Qualitative analysis on a diffusive SIRS epidemic model with standard incidence infection mechanism
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Qualitative analysis on a diffusive SIRS epidemic model with standard incidence infection mechanism Shuyu Han, Chengxia Lei and Xiaoyan Zhang Abstract. In this paper, we are concerned with an SIRS epidemic reaction–diffusion system with standard incidence infection mechanism in a spatially heterogeneous environment. We first establish the uniform bounds of solutions and then derive the threshold dynamics in terms of the basic reproduction number R0 . Our main focus is on the asymptotic profile of endemic equilibria (when exists) if the diffusion (migration) rate of the susceptible, infected or recovered population is small, and our results show that the disease always exists in the entire habitat in each case of small diffusion rate. This suggests that restricting the diffusion (migration) rate of population is not an effective strategy of disease eradication. Mathematics Subject Classification. 35K57, 35J57, 35B40, 92D25. Keywords. SIRS epidemic reaction–diffusion model, Basic reproduction number, Endemic equilibria, Small diffusion (migration) rate, Asymptotic profile.
1. Introduction In 1927, in the pioneering work [18], Kermack and McKendrick applied the mass action infection mechanism to a SIR epidemic model. Mathematically, this type of disease is represented by the bilinear function βSI. Later, in 1995, de Jong et al. [9] proposed another type of infection mechanism, taking the form βSI , to describe the transmission of epidemic diseases. Such an infection mechanism is now called as of S+I standard incidence infection mechanism. From then on, infections affected by these two mechanisms have been extensively studied by researchers; one may refer to [12,24,25,37,44]. In [31], McCallum, Barlow and Hone concluded that either mass action infection mechanism or standard incidence infection mechanism has its advantages and disadvantages by comparing the known results from field and experimental data. Recently, Li and Bie [21] discussed this SIRS epidemic model with mass action mechanism:
S. Han and C. Lei were partially supported by NSF of China (Nos. 11671175, 11801232, 11971454), the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Natural Science Foundation of the Jiangsu Province (No. BK20180999), the Foundation of Jiangsu Normal University (No. 17XLR008). X. Zhang was partially supported by NSF of China (No. 11571200). 0123456789().: V,-vol
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⎧ ∂S ⎪ ⎪ − dS ΔS = Λ(x, t) − β(x, t)SI − μ(x, t)S + γ(x, t)R, ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ∂I ⎪ ⎪ − dI ΔI = β(x, t)SI − [δ(x, t) + μ(x, t) + α(x, t)] I, ⎪ ⎪ ⎨ ∂t ∂R − dR ΔR = δ(x, t)I − [μ(x, t) + γ(x, t)] R, ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ∂S ∂I ∂R ⎪ ⎪ = = = 0, ⎪ ⎪ ⎪ ∂ν ∂ν ∂ν ⎪ ⎩ S(x, 0) = S0 (x) ≥ 0, I(x, 0) = I0 (x) ≥, ≡ 0, R(x, 0) = R0 (x) ≥ 0,
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x ∈ Ω, t > 0, x ∈ Ω, t > 0, x ∈ Ω, t > 0,
(1.1)
x ∈ ∂Ω, t > 0,
where S(x, t), I(x, t) and R(x, t), represent the densities of susceptible, infected and recovered individuals, respectively, at location x and time t; the po
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