Dynamics of a nonlocal epidemic model with a new free boundary condition, part 1: Spreading-vanishing dichotomy
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| Format: | Preprint |
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2026
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| _version_ | 1866914306891710464 |
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| author | Chen, Yao Du, Yihong Li, Wan-Tong Wang, Rong |
| author_facet | Chen, Yao Du, Yihong Li, Wan-Tong Wang, Rong |
| contents | This paper investigates the long-time dynamics of a nonlocal epidemic model with free boundaries, where a pathogen with density $u(t,x)$ and the infected humans with density $v(t,x)$ evolve according to a reaction-diffusion system with nonlocal diffusion over a one dimensional interval $[g(t), h(t)]$, which represents the epidemic region expanding through its boundaries $x=g(t)$ and $x=h(t)$, known as free boundaries. Such a model with free boundary conditions based on those of Cao et al. \cite{fb27} was considered by several works. Inspired by recent works of Feng et al. \cite{fb20} and Long et al. \cite{fb5}, we propose a new free boundary condition, where the expansion rate of the epidemic region, determined by $h'(t)$ and $g'(t)$, is proportional to a linear combination of the outward flux of the pathogen \(u\) through the range boundary (as in \cite{fb27}) and the weighted total population of infected individuals \(v\) within the region (as in \cite{fb5}). We prove that the system under this new free boundary condition is well-posed, and its long-time dynamical behavior is characterized by a spreading-vanishing dichotomy. Moreover, we obtain sharp criteria for this dichotomy, including a sharp threshold in terms of the initial data $(u_0,v_0)$; and by studying a related eigenvalue problem, we also find a sharp threshold in terms of the diffusion rate, which complements related results in Nguyen and Vo \cite{fb7}. This is Part $1$ of a two part series. In Part $2$, we will determine the spreading speed of the model when spreading occurs, and for some typical classes of kernel functions, we will obtain the precise rates of accelerated spreading. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_05331 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Dynamics of a nonlocal epidemic model with a new free boundary condition, part 1: Spreading-vanishing dichotomy Chen, Yao Du, Yihong Li, Wan-Tong Wang, Rong Analysis of PDEs 35K57, 35R35, 92D30 This paper investigates the long-time dynamics of a nonlocal epidemic model with free boundaries, where a pathogen with density $u(t,x)$ and the infected humans with density $v(t,x)$ evolve according to a reaction-diffusion system with nonlocal diffusion over a one dimensional interval $[g(t), h(t)]$, which represents the epidemic region expanding through its boundaries $x=g(t)$ and $x=h(t)$, known as free boundaries. Such a model with free boundary conditions based on those of Cao et al. \cite{fb27} was considered by several works. Inspired by recent works of Feng et al. \cite{fb20} and Long et al. \cite{fb5}, we propose a new free boundary condition, where the expansion rate of the epidemic region, determined by $h'(t)$ and $g'(t)$, is proportional to a linear combination of the outward flux of the pathogen \(u\) through the range boundary (as in \cite{fb27}) and the weighted total population of infected individuals \(v\) within the region (as in \cite{fb5}). We prove that the system under this new free boundary condition is well-posed, and its long-time dynamical behavior is characterized by a spreading-vanishing dichotomy. Moreover, we obtain sharp criteria for this dichotomy, including a sharp threshold in terms of the initial data $(u_0,v_0)$; and by studying a related eigenvalue problem, we also find a sharp threshold in terms of the diffusion rate, which complements related results in Nguyen and Vo \cite{fb7}. This is Part $1$ of a two part series. In Part $2$, we will determine the spreading speed of the model when spreading occurs, and for some typical classes of kernel functions, we will obtain the precise rates of accelerated spreading. |
| title | Dynamics of a nonlocal epidemic model with a new free boundary condition, part 1: Spreading-vanishing dichotomy |
| topic | Analysis of PDEs 35K57, 35R35, 92D30 |
| url | https://arxiv.org/abs/2602.05331 |