Abstract
Classical epidemic models assume that the size of the total population is constant. More recent models consider a population size variable to take into account a longer period with death and disease causing reduced reproductive. The model contains a disease-free equilibrium and one or multiple equilibria are endemic. The stability of a disease-free status equilibrium and the existence of other nontrivial equilibria can be determined by the ratio called the basic reproductive number, which quantifies the number of secondary infections arise from a simple put infected in a population of sensitive. First, the local stability of the infection-free equilibrium and endemic equilibrium were analyzed, respectively. Second, the endemic equilibrium was formulated in terms of the incidence rate and local asymptotic stability. Finally we applied the adomian decomposition method to the system Epidemiologic. This method yields an analytical solution in terms of convergent infinite power series.
Highlights
Model equations: The system is described by equations which are defined as follows Eq 1: Classical epidemic models assume that the size of the total population is constant
Sufficient conditions were given to ensure the existence of the endemic equilibrium for the system and the stability of the endemic equilibrium is studied, we have shown that under certain restrictions on the parameter values and the infectious period
The ‘endemic equilibrium is locally asymptotically stable, epidemiologically, this means that the disease will prevail and persist in a population
Summary
Model equations: The system is described by equations which are defined as follows Eq 1: Classical epidemic models assume that the size of the total population is constant. The positive constants α, β is the average numbers of contacts infective for S and I. We consider the operator equation Fu = g, when F is the operator represents a general nonlinear ordinary differential and G is a given function. We decompose the nonlinear term, N as a series of number of infected individuals. The frequency- special polynomials called Adomian polynomials Eq 19: dependent transmission is common for the diseases directly transmitted for which the number of contacts is fixed. With applying the differential operator inverse L−1 we have Eq 29: Xi (t) = Xi (t*) + L−1Ni + L−1Ri , i = 1, 2,3,..., n (29) It was the first nonlinear term is Eq 31:. We note that the system is a more general homogeneous system of ordinary differential equations where the nonlinear term is the product of two variables. Solve the system using the method (MADM): The solution explicite Eq 43-45:
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