just returned from leave becomes ill and is diagnosed with Jade fever -- an exotic, dangerous and highly contagious variety of flu.

• Will there be a flu epidemic on the base?

• If so, does the base hospital have enough beds?

• How many of the base staff will get the flu?

Dengue Fever Epidemic

• A suitable model

• Must include vector dynamics

• Control Strategies

• Retrospective analysis

Mathematical Epidemiology

• What are epidemics?• Mathematical Models

• Statistical – Finding ‘trends’ in available data

• Mechanistic– Incorporating the mechanics of transmission

• Retrospective Analysis• Data analysis

Dynamic / Mechanistic Models

• Deterministic Models– Differential Equations– Difference Equations– Delay Differential Equations

• Stochastic Models– Markov Chains – Stochastic Differential Equations

Why Dynamic Models

• Models allow you to predict (estimate) when you don’t KNOW– What are the costs and benefits of different control

strategies?– When should there be quarantines?– Who should receive vaccinations?– When should wildlife or domestic animals be killed?– Which human populations are most vulnerable?– How many people are likely to be infected? To get

sick? To die?

A Basic SIR Model• Susceptible Population (S)

– people who are able to catch the disease• Infectious Population (I)

– people who have the disease and can transmit it• Removed Population (R)

– people who have recovered and are no longer susceptible

– People who are naturally immune or isolated etc.

SIR Model Schematic

The Model Assumptions

• Susceptible individuals become infected upon coming in contact with infectious individuals.

• Each infected individual has a fixed number, r, of contacts per day that are sufficient to spread the disease– The parameter r contains information about the number

of contacts and probability of infection

• Infected individuals recover from the disease at rate a– 1/a is the average recovery time

The Model Assumptions• The incubation period of the disease is

short enough to be neglected

• All population classes are well mixed

• Births, deaths, immigration, and migration can be ignored on the timescale of interest

The Model Equations

The rate of transmission of the disease is proportional to the rate of encounter of susceptible and infected individual.

dI

dtrSI aI

dS

dt rSI

dR

dtaI

S(0) S0

I(0) I0

R(0) 0

Gaining Insight into the Model• Epidemiological Questions

– Will an epidemic occur?

– If an epidemic does occur• how severe will it be? • will the disease eventually die out or persist in the

population?• how many people will get the disease during the

course of the infection?

Gaining Insight into the Model

• Definition: We will say that an epidemic occurs if the number of infectious individuals is greater than the initial number I0 for some time t

• Epidemiological Question: Given r, S0 , I0, and a, when will an epidemic occur?

• Mathematical Question: Is I(t) > I0 for any time, t?

When Will an Epidemic Occur?• Consider the I-equation first

• Therefore, the I population will increase initially if and the I population

will decrease initially otherwise. Therefore, this condition sufficient for an epidemic to occur.

dI

dtrSI aI At t = 0 arSI

dt

dI 00

S0 a r

When Will an Epidemic Occur?

• Now consider the S-equation

• Therefore, if there will not be an epidemic.

• However if, then an epidemic will occur.

dS

dt rSI

dS

dt 0,t

S S0,t

So if

S0 a r We know

dI

dtI rS a 0,t

S0 a r

S0 a r

Gaining Insight into the Model

• Epidemiological Question: If an epidemic occurs, will the disease eventually die out or will it persist in the population.

• Mathematical Question: What are the steady states and their stability.– More specifically, is I = 0 a steady state and if so is

it stable?

Steady States

• Note: I = 0 makes all three equations zero.

• Therefore I = 0 represents an entire line (or plane) of steady states

• Traditional stability analysis leads to a zero eigenvalue.

dI

dtrSI aI

dS

dt rSI

dR

dtaI

Phase Plane -- Analytically

• Recall that R is decoupled

dI

dtrSI aI

dS

dt rSI

rS

a

dS

dI 1

I S a

rlnS C

000 lnln Sr

aSIS

r

aSI

Gaining Insight into the Model Equations

• Epidemiological Question: What will be the final state of the population be after the infection has run its course?

• Mathematical Question: What are the steady states for S and R?

Steady States for S and R

• S* = is the number of people who did not catch the disease.

S* S0eN S*

R* N S*

Gaining Insight into the Model

• How many people will catch the disease before it dies out?

*00 SSII total

Gaining Insight into the Model

• Epidemiological Questions: When an epidemic occurs, how severe will it be?

• Mathematical Question: What is the maximum number of infectious individuals?

Severity of Epidemic

• I achieves its max when S = a/r

r

awhere

S

SSNI

ln0

Imax N lnS0

Severity of the Epidemic

Basic Reproduction Number

• As we’ve seen is an important parameter.

