SIR Analysis Of A Deterministic And Stochastic · mathematical modeling has developed as an...

Researchjournali’s Journal of Mathematics Vol. 1 | No. 5 October | 2014 ISSN 2349-5375

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Nana-Kyere Sacrifice

Department of Mathematics, Ola Girl's Secondary

School, Kenyasi-Ghana

Hoggar Glory

Department of Mathematics, Sunyani Polytechnic,

Sunyani-Ghana

Justice Kwame Appati

Department of Mathematics, Kwame Nkrumah

University of Science and Technoogy, PMB,

Kumasi-Ghana

SIR Analysis Of

A Deterministic

And Stochastic

Differential Equation

Of An HIV Model


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ABSTRACT

In this Paper, deterministic and stochastic differential equation (SIR) models for HIV propagation are

formulated using Ghana data. The stability of the disease free and endemic equilibrium points of the models

is investigated as well as an implementation of numerical simulation of the models to observe the effect of a

decrease in the infection rate. It was found that for both the deterministic and the stochastic models, an

infection rate of or less would cause the number of infective to be permanently less than the number

of susceptible.

Keywords: Basic Reproduction Number; Ito-Formula; Euler- Maruyama Method; Disease-free equilibrium

(DFE); Endemic Equilibrium (EDE); Human Immunodeficiency Virus (HIV)

1. INTRODUCTION

The intuition that transmission of infectious diseases follows certain laws that can be modeled mathematically

has existed long. In 1766, Daniel Bernoulli published an article where he mathematically analyzed the effects

of smallpox variolation on life expectancy (Dietz and Heesterbeek, 2000). Sir Ronald Ross, who received the

Nobel Prize award for his contribution on elucidating the life cycle of the Malaria parasite, used mathematical

modeling to investigate the effectiveness of various intervention strategies for Malaria (Ross, 1911). Kermack

and Meckendrick (1991a, 1991b, 1991c), described the kinetics of disease transmission in terms of a system

of differential equation, and opened up the concept of threshold quantities. However the nonlinear dynamics

of infectious disease transmission came into the scene somewhere early twentieth century. Since then

mathematical modeling has developed as an interesting area in the area of applied mathematics, and has been

of great avail to public health in policy making. The study of epidemics has come up with an astonishing

number of variety of models and explanations for the spread and cause of the epidemic outbreak. In McNeil

(1989), he explains the relation between disease and the great unwashed.

Another astonishing work in regards to epidemiological modeling is that of Oldstone (1998). Oldstone

described the several views of diseases from the triumphs of medicine to socioeconomic. Clay sculpture has

been helpful in making estimates for the level of inoculation for the control of transmitting infectious

diseases. Anderson and May (1982, 1985, 1991) discussed and calculated by the models the effects of

different vaccination programs. A compartmental model is one for which the mortals in a population are

separated into compartments depending on the disease status with regards to the infection under study.

Therefore the soul may be classified as susceptible, S, infected, I and removed, R, (SIR) based on their status

of the disease under consideration. For instance, an SIR model describes a disease history of susceptible

individual becoming infectious through interactions with an infected individual, and infectious individual

moving into the removal class by either immunity or death. A compartmental model for infection


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transmission with an unwrapped (or latent) compartment (explicitly containing those infected, but not yet

infectious) and lasting immunity would be called an SEIR model, and situations where susceptibility can

return after infection (or after immunity) would be called an SIS (or an SIRS) model (Busenberg and Cooke,

1993). In this report, we try to develop a deterministic and stochastic differential equation model for HIV

propagation in Ghana. The Stabilities of the disease-free equilibrium and the endemic equilibrium of the

model were investigated, and the Ito-formula was utilized in defining the stability of the disease-free of the

stochastic model.

The model is acquired in section 2. The basic reproduction number is specified and shown to be a threshold

parameter, and the deterministic and Stochastic SIR model of HIV is introduced and dissected in this

department. The Section 3 deals with the simulation and discussion of the mannequin. The section 4 provides

the summary, the concluding remarks and good words.

2. THE MODEL

2.1 THE BASIC REPRODUCTION NUMBER

A quantity of central importance in epidemiology is the basic reproduction number denoted by . From time

to time, it is also called the basic reproduction ratio. is defined as the mean number of secondary infections

produced when a single infected individual is introduced into a host population where everyone is susceptible.

Thus, is a threshold parameter that determines when an infection invades a host population and when it

does not.

If then a single infected individual introduced into a wholly susceptible host population will run to

an epidemic. Conversely, if , an infection cannot invade the host population, but will die out. For

models with a single infected compartment, is a product of the infection rate, and the mean duration of

the infection,

. Thus ,

.

