Dynamics of SVIS Model with Holling Type IV Functional Response

In this paper, we will study the effect of some epidemic concepts such as immigrants and vaccine on the dynamical behaviour of epidemic models. The existence, uniqueness and boundedness of the solution are investigated. The local stability analyses of the system is carried out .The global dynamics of the system is investigated numerically.

ISSN 2347-1921 5753 | P a g e F a b r u a r y 0 2 , 2 0 1 6

1.INTRODUCTION
Because of many infectious disease causing death and varying total population, the study of epidemic models become one of the important area in the mathematical theory of epidemiology. Epidemic model with vaccination is one of the most important models in decreasing the spread of many diseases. In [8] Kribs-Zaleta and Velasco-Hernandez presented a simple two dimensional SIS model with vaccination exhibiting backward bifurcation. Farringten [2] derived relation between vaccine efficacy against transmission and analyzed the impact of vaccination program on the transmission potential of the infection in large populations. In [3] Gumel and Moghadas proposed a model for the dynamics of an infectious disease in the presence of a preventive vaccine considering non-linear incidence rate I cI  1 . Liu et al. [4,13] proposed more realistic models that assume non-linear incidence rate given by Shim [11] assumed that the total population is asymptotically constant, and supposed the incidence rate N  where  is the transmission rate. In this paper, we will study the SVIS model with non-linear incidence rate

Model formulation
Consider an SIS disease when a vaccination program is in effect and there is a constant flow of incoming immigrants. A population of size at time t is partition into three classes of individuals; susceptible, infections and vaccinated, with sizes denoted by and , respectively which represented in the block diagram given by Fig. (1) can be represented by the following system of non-linear ordinary differential equations. The assumption we have in this model is as follows: is the constant natural birth rate, with all newborns coming into the susceptible class.
which is positively invariant for system (1) and all the solutions of system(1) with non-negative initial conditions are uniformly bounded as it is proved in the following theorem.
Theorem (2.1): All solution of system (1) with non-negative initial condition are uniformly bounded.
Proof: Let be any solution of the system (1) with non-negative initial condition Since then so Which has an integrating factor and hence a solution is where that means . Therefore, as , hence all solutions of system (1) that initiate in the region are eventually confined in the region: Thus these solution are uniformly bounded and then the proof is complete.
3 Existence of Equilibrium point of system (1) In this section, we find all possible equilibrium points of system (1) shows that there are at most two non-negative equilibrium points, the existence conditions for each of these equilibrium points are discussed in the following:

Local Stability of system (1)
In the following section the local stability analysis for the above equilibrium points is studied as shown in the following theorems.
Theorem Theorem (4.2): Assume that the positive equilibrium point of the system (1) exist and let the following inequalities hold: Here we have: Then it is locally asymptotically stable in the .
Proof: The linearized system of the system (1)  Hence is a positive definite function. Now, by differentiating with respect to time t, gives: Substituting the value of in the above equation, and after doing some algebraic manipulation, it gives that: Obviously, due to condition (9.a)-(9.d), it is obtained that , therefore the origin and then is locally asymptotically stable point in the .

Globally stability of all equilibrium point
In this section, the global dynamics of system (1) is studied with the help of Lyapunov function as shown in thefollowing theorems.
Theorem (5.1): The disease free equilibrium point of system (1) is globally asymptotically stable in the sub region: Where and Proof: Consider the function By differentiating with respect to t along the solution of system (1), we get: Now for any in and by Eq. (11) we get F a b r u a r y 0 2 , 2 0 1 6 is negative definite and hence is a Lyapunov function with respect to hence, is globally asymptotically stable in the sub region .
Theorem (5.2):Assume that the endemic equilibrium point of system (1) is locally asymptotically stable then it is globally asymptotically stable in the sub region that satisfies the following conditions:

Where and
Proof: Consider the function By differentiating with respect to t along the solution of system (1), we get: Now from Eq. (14) and Eq. (12.a)-(12.c) we have: So is negative definite and is a Lyapunov function with respect to hence, is globally asymptotically stable in the sub region .

Numerical analysis
In this section the global dynamics of system (1) (1) is solved for parameters values and respectively keeping other parameters fixed as given in equation (15) with and then the trajectories of system (1) are drawn in Figure 6. Also, in order to discuss the effect of varying the vaccination converge rate on the dynamical behavior of system (1) is studied too. The system is solved numerically for different value of and keeping the rest of parameters fixed as given in equation (15) with and time series of the solution of system (1) are drawn in Figure  8.a-c. a. For the system approaches asymptotically to b. For the system approaches asymptotically to c. For the system approaches asymptotically to From Figure 8.a-c. we note that the system (1) still approaches to endemic equilibrium point.
Similarly, the effect of varying the number of individuals who lose vaccine immunity and return to susceptible on the dynamical behavior of system (1) is studied the system is solved for the value and keeping other parameters as given in equation (15) with and then the solution of system (1) are drawn in Figure  9.a-c. respectively. a. For the system approaches asymptotically to b. For the system approaches asymptotically to c. For the system approaches asymptotically to F a b r u a r y 0 2 , 2 0 1 6 From Figure 9.a-c it is observed that as increases the system (1) still approaches to endemic equilibrium point and increasing causes increasing in the susceptible and infected but the number of vaccinated decreases.
Finally, the effect of vaccine efficiency against the disease on the dynamical behavior of system (1) is investigated. The system is solved for different values of and keeping other parameters as given in equation (15) with and then the solution of system (1) are drawn in Figure 10.a-c. respectively. a. For the system approaches asymptotically to b. For the system approaches asymptotically to c. For the system approaches asymptotically to From Figure 10.a-c. we conclude that as the vaccine efficiency increases the endemic equilibrium point of system (1) still coexists and stable.

Discussion and conclusions
In this section, we have analyzed a SIS epidemic model with the effect of vaccine and immigrants on the dynamical behavior on it. The local as well as global stability analysis of each possible equilibrium point are studied analytically as well as numerically. From numerically simulation (section 6) the following results are obtained the SVIS system (1) is approaches either to the disease free equilibrium point or to endemic equilibrium point.

As the fraction of infected immigrant individuals
increases then the number of susceptible and vaccinated individuals increase but the number of infected individual's increases.
If the infection rate increases then the number of susceptible and vaccinated individuals decrease but the number of infected individual's increases increasing of causes increasing in the vaccinated but the number of susceptible infected decrease (very slowly).