• It is called the infectious contact number or basic reproduction number of the of the infection.

R0 S0r

a

Basic Reproduction Number: Breakdown

• A given infective will, on average, be infectious for 1/a units of time.

• The number of susceptibles infected by one infectious individual per unit time is rS.

• Therefore, the number of infections produced by one infective is rS/a.

• If R0>1, then an epidemic will occur

Importance of R0

• R0 is not actually a characteristic of the disease) but of the virus in a specific population at a specific time and place. By altering some or all of the components you can also alter R0.

Fit of the Model to Data for Influenza

In 1978, a flu epidemic occurred in a boys boarding school in the north of England. There were 763 resident boys, including one initial infective. The school kept records of the number of residents confined to bed, and we will assume those to be the infectives. The data for the two-week can be fit to the SIR model.

Fit of the Model to Data for Influenza

Time Course of the Influenza Epidemic

Results of the SIR Model

• The occurrence of an epidemic depends solely on the number susceptibles, the transmission rate, and recovery rate.

– The initial number of infectives plays no role in whether or not there is an epidemic.

– That is, no matter how many infectives there are, an epidemic will not occur unless S0 > a/r.

Modifications of the SIR Model

• Other considerations, such as vital dynamics (births and deaths), length of immunity, the incubation period of the disease, and disease induced mortality can all have large influences on the course of an outbreak.

Another Example

• An Army base has a total staff of 8342. An individual who has just returned from leave becomes ill and is diagnosed with Jade fever -- an exotic, dangerous and highly contagious variety of flu.

• All individuals getting this flu must be hospitalized. The base hospital has 240 beds. The transmission parameter for this flu at this base is r = 5x10-5 per day and the recovery rate is a = .32 per day.

Questions For You

• Is the condition for an epidemic satisfied?

• Does the base hospital have enough beds?

• How many of the base staff will get the flu?

Answers

• Is the condition for an epidemic satisfied?– S0 =8341, I0 = 1, R0 = 0

–

– Therefore an epidemic will occur.

a

r

0.32

5e 56,400

R0 S0r

a1.3 1

Answers

• Does the hospital have enough beds?

• Therefore if the hospital has 245 beds, it will need more.

Imax N lnS0

268

Answers

• How many of the base staff will get the flu?

• S* = 4783, so about 3649 people (43% of the population) will catch the disease.

Itotal I0 S0 S*

Time Course of the Jade Fever Epidemic

S.I.R.S Model

• The model equations are

• This allows for endemic steady states

Zombie Epidemic Model• The zombie epidemic spreads by zombie to person

contact and resurrection as a zombie• We consider three classes

– Susceptibles (S)– Zombies (Z)– Removed (R)

• Zombies are created by– Zombies attacking a human successfully– The Removed Humans being resurrected as Zombies

Mathematical formulation

• The dynamics are given by an SIR type model

Analysis of the Model

• When time period is long the vital dynamics cannot be ignored

• In this case note

• So that as

• As this is DOOMSDAY for humanity

Short Time Dynamics

• When the time period is short as compared to human lifetimes

• There are two equilibria

• Unfortunately the zombie takeover is asymptotically stable

But what about Latency?

• All zombie flicks teach us that there is latency• Lets include an ‘infected (I)’ latent class

Any Good News?

• Long term dynamics still spell doomsday• Short term dynamics have two equilibria

• Unfortunately once again the Zombie free equilibrium is unstable

Will Quarantine Help?

• Lets quarantine the infected population• Mathematically we have

Analysis of Quarantine

• In a short outbreak we have two equilibria

• Stability depends on

• The DFE is stable for

• Else the co existence equilibrium is stable

Perhaps!!!

• We obtain the basic reproductive number

• Realistically

• In this case need to be very large for

• Need a large percentage of infected to be quarantined for co-existence

Does Salvation Lie in Treatment?

• Suppose a cure becomes available

• The mathematical model is

Analysis

• Besides the DFE we have the coexistence equilibrium

• Stability analysis show that this ‘endemic’ equilibrium is stable

• With our parameters this results in some humans and lots of zombies coexisting

A Model for the Spread of Dengue Fever

• We have developed a model involving two strains of the DF

• This is relevant in Pakistan where an overwhelming majority of cases are type II and type III

• Data available form previous outbreaks will be used to predict R0

Model Assumptions• Constant recruitment rate into susceptibles

• Constant natural death rate

• Two strains prevalent in the population

• Rate of infection incorporates Antibody Dependent Enhancement

• Recovery from a strain of dengue results in permanent immunity

The Mathematical Model

The Model Variables

The Model Parameters

The Basic Reproductive Number

Data (Lahore 2011)

Estimating

QUESTIONS