But for more complex models with several infected compartments, a precise definition is made by (Driessche,

Watmough, 2002) as the base number of secondary infections produced by a typical person in a population at

a disease-free equilibrium (DFE). Thus, Is demonstrated by investigating the stability of the model at the

disease-free country. If , then, the is locally asymptotically stable and the disease cannot invade

the population. On the other hand, if then then DFE is unstable and there can be an epidemic. But

when there disease becomes endemic and the disease will persist in the population at a constant pace.

This indicates that a prerequisite for the existence of an endemic sense of balance is a stream of new

susceptible either through recovery without immunity against reinfection or through births (Fred Brauer,

2012).


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2.2 DETERMINISTIC SIR EPIDEMIC MODEL

Take a simple SIR model for HIV in a mixing homogeneous population, which can be grouped into three

distinct compartments of Susceptible (S), infective (I), and Removed (R).

The model considers demographic turnovers (birth and destruction) and wear that the birth rate equal death

rate so that the total size of the population will still remain constant, and all infections are assumed to end

with recovery. The entire size of the population is constant and is denoted by , where

, because they represent the numbers of people.Further we assume that;

Encounters between infected and susceptible individuals occur at a rate relative to their respective

numbers in the population. Hence the pace of new infection is defined as , where >0 is a parameter

for infectivity

The rate of removal of infective to the removed class is proportional to the number of infective, thus

where is a constant

The brooding period is short enough to be trifling; that is a susceptible who contracts the disease is

infective right away.

Hence

β β γ

µ µ µ

Figure (2.1). A Schematic of system (2.1), where is the susceptible, is the infection and R is the removed. β

and γ have the same signification as in the epidemic SIR model.

The differential equations for the system are;

(2.1)

Where

Introducing fractional variables

S I R


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,

,

into the system and since the arrangement can be boiled down to two

dimensional since has no effect on and .

(2.2)

(2.3)

Dividing equation (2.2) by (2.3) gives the reproduction number of the model

If and , then .

If and then .

In the respect is an epidemic if

.

If then

Figure (2.2): Phase portrait of Solution of the SIR epidemic model, with initial conditions (

2.3 EQUILIBRIUM POINTS

Evidently, the system ((2.2) and (2.3)) has two equilibrium points. Hence the disease-free, where =0 and

endemic, where ≠0.


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Equating the right-hand side of the equation (2.2) and (2.3) to zero and solving the systems simultaneously for

and turns over the equilibrium points

and

.

Understandably, the equilibrium point is the disease-free equilibrium, since and the

point

is endemic equilibrium since

2.4 DISEASE-FREE EQUILIBRIUM

The stability of the system at disease-free equilibrium is found by evaluating the jacobian of the systems (2.2)

and (2.3) at the equilibrium point

Let

Hence the Jacobian of and is given by

Therefore

So by using the characteristic equation, formula for the (2×2) matrix

Since the Jacobian matrix is diagonal, it is clean that the eigenvalues are

and


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In parliamentary procedure for the equilibrium point to be asymptotically stable, both eigenvalues must be

minus. It is clear that is negative and that if

then both eigenvalues are negative and the equilibrium point is asymptotically stable.

This means a small population of infectives introduced into the system would not cause a persistent infection

and that the population would return to disease-free state after some time.

On the other hand, if

then the equilibrium point is unstable and an introduction of infectives will result in a persistent infection.

Hence, in that location will be endemicity.

2.5 ENDEMIC EQUILIBRIUM

Evaluating the Jacobian matrix at the endemic equilibrium

produces the characteristic equation

.

Since the trace is less than zero and the determinant,

it satisfies that the endemic equilibrium is asymptotically stable (thus makes it stable). If it

becomes unstable.

2.6 STOCHASTIC SIR EPIDEMIC MODEL

Immediately that the deterministic model is understood, a stochastic variant of the SIR model is obtained by

random perturbation of the deterministic model with white noise. We Replace the contact rate in the system

(2.1) by

,

where

is a white noise (i.e. is a Brownian motion [2], [3], [7]).


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Hence

(2.4)

Substituting the fractional variables into (2.4) and re-arranging gives the stochastic version of the

system (2.1) as

(2.5)

3. NUMERICAL IMPLEMENTATION

3.1 THE ITO-FORMULA

The Ito-formula is used to solve stochastic differential equations which are difficult to be integrated by the

normal integration. Let the Stochastic process be a solution of the stochastic differential equation

,

for some suitable functions .

Let also be a twice continuously differentiable function. Then

, is a stochastic process for which

(3.1)

Where

is computed according to the rules

Equation (3.1) is called Ito’s formula [2], [7].

