Abstract
The purpose of this work is to compare the stochastic and deterministic versions of an SIRS epidemic model. The SIRS models studied here include constant inflows of new susceptibles, infectives and removeds. These models also incorporate saturation incidence rate and disease-related death. First, we study the global stability of deterministic model with and without the presence of a positive flow of infectives into the population. Next, we extend the deterministic model system to a stochastic differential system by incorporating white noise. We show there is a unique positive solution to the system, and the long-time behavior of solution is studied. Mainly, we show how the solution goes around the endemic equilibrium and the disease-free equilibrium of deterministic system under different conditions on the intensities of noises and the parameters of the model. Finally, we introduce some numerical simulation graphics to illustrate our main results.
Highlights
Mathematical modeling is an essential tool in studying the spread of infectious diseases
E0∗(S0∗, I0∗, R0∗) is the endemic equilibrium state of the model (1) without new infective members of immigration which exists provided that the reproduction number R0 > 1 [19]
If the constant inflow of the infectives is null in the deterministic model (1), we find the basic reproduction number R0
Summary
Mathematical modeling is an essential tool in studying the spread of infectious diseases. We construct an SIRS model with saturated incidence rate that includes the immigration of distinct compartments, that is, there are susceptible, infective, and removed individuals in the new number of immigration. For the bilinear incidence rate (when = 0) and p2 = 0, Brauer and Van den Driessche [3] studied the global behavior of the SIR model (γ = 0) that includes immigration of infective individuals and variable population size. In the special case where there is no input of infective and removed individuals, Mena-Lorca and Hethcote [15], constructed a Lyapunov function for an SIR epidemic model. Lahrouz et al [13] introduced noise into an SIRS model by perturbing the contact rate Both of them show that the models established in their paper possess non-negative solutions. A discussion and numerical simulations are presented to confirm our mathematical findings
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