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

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.


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.

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

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

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.

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]