High-order reliable numerical methods for epidemic models with non-constant recruitment rate

TL;DR

The paper addresses reliable numerical simulation of an SEIR epidemic model with non-constant recruitment by studying positivity-preserving properties of both explicit Euler and high-order SSP Runge-Kutta methods. It formulates a generalized SEIR system with a general incidence function $f(I)$ and a time-dependent recruitment $\Pi(t)$, proving existence, positivity, and boundedness results for the continuous model and deriving discrete-time step-size conditions that preserve these properties. The authors establish explicit nonnegativity and boundedness criteria for the Euler method and extend these results to SSP Runge-Kutta schemes, including an analysis of the optimal SSP coefficient $\mathcal{C}$ and related convergence behavior. Numerical experiments confirm the theoretical bounds and demonstrate order accuracy, highlighting practical guidelines for choosing time steps to ensure biologically meaningful simulations in scenarios with migration and births.

Abstract

The mathematical modeling of the propagation of illnesses has an important role from both mathematical and biological points of view. In this article, we observe an SEIR-type model with a general incidence rate and a non-constant recruitment rate function. First, we observe the qualitative properties of different methods: first-order and higher-order strong stability preserving Runge-Kutta methods \cite{shu}. We give different conditions under which the numerical schemes behave as expected. Then, the theoretical results are demonstrated by some numerical experiments. \keywords{positivity preservation, general SEIR model, SSP Runge-Kutta methods}

Figures (3)

• Figure 1: The function $\Pi(t)$ (orange) and the numerical solution produced by the explicit Euler method for $N(t)$ (blue). As we can see, the difference of these functions does not converge to zero.
• Figure 2: The second order method with timestep $\tau=4.8$ (left) and $\tau=3.3$ (right). The method violates the property on the left figure but behaves as expected on the right one.
• Figure 3: The orders of the different methods. As we can see, they behave as expected, i.e. the second order method attains an order of two, and so on.

Theorems & Definitions (13)

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