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Kermack-McKendrick type models for epidemics with nonlocal aggregation terms

Marco Di Francesco, Fatemeh Ghaderi Zefreh

TL;DR

This work develops a rigorous framework for Kermack–McKendrick type SIR models with nonlocal aggregation terms modeling spatial heterogeneity in epidemics. The authors construct a well-posedness theory by regularizing with diffusion, obtaining uniform estimates, and passing to the vanishing-diffusion limit to recover the original nonlocal system, including a uniqueness result. They also analyze steady states, demonstrating how nonlocal transport can yield space-dependent equilibria, with a concrete SIS example revealing a threshold driven by a space-dependent reproduction number $\mathcal{R}_0$, and they validate the theory with numerical simulations. Overall, the paper provides a solid mathematical foundation for nonlocal-transport epidemic models and highlights how spatial interactions can generate rich long-time behavior beyond classical homogeneous or diffusive models.

Abstract

We propose an approach to model spatial heterogeneity in SIR-type models for the spread of epidemics via \emph{nonlocal aggregation terms}. More precisely, we first consider an SIR model with spatial movements driven by nonlocal aggregation terms, in which the inter-compartment and intra-compartment interaction terms are distinct, and modelled through smooth interaction kernels. For the Cauchy problem of said model we provide a full well-posedness theory on $\R^2$ for $L^1\cap L^\infty \cap H^1$ initial conditions. The existence part is achieved by considering an approximated model with artificial linear diffusion, for which existence and uniqueness is proven via Duhamel's principle and Banach fixed point, and by providing suitable uniform estimates on the approximated solution in order to pass to the limit via classical compactness techniques. To prove uniqueness, we use classical $L^2$-stability which relies on the $H^1$-regularity of the solution. In the second part of the paper we provide a brief, general discussion on the steady states for these type of models, and display a specific example of non-trivial steady states for an SIS model with aggregations (driven by a single repulsive-attractive potential), the existence of which is determined by a threshold condition for a suitable "space-dependent" basic reproduction rate. We complement the analysis with numerical simulations on the SIS model.

Kermack-McKendrick type models for epidemics with nonlocal aggregation terms

TL;DR

This work develops a rigorous framework for Kermack–McKendrick type SIR models with nonlocal aggregation terms modeling spatial heterogeneity in epidemics. The authors construct a well-posedness theory by regularizing with diffusion, obtaining uniform estimates, and passing to the vanishing-diffusion limit to recover the original nonlocal system, including a uniqueness result. They also analyze steady states, demonstrating how nonlocal transport can yield space-dependent equilibria, with a concrete SIS example revealing a threshold driven by a space-dependent reproduction number , and they validate the theory with numerical simulations. Overall, the paper provides a solid mathematical foundation for nonlocal-transport epidemic models and highlights how spatial interactions can generate rich long-time behavior beyond classical homogeneous or diffusive models.

Abstract

We propose an approach to model spatial heterogeneity in SIR-type models for the spread of epidemics via \emph{nonlocal aggregation terms}. More precisely, we first consider an SIR model with spatial movements driven by nonlocal aggregation terms, in which the inter-compartment and intra-compartment interaction terms are distinct, and modelled through smooth interaction kernels. For the Cauchy problem of said model we provide a full well-posedness theory on for initial conditions. The existence part is achieved by considering an approximated model with artificial linear diffusion, for which existence and uniqueness is proven via Duhamel's principle and Banach fixed point, and by providing suitable uniform estimates on the approximated solution in order to pass to the limit via classical compactness techniques. To prove uniqueness, we use classical -stability which relies on the -regularity of the solution. In the second part of the paper we provide a brief, general discussion on the steady states for these type of models, and display a specific example of non-trivial steady states for an SIS model with aggregations (driven by a single repulsive-attractive potential), the existence of which is determined by a threshold condition for a suitable "space-dependent" basic reproduction rate. We complement the analysis with numerical simulations on the SIS model.
Paper Structure (13 sections, 11 theorems, 157 equations, 2 figures)

This paper contains 13 sections, 11 theorems, 157 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The approximated problem
  3. Local existence and uniqueness for the approximated problem
  4. Non-negativity of local solutions
  5. Global existence for the approximated problem
  6. Existence via vanishing diffusion
  7. Uniform $H^1$ estimates
  8. Vanishing viscosity limit
  9. Uniqueness
  10. Steady states
  11. A general discussion
  12. A specific repulsive-attractive potential
  13. Numerical Simulations

Key Result

Theorem 1.1

Assume the interaction kernels $W_{\xi \eta}$ satisfy eq:ass_reg_W and eq:ass_sym_W. Assume the initial data $S_0, I_0, R_0$ belong to $L^1(\mathbb{R}^2)\cap L^\infty(\mathbb{R}^2) \cap H^1(\mathbb{R}^2)$ and are nonnegative. Then, for any arbitrary time $T\geq 0$, there exists one and only one solving mainsystem_intro in the weak sense and having $(S_0, I_0, R_0)$ as initial datum.

Figures (2)

  • Figure 1: This figure shows the evolution of the epidemic in time, when $\mathcal{R}_0<1$, $\beta=0.5$. Here, the number of infected individuals is decreasing in time. Eventually, there are no infected individuals left, and we reach the disease-free equilibrium, and $S$ is supported in an interval of size $\gamma$.
  • Figure 2: This Figure shows the evolution of the epidemic in time, when $\mathcal{R}_0>1$, $\beta=1$. Here, $I$ grows in time and spreads until both $S$ and $I$ are supported in the same interval of size $\gamma$, and they are both constant in this interval.

Theorems & Definitions (24)

  • Theorem 1.1: Existence and uniqueness of solutions
  • Remark 2.1: Use of generic constants
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.2: Non-negativity of solutions
  • proof
  • Lemma 2.3: Conservation of the total mass
  • ...and 14 more