Stochastic Stability and Analytical Solution with Homotopy Perturbation Method of Multicompartment Non-Linear Epidemic Model with Saturated Rate

In this work, we consider a nonlinear epidemic model with a saturated incidence rate. we consider a population of size N(t) at time t, this population is divided into six subclasses, with N(t)=S(t)+I(t)+I1(t)+I2(t)+I3(t)+Q(t). Where S(t), I(t), I1(t), I2(t), I3(t), and Q(t) denote the sizes of the population susceptible to disease, infectious members, and quarantine members, respectively. We have made the following contributions: 1.The local stabilities of the infection-free equilibrium and endemic equilibrium are; analyzed, respectively. The stability of a disease-free equilibrium and the existence of other nontrivial equilibria can be determined by the ratio called the basic reproductive number. 2. We find the analytical solution of the nonlinear epidemic model by Homotopy perturbation method. 3. Finally the stochastic stabilities. The study of its sections are justified with theorems and demonstrations under certain conditions. In this work, we have used the different references cited in different studies in the three sections already mentioned.


Introduction
This paper considers the following epidemic model with saturated incidence rate.
 Consider a population of size N(t) at time t, this population is divided into six subclasses, with N(t)=S(t)+I(t)+I₁(t)+I₂(t)+I₃(t)+Q(t). Where S(t), I(t), I₁(t), I₂(t), I₃(t) and Q(t) denote the sizes of the population susceptible to disease, infectious members and quarantine members, respectively.  The positive constants μ represent rate of incidence. The positive constant β is the average numbers of contacts infective for S to I. The positive constant ν is the parameter of emigration.  The positive constant r is the parameter of immigration.
 The positive constants γ₁, γ₂, and γ₃, are the numbers of transfer or conversion of infected people quarantined. The positive constant α₁, α₂ and α₃ are the average numbers of contacts for I to I i, i=1,2,3.  The positive constants μ, μ 0 , μ₁, μ₂, μ₃ and μ₄ represent the death rates of susceptible, infectious and quarantine.  Biologically, it is natural to assume that μ ≤ min {μ₀, μ₁, μ₂, μ₃, μ₄}. The positive constant d is natural mortality rate.
The region is positively invariant set of (1).

Equilibrium and Local Stability
An equilibrium point of system (3) satisfies.
We calculate the points of equilibrium in the absence and presence of infection.
The system (3) has a disease-free equilibrium E₀.

Theorem 2.1
The disease-free equilibrium E₀ of the system (3) is locally asymptotically stable if R₀<1.
So * E is the unique positive endemic equilibrium point which exists if R₀>1.

Proof
The eigenvalues can be determined by solving the characteristic equation of the linearization of (3) near E₀. Therefore, the eigenvalues are: In order to A 6 will be negative, and then we define the basic reproduction number of the infection R₀ as follows: If R₀<1, A 6 <0. We have A i <0, i=1, 2, 3, A₄<0, A₅<0 and A₆<0, if R₀<1. Then E₀ of the system (3) is locally asymptotically stable.
In the presence of infection , substituting in the system, Ω also contains a unique positive, endemic equilibrium. Where So is the unique positive endemic equilibrium point which exists if R₀>1. W

Theorem 2.2
If R₀>1, the system (3) has a unique non-trivial equilibrium * E which is locally asymptotically stable.

Solution of Model by HPM
We define the operator d L dt = , By applying the homotopy perturbation method to system, (3) we obtain the following form: The initial condition is We assume the solution for system (8)

Stochastic Stability
The system (3), is transformed to the Itô Stochastic differential equations. We replace β by β+ab (t) where b (t) is white noise.

Conclusion
This paper addresses a the equilibrium and local stability of the epidemic model with saturated incidence rate, in the absence of infection, the system has a disease-free equilibrium, in the presence of infection the system, has a unique positive, endemic equilibrium. Then we applied the Homotopy perturbation method, we obtained The zero, first and second order solutions.