Conditions for Permanence and Ergodicity of Certain SIR Epidemic Models
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Conditions for Permanence and Ergodicity of Certain SIR Epidemic Models Nguyen Huu Du1 · Nguyen Thanh Dieu2 · Nguyen Ngoc Nhu1
Received: 23 April 2016 / Accepted: 10 June 2018 © Springer Nature B.V. 2018
Abstract In this paper, we study sufficient conditions for the permanence and ergodicity of a stochastic susceptible-infected-recovered (SIR) epidemic model with BeddingtonDeAngelis incidence rate in both of non-degenerate and degenerate cases. The conditions obtained in fact are close to the necessary one. We also characterize the support of the invariant probability measure and prove the convergence in total variation norm of the transition probability to the invariant measure. Some of numerical examples are given to illustrate our results. Keywords SIR model · Extinction · Permanence · Stationary distribution · Ergodicity Mathematics Subject Classification 34C12 · 60H10 · 92D25
1 Introduction The SIR epidemic model consists of three groups of individuals, the susceptible individuals, the infected individuals and the recovered individuals, whose densities at the time t are denoted by S(t), I (t) and R(t) respectively. The relations between these quantities are in Authors would like to thank Vietnam Institute for Advance Study in Mathematics (VIASM) for supporting and providing a fruitful research environment and hospitality.
B N.T. Dieu
[email protected] N.H. Du [email protected] N.N. Nhu [email protected]
1
Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
2
Department of Mathematics, Vinh University, 182 Le Duan, Vinh, Nghe An, Vietnam
N.H. Du et al.
general described by the following equations: ⎧ ⎪ ⎨dS(t) = [α − μS S(t) − f (S(t), I (t))]dt dI (t) = [−(μI + γ )I (t) + f (S(t), I (t))]dt ⎪ ⎩ dR(t) = [−μR R(t) + γ I (t)]dt, where α > 0 is the recruitment rate of the population; μS , μI and μR are respectively the death rate of susceptible, infected and recovered individuals; γ is the recovery rate of the infected individuals and f (S(t), I (t)) is the incidence rate. The SIR epidemic models are widely considered with variety of incidence rates as in [4, 5, 9, 11, 17, 21, 22, 27–31, 35– 38] and references therein. For example, in 1975, Beddington [2] researched the interference between predators or parasites and by considerations of time utilization, the incidence rate (t) was derived mechanistically. This functional response was f (S(t), I (t)) = 1+mβS(t)I 1 S(t)+m2 I (t) also introduced by DeAngelis et al. in [8]; therefore it is usually named as BeddingtonDeAngelis incidence rate or Beddington-DeAngelis functional response. This incidence rate allows a simple physical interpretation of the effect on predator efficiency of prey and predator density in predator-prey models or on parasite density in parasite-host models as well (see [2] in more details). It is similar to the well-known Holling type II functional response, which was introduced by Capasso and Serio [4] in their study of cholera epidemic in Bari, Italy
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