3.2 SOLVING THE STOCHASTIC SIR EPIDEMIC MODEL

Immediately that the deterministic model is understood, consider the stochastic SIR model of the system

(2.5). Again the term is ignored, since it has no consequence on the dynamics of and . Hence the

equivalent system is


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(3.2)

3.3 THE DISEASE-FREE EQUILIBRIUM

In the absence of infection, Hence the system (3.2) reduces to

(3.3)

3.4 SOLVING THE STOCHASTIC DISEASE-FREE EQUILIBRIUM BY ITO-FORMULA

In preliminary law to solve the system (3.3), we set so that system (3.3) becomes

(3.4)

Now applying the Ito Formula (3.1) to (3.4), we denote

,

and compute its derivative at point using the Ito Formula.

(3.5)

= (3.6)

Integrating equation (3.6) and substituting the initial conditions into (2.4) gives the equilibrium

point of the disease-free equilibrium

.

3.5 NUMERICAL SOLUTION OF SDES-EULER-MARUYAMA METHOD

A scalar, autonomous SDE can be written in integral form as

(3.7)

Where, and are scalar functions and the initial condition is a random variable.

If is the solution to then the solution x(t) is a random variable that goes up when we assume the

zero step size limit in the numerical method.

Hence the differential equation form of (3.7) can be written every bit

(3.8)


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Hence, from equation (3.8), if and Is a constant, then the problem becomes deterministic and (3.8)

reduces to an ordinary differential equation

(3.9)

with

To use the Euler-Maruyama method to (3.8) over we first discretize the interval. Thus we let

for some positive integer and

Hence our numerical approximation to will be denoted by Therefore the Euler-Maruyama (EM)

method contains the strain

(3.10)

The equation (3.10) comes from the integral form

(3.11)

Again if and constant, then (3.8) reduces to Euler’s Method. (See [16], for more on Euler –

Maruyama method).

4. RESULT AND DISCUSSION

4.1 NUMERICAL SIMULATION

It is important to emphasize that the parameter values death rate ( ) and the removal rate (γ) were unchanged

throughout our simulation, with the exclusion of the infection rate ( ) which was changed in order to explore

the behavior of the mannequins.

The parameters Definition Parameter Values Parameter Source

µ Mortality Rate 0.0875 CIA world factbook

Ghana Sentinel Survey

Ghana Sentinel Survey Infection Rate 1.5

γ Removal Rate 0.029

Table 4.1: Table of parameter values

4.2 SIMULATIONS OF THE DETERMINISTIC AND THE STOCHASTIC MODELS


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Figure 4.1: A Computer Simulation of the Deterministic and Stochastic models with parameter values

, ,

Figure 4.2: A computer simulation of the Deterministic and the Stochastic Models with a lower infection rate

of , ,


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Figure 4.3: A computer simulation of the Deterministic the Stochastic Models with a lower infection rate of

, ,

4.3 DISCUSSION

The stochastic model simulations are generally in accord with the deterministic model ones in terms of the

population crossovers. The kinetics of the HIV disease are represented by deterministic and the stochastic

model figures 4.1, 4.2, 4.3. The graphs showed that at the outbreak of the disease, the infected population

starts small, and then equally we have random and constant interaction between the infection and the

susceptible, more people get infected and hence the infects curve rises rapidly. The susceptible curve

decreases as a consequence of more people getting infected and running from the susceptible class to the

infective class. The graphs showed that in the long term the infected individuals do not go away from the

population, and this corresponds to the endemic equilibrium. To investigate the effect of varying the infection

rate on our model, was given different sets of values. The plots showed that an increase in the infection

rate increases the number of infections in the population, and a diminution in the infection rate decreases the

number of infections.

With the infection rate of =0. 2, both the deterministic and the stochastic model simulations indicated that

over 80 percent of the population would be susceptible, and below 10 percent of the population would be

infected, with close to 10 percent of the population murdered, and the curves would not traverse over.


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From our epidemiological perspective Is an important index of the initial cases of HIV, and the infection

rate has a big effect on the spread of HIV in Ghana. The computer-simulated results can be adapted to project

future occurrence of the disease.

Our deterministic plot of figure 4.1 showed that nearly 72 percent of the population would be infected, 8

percent would be susceptible and about 20 percent would be removed, within the time frame of over 25 years.

The endemic equilibrium point confirmed this effect with the same lot of parameter values. The endemic

equilibrium estimates that close to 8 percent would be susceptible and 70 percent would be ineffective in the

same time period.

When the infection rate was changed to , the computer-simulated results showed that 62 percent of

the population would be infected, and almost 22 percent would be susceptible, with 16 percent removed

within the time frame of over 25 years.

With sufficiently lower infection rate of =0. 2, the simulation indicated that nearly 92 percent of the

population would be susceptible and about 8 percent of the population would be infected and the population

would not traverse over.

