Dynamics of a nonlocal epidemic model with a new free boundary condition, part 1: Spreading-vanishing dichotomy

Technical Summary: Dynamics of a Nonlocal Epidemic Model with a New Free Boundary Condition, Part 1

Problem Statement
This paper investigates the long-time dynamics of a nonlocal epidemic model describing the interaction between a pathogen (density $u$) and infected humans (density $v$) within a one-dimensional epidemic region $[g(t), h(t)]$. The region expands or contracts over time, with $g(t)$ and $h(t)$ representing free boundaries. The system incorporates nonlocal diffusion mechanisms, where the movement of both species is governed by integral operators involving kernel functions $J_1$ and $J_2$.

The core innovation of this work lies in the formulation of a new free boundary condition. Unlike previous models where the expansion rate of the boundary is determined solely by the outward flux of the species (analogous to the Stefan condition), or solely by the weighted total population, this model proposes a condition where the expansion rate $h'(t)$ (and $g'(t)$) is proportional to a linear combination of:

1. The outward flux of the pathogen $u$ through the boundary (representing environmental dispersal, e.g., via water).
2. The weighted total population of infected individuals $v$ within the region (representing human-driven expansion).

Mathematically, the free boundary conditions are given by:
$$h'(t) = \mu \int_{g(t)}^{h(t)} \left[ u(t, x)W_{J_1}(h(t)-x) + \rho v(t, x)W(h(t)-x) \right] dx$$
$$g'(t) = -\mu \int_{g(t)}^{h(t)} \left[ u(t, x)W_{J_1}(h(t)-x) + \rho v(t, x)W(h(t)-x) \right] dx$$
where $W_{J_1}$ is the tail integral of the pathogen kernel, and $W$ is a weight function independent of the dispersal kernel for humans.

Methodology
The authors employ a rigorous analytical framework combining fixed-point theory, comparison principles, and spectral analysis:

1. Well-posedness: The existence and uniqueness of global solutions are established using the Contraction Mapping Theorem. The authors define a suitable complete metric space for the boundary functions $(g, h)$ and construct a mapping based on the solution of the associated reaction-diffusion system with fixed boundaries.
2. Comparison Principles: A series of comparison lemmas are developed to bound the solution $(u, v)$ and the boundaries $(g, h)$ using upper and lower solutions. These principles are crucial for analyzing the asymptotic behavior and proving the dichotomy results.
3. Spectral Analysis: The study relies heavily on the principal eigenvalue $\lambda_p$ of a related nonlocal eigenvalue problem defined on a finite interval. The sign of $\lambda_p$ determines the stability of the zero equilibrium and dictates whether the disease spreads or vanishes.
4. Asymptotic Analysis: The long-time behavior is analyzed by distinguishing between two scenarios:
   • Vanishing: The boundaries converge to a finite interval, and the densities decay to zero.
   • Spreading: The boundaries expand to infinity, and the densities converge to a unique positive steady state.

Key Contributions and Results

1. Global Existence and Uniqueness (Theorem 1.1): The paper proves that the proposed nonlocal system with the new mixed free boundary condition admits a unique global solution for all $t > 0$, provided the initial data and kernel functions satisfy standard regularity and positivity conditions.

2. Spreading-Vanishing Dichotomy (Theorem 1.2): The long-time dynamics are characterized by a sharp dichotomy. Depending on the parameters, the system either:

   • Vanishes: The epidemic region remains bounded ($g_\infty, h_\infty$ are finite), and the pathogen and infected population densities converge uniformly to $(0, 0)$.
   • Spreads: The epidemic region expands to the entire real line ($g_\infty = -\infty, h_\infty = +\infty$), and the densities converge locally uniformly to a unique positive steady state $(u^*, v^*)$.

3. Sharp Criteria for Spreading and Vanishing (Theorem 1.3): The authors derive precise conditions determining which outcome occurs:

   • If the basic reproduction number $R_0 < 1$, vanishing always occurs.
   • If $R_0 \geq (1 + d_1/a)(1 + d_2/b)$, spreading always occurs.
   • In the intermediate regime ($1 < R_0 < (1 + d_1/a)(1 + d_2/b)$), the outcome depends on the initial domain size $h_0$ and the expansion coefficient $\mu$:
     • If $h_0$ is sufficiently large ($h_0 \geq L^*$), spreading occurs.
     • If $h_0$ is small, there exists a critical threshold $\mu^*$ (dependent on initial data) such that vanishing occurs for $\mu \leq \mu^*$ and spreading for $\mu > \mu^*$.
     • Similarly, for fixed small domains, there is a threshold for the initial population magnitude $\sigma$.

4. Diffusion Rate Threshold (Theorem 1.4): The paper establishes a sharp threshold for the diffusion rate. By scaling the diffusion coefficients as $(d_1, d_2) = s(d_1^0, d_2^0)$, the authors identify a critical value $d^*$ such that:

   • If $s \leq d^*$ (slow diffusion), spreading always occurs (assuming $R_0 > 1$).
   • If $s > d^*$ (fast diffusion), the outcome depends on the expansion coefficient $\mu$, exhibiting a dichotomy similar to the initial data case.
   • This result complements previous work by Nguyen and Vo [30] by resolving the case where the diffusion rates are equal and providing a threshold for the general case.

5. Critical Case $R_0 = 1$: The paper shows that if $R_0 = 1$, vanishing generally occurs (densities go to zero), though proving the finiteness of the final interval requires additional conditions on the weight function $W$ or the infection rate function $G$.

Significance and Claims
The paper claims to provide a comprehensive theoretical foundation for a new class of epidemic models that integrate nonlocal dispersal with a biologically motivated, hybrid free boundary condition. The significance lies in:

• Unification: It bridges the gap between models driven purely by flux (Cao et al. [4]) and those driven purely by population weight (Long et al. [27]), offering a more flexible framework for modeling diseases with both environmental and human-mediated transmission.
• Sharp Thresholds: The derivation of sharp thresholds for the initial domain size, the expansion coefficient, and the diffusion rate provides precise criteria for predicting disease outcomes, which is vital for control strategies.
• Mathematical Rigor: The work extends the theory of nonlocal free boundary problems, specifically addressing the well-posedness and asymptotic dynamics under the new boundary conditions, filling gaps in the existing literature regarding the interplay between nonlocal diffusion and mixed boundary drivers.

This work is presented as Part 1 of a two-part series, with Part 2 dedicated to determining the precise spreading speeds and rates of accelerated spreading when spreading occurs.