Global $L^\infty$-bounds and Long-time Behavior of a Diffusive Epidemic System in A Heterogeneous Environment

Peng, Rui; Wu, Yixiang

doi:10.1137/19m1276030

Cited by 21 publications

(59 citation statements)

References 53 publications

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“…On the other hand, the following PDE model (and its variant) with mass action infection function was studied in Refs. 29, 31, 35, 39, 40, 43–45:

$$\begin{align} {\left\lbrace \def\eqcellsep{&}\begin{array}{llll}\displaystyle S_t - d_S \Delta S = -\beta (x) SI + \gamma (x)I, & x\in \Omega ,\,t&gt;0, \\[3pt] \displaystyle I_t - d_I \Delta I = \beta (x) SI - \gamma (x)I, & x\in \Omega ,\,t&gt;0,\\[3pt] \displaystyle \frac{\partial S}{\partial \nu } = \frac{\partial I}{\partial \nu } =0,&x\in \partial \Omega ,\,t&gt;0,\\[3pt] \displaystyle S(x,0) = S_0(x)\ge 0,\quad I(x,0) \ge 0,\not\equiv 0,&x\in \Omega . \end{array} \right.}

…”

Section: Discussionmentioning

confidence: 99%

“…To mention a few, we refer interested readers to Refs. 5, 10-26 for discrete diffusion models or patch models, [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45] for continuous diffusion models or reaction-diffusion(-advection) partial differential equations (PDE) models, [46][47][48][49] for nonlocal diffusion models, and Refs. 50-52 for other types of mathematical models.…”

Section: Introductionmentioning

confidence: 99%

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An SIS epidemic model with mass action infection mechanism in a patchy environment

Peng

2022

Stud Appl Math

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In this paper, we study an SIS (susceptible–infected–susceptible) epidemic model with mass action infection mechanism in a patchy environment. We first analyze the long‐time dynamics of the model in terms of the basic reproduction number scriptR0$\mathcal {R}_0$, and prove under certain conditions that the disease‐free equilibrium (DFE) is globally attractive if R0≤1$\mathcal {R}_0\le 1$ whereas the endemic equilibrium (EE) is globally attractive if R0>1$\mathcal {R}_0>1$. Then we establish the existence and uniqueness of EE provided R0>1$\mathcal {R}_0>1$, which is achieved, among other ingredients, by reducing the equilibrium problem into the one involving only the component of infected population. We further investigate the asymptotic profile of EE as the motility rate of the susceptible and/or infected population becomes small. We also perform the theoretical analysis on the corresponding model with linear birth/death effect. By comparing the findings of the models considered here with those of other closely related ones, our results suggest that the variation of total population, individual motility, infection mechanism, and spatial heterogeneity all can play significant roles in the dynamics of disease transmission, which may shed light on the practical implications for predicting the patterns of disease outbreak and designing optimal control strategies.

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“…On the other hand, the following PDE model (and its variant) with mass action infection function was studied in Refs. 29, 31, 35, 39, 40, 43–45:

$$\begin{align} {\left\lbrace \def\eqcellsep{&}\begin{array}{llll}\displaystyle S_t - d_S \Delta S = -\beta (x) SI + \gamma (x)I, & x\in \Omega ,\,t&gt;0, \\[3pt] \displaystyle I_t - d_I \Delta I = \beta (x) SI - \gamma (x)I, & x\in \Omega ,\,t&gt;0,\\[3pt] \displaystyle \frac{\partial S}{\partial \nu } = \frac{\partial I}{\partial \nu } =0,&x\in \partial \Omega ,\,t&gt;0,\\[3pt] \displaystyle S(x,0) = S_0(x)\ge 0,\quad I(x,0) \ge 0,\not\equiv 0,&x\in \Omega . \end{array} \right.}

…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An SIS epidemic model with mass action infection mechanism in a patchy environment

Peng

2022

Stud Appl Math

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“…Here we used (37). For any fixed small ε ∈ (0, L), we integrate the equation of I-component over [0, ε] to deduce…”

Section: Profile Of Ee Asmentioning

confidence: 99%

“…To understand the impact of spatial heterogeneity on the dynamics of a disease, an increasing number of reaction-diffusion models have been proposed and studied in recent years. In particular, the SIS epidemic reaction-diffusion models have been developed to investigate the effects of the spatial heterogeneity and the movement of populations on the persistence or extinction of diseases, see, for instance, [1,4,10,13,15,18,22,24,25,26,35,36,37,38,39,41,43].…”

mentioning

confidence: 99%

Concentration phenomenon of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with spontaneous infection

Lei

2022

DCDS-B

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<p style='text-indent:20px;'>In this paper, we investigate the effect of spontaneous infection and advection for a susceptible-infected-susceptible epidemic reaction-diffusion-advection model in a heterogeneous environment. The existence of the endemic equilibrium is proved, and the asymptotic behaviors of the endemic equilibrium in three cases (large advection; small diffusion of the susceptible population; small diffusion of the infected population) are established. Our results suggest that the advection can cause the concentration of the susceptible and infected populations at the downstream, and the spontaneous infection can enhance the persistence of infectious disease in the entire habitat.</p>

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“…In the face of infectious diseases, many mathematical models have been proposed, especially the susceptible-infected-susceptible (SIS) epidemic reaction-diffusion models; refer to [3,6,8,10,15,18,19,20,25,21,23,24,33,36,49,53] without advection term, and [4,5,14] with advection term as well as [35,37,38,50] in the timeperiodic environment. In the more commonly studied situation, we usually ignore the immigration of infected individuals.…”

mentioning

confidence: 99%

Analysis on a diffusive SEI epidemic model with/without immigration of infected hosts

Lei¹,

Yi²,

Zhang³

et al. 2021

DCDS-S

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<p style='text-indent:20px;'>In this paper, we study a reaction-diffusion SEI epidemic model with/without immigration of infected hosts. Our results show that if there is no immigration for the infected (exposed) individuals, the model admits a threshold behaviour in terms of the basic reproduction number, and if the system includes the immigration, the disease always persists. In each case, we explore the global attractivity of the equilibrium via Lyapunov functions in the case of spatially homogeneous environment, and investigate the asymptotic behavior of the endemic equilibrium (when it exists) with respect to the small migration rate of the susceptible, exposed or infected population in the case of spatially heterogeneous environment. Our results suggest that the strategy of controlling the migration rate of population can not eradicate the disease, and the disease transmission risk will be underestimated if the immigration of infected hosts is ignored.</p>

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Global $L^\infty$-bounds and Long-time Behavior of a Diffusive Epidemic System in A Heterogeneous Environment

Cited by 21 publications

References 53 publications

An SIS epidemic model with mass action infection mechanism in a patchy environment

An SIS epidemic model with mass action infection mechanism in a patchy environment

Concentration phenomenon of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with spontaneous infection

Analysis on a diffusive SEI epidemic model with/without immigration of infected hosts

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