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 Email: 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.
This paper examines an SIR epidemic model with a nonmonotonic 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 diseasefree 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; NonMonotonic 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 nonlinear 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 diseasefree 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 diseasefree 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 diseasefree equilibrium (0, 0) of a system (7) is
i) a stable hyperbolic node if ;
ii) a saddlenode 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
[1] Anderson RM. Transmission dynamics and control of infectious disease agents. In: Anderson RM, May RM, eds.Population Biology of Infectious Diseases. SpringerVerlag, Berlin, 1982, pp. 149–76. http://dx.doi.org/10.1007/9783642686351_9.
[2] Anderson RM, May RM, Infectious Diseases of Humans. Dynamics and Control. Oxford University Press, Oxford, 1991.
[3] Diekmann O, Heesterbeek JAP.Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation. Wiley and Sons, Chichester, 2000.
[4] D. Xiao and S. Ruan: Math. Biosci. 208, 2007, 419429. http://dx.doi.org/10.1016/j.mbs.2006.09.025.
[5] H.W. Hethcote: The Mathematics of Infectious Disease, SIAM Rew. 42, 2000, 599633. http://dx.doi.org/10.1137/S0036144500371907.
[6] H.W. Hethcote and P. Van Den Driessche: J. math. Boil. 29, 1991, 271287.
[7] S Khekare et al., Global Dynamics of an Epidemic Model with a Nonmonotonic Incidence Rate, IJOSRJM, 2014, 7177.
[8] L.Perko,Differential Equations and Dynamical Systems, Texts in Applied Mathematics,Vol. 7,SpringerVerlag,New work, 1991
[9] S. Ruan and W. Wang: J. Differential equations, 188, 2003, 135163. http://dx.doi.org/10.1016/S00220396(02)00089X.
[10] V. Capasso and G. Serio: Math. Biosci. 42, 1978, 4361. http://dx.doi.org/10.1016/00255564(78)900068.
[11] W. M. Liu, S. A. Levin and Y. Iwasa: J. Math. Biol., 1986, 187204. http://dx.doi.org/10.1007/BF00276956.
[12] W. M. Liu, H. W. Hethcote and S. Levin: J. Math. Biol., 25, 1987, 359380. http://dx.doi.org/10.1007/BF00277162.
[13] W.M. Liu, P. van den Driessche, Math. Biosci. 128, 1995, 5769. http://dx.doi.org/10.1016/00255564(94)00067A.
[14] W.R. DERRICK AND P.VAN DEN DRIESSCHE: J MATH. BIOL. 31, 1993, 495512. http://dx.doi.org/10.1007/BF00173889.
[15] W. Wang: Math.Biosci. 201, 2006, 5871. http://dx.doi.org/10.1016/j.mbs.2005.12.022.