Abstract
We generalize the deterministic and the stochastic single-group SIRS epidemic models with saturated incidence rate introduced by Lahrouz, Omari, and Kiouach to the multi-group versions. In the deterministic multi-group model, the fact is highlighted that if the threshold $\mathscr{R}_{0}\leq1$ , then the infective condition disappears and it means the extinction of the disease. If $\mathscr{R}_{0}>1$ , then there exists an endemic equilibrium in a feasible region. Allowing the noise perturbation, for the stochastic version, we utilize stochastic Lyapunov functions to show the stability of the disease-free equilibrium of system. A detailed analysis is performed on almost surely exponential stability and pth moment exponential stability of the disease-free equilibrium. We also go into several numerical simulations to illustrate how exactly the theoretical results are verified. Good agreement was observed between our theoretical results and numerical simulations. A comprehensive conclusion is provided.
Highlights
Epidemiology models have been widely studied by many mathematicians and biologists [ – ]
Considering a heterogeneous population whose individuals are distinguishable by age, geography, and/or stage of disease and so on, it will be more realistic and reasonable to divide the individual hosts into groups
5 Conclusion In this paper, we propose deterministic and stochastic multi-group SIRS models with a saturated incidence rate
Summary
Epidemiology models have been widely studied by many mathematicians and biologists [ – ]. We in [ ] discussed the global stability of the multi-group SEIQR model with random perturbation around the positive equilibrium in computer network. ). Globally asymptotic stability of the disease-free equilibrium for deterministic Since the nonnegative matrix M(S, I) + V – F is irreducible, it follows from the Perron-Frobenius theorem (see [ ]) that ρ(M(S, I)) < ρ(V – F ) ≤. This implies that the equation M(S, I)I = I has only the trivial solution I = , where I = If R > , ( . ) has at least one endemic equilibrium
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