Global, stochastic stability and (NSFD) method of a discrete multicompartment non-linear epidemic model.

Abstract

In this paper we consider the multi compartment epidemiological model SIIiQRC
with temporary immunity and saturated incidence rate. Size N(t) at time t, is divided into nine sub classes.
We have N(t) = S(t) + I(t) + Ii(t) + Q(t) + R(t) + C(t),i = 1, 2, 3, 4 where S(t), I(t), Ii(t), Q(t), R(t) and C(t) denote the sizes of the population
susceptible to disease, infectious members, quarantine members, recovered and cross-immune, respectively. We have made the following contributions: 1. The local stability of the infection-free equilibrium and endemic equilibrium are analyzed, respectively. The local stability of E0 if R0 < 1 and if
R0 > 1, the unique non-trivial equilibrium E is locally asymptotically stable witch can be determine by the ratio called the basic reproductive number.
2. This paper study the discretization by employing a nonstandard finite difference (NSFD) method to discretize the epidemic model such that the scheme maintains the population conservation law.
3. The dynamics of discrete system is analyzed and study the global asymptotically stability of disease-free equilibrium E0 if R0 < 1; and the endemic equilibrium E  if R0> 1 with theorems and proof under some conditions.
4. Finally the stochastic stability with the proof of theorem. In this work, we have used the different references cited in different studies.