The stochastic simulations have apparent stochastic behavior and that the overall movement of the trajectories

follows the same route as that of the deterministic ones. This implies that even though fluctuations occur, they

result in the same conduct as the deterministic model. The deterministic approach has limitations that the

stochastic approach is managed in a more naturalistic manner.

The deterministic approach gives the same solution every time the simulation is run with the same initial

values. This might be mathematically correct, but this is not the case in a real epidemic situation. This is

imputable to the fact that on that point may exist many parameters which we cannot model entirely

realistically; by modeling them deterministically we lose some of the complexity of the organization. It is thus

appropriate to put on a stochastic behavior.

Our stochastic simulation of figure 4.3 shows that 90 percent of the population would be infected, and 8

percent would be susceptible and about 2 percent would be withdrawn from the same time frame as the

deterministic one.

The stochastic plot again shows that approximately 60 percent of the population would be infected, 18 percent

would be susceptible and about 22 percent would be withdrawn when the infection rate is =0. 5 in the same

time frame.


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With sufficiently lower infection rate of =0. 2, the stochastic simulation showed that nearly 82 percent of the

population would be susceptible, about 3 percent of the population would be infected, 15 percent of the

population would be withdrawn, and the population would not traverse over.

The stochastic simulations seem to devote a higher percentage of infective population with the same lot of

parameter values as compared to the deterministic simulations. This is imputable to the fact that the

deterministic model is insensitive to stochastic variation which occurs in actual population of course.

5. CONCLUSION

In this report, we studied epidemiological models of HIV for both Deterministic and Stochastic approaches.

Since equilibrium points are significant instruments in performing stability analysis of infectious disease

models, we found the two balance points. Hence the disease-free equilibrium and the endemic equilibrium

point of the deterministic model and the disease-free equilibrium point of the stochastic model. The constancy

of these equilibrium points was then settled.

In parliamentary law to create our model reflect reality, our parameter values were obtained from Ghana and

were gone into the manikin, and their stability at the equilibrium points was then settled.

The outcomes showed an unstable disease-free equilibrium and a stable endemic equilibrium. This is true in

this case, since for the disease-free to be stable and for the endemic equilibrium to be stable.

Our results showed for both the disease-free and endemic to be greater than 1.

Hence, we consider the importance of the mathematical model as they can be used to search and identify the

types of information that needs to be gathered and the parameter values that need to be accessed.

6. REFERENCES

[1] Murray, J.D (2002)., Mathematical Biology, Springer-Verlag, New York.

[2] Allen, E (2007)., Modeling with Ito Stochastic Differential Equations, Springer-Verlag.

[3] Jacobs, J (2010)., Stochastic processes for Physicists; Understanding Noisy System. Cambridge University Press. [

www.cambridge.org/9780521765428].

[4] Anderson, R.M Ed (1982)., Population Dynamic of Infectious Diseases, Chapman and Hall, London.

[5] Anderson, R.M and May, R.M EDS(1991)., Infectious Diseases of Humans: Dynamics and Control, Oxford University Press,

Oxford, UK.

[6] Anderson, R.M and May, R.M EDS (1985)., Age related changes in the rate of disease transmission: Implication for the design of

vaccination programs, J.Hyg.Camb, 94, pp.365-436.

[7] Arnold, L (1972)., ‘Stochastic Differential Equations: Theory and Applications, John Wiley and Sons,.

[8] Oldstone, M (1998)., ‘Viruses, Plagues, and History. Oxford University Press, New York.

[9] KermackWO, and McKendrick AG (1991a) Contributions to the mathematical theory of epidemics–II.The problem of

endemicity.1932. Bull Math Biol; 53(1–2):57–87.

http://www.cambridge.org/9780521765428


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[10] Kermack WO, and McKendrick AG (1991b). Contributions to the mathematical theory of epidemics –I. 1927. Bull Math Biol;

53(1–2):33–55.

[11] Kermack WO, and McKendrick AG (1991c) .Contributions to the mathematical theory of epidemics–III. Further studies of the

problem of endemicity.1933. Bull Math Biol; 53(1–2):89–118.

[12] Ross, R (1911); The Prevention of malaria, John Murray, London.

[13] www.globalhealthfacts.org/data/topic/m. ( last accessed on January, 2013).

[14] Busenburg S. and Cooke K (1993)., ‘Vertically Transmitted diseases, Models and dynamics. Berlin : Springer-Verlag.

[15] Dietz K.and Heesterbeek JA (2000); Bernoulli was a head of modern epidemiology. Nature; 408(6812):513–4.

[16] Higham, DJ (2001),. An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, Society for

Industrial and applied Mathematics, Vol. 43, No . 3, pp. 525–546 (http://www.siam.org/journals/sirev/43-3/37830.html).

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http://www.siam.org/journals/sirev/43-3/37830.html

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