Stability analysis of an SIR model with immunity and modified transmission function

 

Nidhi Nirwani *, V.H.Badshah, R.Khandelwal

 

School of Studies in Mathematics Vikram University, Ujjain (M.P.) India

*Corresponding author E-mail: nd.mathematics2009@gmail.com

 

 

Copyright © 2015 Nidhi Nirwani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

 

Abstract

 

This paper examines an SIR epidemic model with a non-monotonic incidence rate. We analyzed the model by considering after infection, only a fraction of transmitted part is shifted to infectious and remaining part gets recovered without becoming infectious. We also analyze the dynamical behavior of the model and derive the stability conditions for the disease-free and the endemic equilibrium. We have found a threshold condition, in terms of basic reproduction number  which is, less than one, the disease free equilibrium is globally attractive and if more than one, the endemic equilibrium exists and is globally stable. We illustrate theoretical results by carrying numerical simulation.

 

Keywords: Basic Reproduction Number; Disease Frees Equilibrium; Endemic Equilibrium; Epidemiology; Non-Monotonic Incidence; SIR Model; Stability.

 

1.         Introduction

The concealed and apparently unpredictable nature of infectious diseases has been a source of fear and superstition since the first ages of human civilization. The asymptotic behavior of solution of an infectious disease transmission model depends not only on epidemiological formation, but also on the demographic process incorporated into the model. Many authors have proposed various kinds of epidemic models to understand the mechanism of disease transmission. Anderson R M [1] proposed transmission and control phenomena of infectious diseases. Diekmann O. Heesterbeek [3] proposed and analyzed the evidence of the increasing diversification of infectious diseases. Several studies are there for the treatment of epidemics with different kind of incidence rates which measures the transfer rate of susceptible to get infected [4,5,6,9,11,15].Thus the incidence is number of new infectious per day or per other time unit.

We have several different incidence rates which have been proposed by many researchers in epidemic model. Capasso and Serio [10] introduced a saturated incidence rate  into epidemic models. Where  tends to a saturation level when  gets large.

Nonlinear incidence rates of form  were investigated by Lui et. al.[12,13]. A very general form of non-linear incidence rate was considered by Derrick and Driessche [14].One of the most fundamental quantities used by epidemiologists is certainty the basic reproduction number consider and analyzed by Anderson R M, May R M[2].In this paper the result is written in terms of basic reproduction number and stability of the equilibriums are investigated. 

2.         The mathematical model

The model we analyze in this paper is considered within the framework of the following nonlinear ordinary differential equations.

 


                                                                                                                                                  

 

Table 1: Variables and Parameters in the Model

Symbol

Description

Number of Susceptible individuals at time t

Number of Infective individuals at time t

Number of Recovered  individuals at time t

The recruitment rate of the Population

The natural death rate of the population

A positive constant where.

The natural recovery rate of the infective Individuals

The rate at which recovered Individuals lose immunity

The proportionality constant

The parameter which measure the affects of medical infrastructure

The parameter which measure awareness of people through education.

 

Existence of Equilibria

For the system (1), for any values of parameters, it always has a disease free- equilibrium.

 

 

Define the basic reproduction number as follows:


                                                                                                                                                                                            (2)

Then we have the following:

 

Theorem 2.1:

1)           If, then there is no positive equilibrium.

2)           If , then there is a unique positive equilibrium  called the “endemic equilibrium”. Given by

 

 

 

 

 

 

Now we check the dynamical behavior of disease free equilibrium point  and endemic equilibrium point  . The variation matrix of  is

 

Its characteristics equation is given by

 

 

 

Clearly,   is stable is

 

 

Its characteristic equation is given by

 

                                                                                                                                             

 

 

 

Here  provided

 

 

 

Performing simple calculations it can be easily be verified that   under the above conditions.

Thus by Routh Hurwitz criterion, all Eigen values of (3) will have negative real part. Hence   is asymptotically stable.

3.         Global analysis

In this section, we study the properties of the equilibriums and derive the stability condition for the disease- free and the endemic equilibrium of model (1).

 

Lemma 3.1: The plane    is an invariant manifold of system (1) which is attracting in the first octant.

 

Proof: Summing up the three equations in (1) and denoting

, we have

 


                                                                                                                                                                    

 

For the equilibrium point, set

From the above equation, it is clear that

 

 

Is one solution of (4) and for any  ,the general solution of equation (4) is

Also,

 

 

This completes the proof.

 

Clearly limit set of a system (1) is on the plane.

 

 

Thus we focus on the reduced system.

 


                                                                                                           

 

 

Theorem3.2: System (5) does not have nontrivial periodic orbits.

 

Proof: Consider system (5) for  and  . Take Dulac function [8] as

 

 

Then we have

 


                                                                                   

 

Thus the expression (6) is negative for. Hence, the conclusion follows. 

In order to study the properties of the disease-free equilibrium  and the endemic equilibrium  we rescale (5) by taking

 

 

Then we obtain,

 


                                                                                                                                     

 

 

Where

 

 

The trivial equilibrium (0, 0) of the system (7) is the disease-free equilibrium  of the model (1) and the unique positive equilibrium  of a system (7) is the endemic equilibrium  of a model (1) where

 

 

Which is positive if 

And  is the positive solution of the following quadratic equation


                                                                                                                                                     

Where

 

 

 

 

 

 

 

Obviously, equation (8) has a positive root if    .   Obviously Thus, the equilibrium points     exists.

We first determine the stability and topological type of (0, 0). The jacobian matrix of a system (7) at (0, 0) is.

 

 

 

If  = 0, then there exists a small neighborhood of (0, 0) such that the dynamic of a system (7) are equivalent to that of

 

 

 

Theorem 3.3: The disease-free equilibrium (0, 0) of a system (7) is

 

i)             a stable hyperbolic node if  ;

ii)           a saddle-node if   ;

iii)          a hyperbolic saddle if 

 

When   , we discuss the stability and topological type of the endemic equilibrium  is

The jacobian of the system (7) at  is

 

 


 Thus, if the above condition holds then  is a node or a focus or a centre.

The trace of matrix M, is given by

 

 

Thus,   if

 


                                                                                                                                           

Where

 

 

Thus, if both the conditions (9) and (10) are satisfied, we get a unique equilibrium  of model (7), which is stable.

4.         Numerical results

Consider following values of parameters

 

 

Note that at  we get  ,when  the value of and when  the value of  and therefore these exists unique equilibrium.

 

0.1

0.2

0.3

0.4

0.5

0.6

0.1899

0.4980

0.7470

1

1.2450

1.5192

5.         Conclusion

In this paper we studied global quantative analysis of a realistic SIR model. In terms of the basic reproduction  our main result indicate that when  the disease free equilibrium is globally attractive. When  , the endemic equilibrium exists and is globally stable. Though the basic reproduction number  does not depend on a and b.This implies that the spread of disease decreases as the social or psychological protective measures for the infective increases.This implies for  the model coincides with that of Khekare et al. [7]

References

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