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ODE and PDE models for COVID-19, with reinfection and vaccination process for Cameroon and Germany

Hamadjam Abboubakar, Reinhard Racke, Nicolas Schlosser

TL;DR

The paper develops and analyzes both an ODE and a reaction-diffusion PDE model for COVID-19 that incorporates reinfection and vaccination, applied to Germany and Cameroon. It introduces a control reproduction number $\mathcal{R}_c$, proves global stability of the disease-free equilibrium for $\mathcal{R}_c<1$, and demonstrates the existence of endemic equilibria when $\mathcal{R}_c>1$, including potential backward bifurcation. Parameter estimation using German and Cameroonian data yields $\mathcal{R}_c$ values around $1.13$ and $1.2554$, respectively, indicating disease persistence in both contexts. The PDE extension preserves well-posedness and mirrors the ODE dynamics while capturing spatial spread, with numerical simulations validating theoretical results and illustrating diffusion-driven propagation across Germany and Cameroon. Overall, the work provides a rigorous framework for assessing vaccination and reinfection effects in space-time COVID-19 dynamics and offers insights for region-specific control strategies.

Abstract

The goal of this work is to develop and analyze a reaction-diffusion model for the transmission dynamics of the Coronavirus (COVID-19) that accounts for reinfection and vaccination, as well as to compare it to the ODE model. After developing a time-dependent ODE model, we calculate the control reproduction number $\mathcal{R}_c$ and demonstrate the global stability of the COVID-19 free equilibrium for $\mathcal{R}_c<1$. We also show that when $\mathcal{R}_c>1$, the free equilibrium of COVID-19 becomes unstable and co-exists with at least one endemic equilibrium point. We then used data from Germany and Cameroon to calibrate our model and estimate some of its characteristics. We find $\mathcal{R}_c\approx 1.13$ for Germany and $\mathcal R_c \approx 1.2554$ for Cameroon, indicating that the disease persists in both populations. Following that, we modify the prior model into a reaction-diffusion PDE model to account for spatial mobility. We show that the solutions to the final initial value boundary problem (IVBP) exist and are nonnegative and unique. We also show that the disease-free equilibrium is stable locally, and globally when $\mathcal{R}_c<1$. In contrast, when $\mathcal{R}_c>1$, the DFE is unstable and coexists with at least one endemic equilibrium point. We ran multiple numerical simulations to validate our theoretical predictions. We then compare the ODE and the PDE models.

ODE and PDE models for COVID-19, with reinfection and vaccination process for Cameroon and Germany

TL;DR

The paper develops and analyzes both an ODE and a reaction-diffusion PDE model for COVID-19 that incorporates reinfection and vaccination, applied to Germany and Cameroon. It introduces a control reproduction number , proves global stability of the disease-free equilibrium for , and demonstrates the existence of endemic equilibria when , including potential backward bifurcation. Parameter estimation using German and Cameroonian data yields values around and , respectively, indicating disease persistence in both contexts. The PDE extension preserves well-posedness and mirrors the ODE dynamics while capturing spatial spread, with numerical simulations validating theoretical results and illustrating diffusion-driven propagation across Germany and Cameroon. Overall, the work provides a rigorous framework for assessing vaccination and reinfection effects in space-time COVID-19 dynamics and offers insights for region-specific control strategies.

Abstract

The goal of this work is to develop and analyze a reaction-diffusion model for the transmission dynamics of the Coronavirus (COVID-19) that accounts for reinfection and vaccination, as well as to compare it to the ODE model. After developing a time-dependent ODE model, we calculate the control reproduction number and demonstrate the global stability of the COVID-19 free equilibrium for . We also show that when , the free equilibrium of COVID-19 becomes unstable and co-exists with at least one endemic equilibrium point. We then used data from Germany and Cameroon to calibrate our model and estimate some of its characteristics. We find for Germany and for Cameroon, indicating that the disease persists in both populations. Following that, we modify the prior model into a reaction-diffusion PDE model to account for spatial mobility. We show that the solutions to the final initial value boundary problem (IVBP) exist and are nonnegative and unique. We also show that the disease-free equilibrium is stable locally, and globally when . In contrast, when , the DFE is unstable and coexists with at least one endemic equilibrium point. We ran multiple numerical simulations to validate our theoretical predictions. We then compare the ODE and the PDE models.
Paper Structure (26 sections, 7 theorems, 40 equations, 30 figures, 2 tables)

This paper contains 26 sections, 7 theorems, 40 equations, 30 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model formulation
  3. The disease-free equilibrium and the control reproduction number
  4. Existence of the endemic equilibrium
  5. Model calibration and forecasting
  6. Model calibration with German data
  7. Model calibration with Cameroon data
  8. Reaction-diffusion model and its analysis
  9. Qualitative analysis of the model
  10. Control reproduction number and stability of the DFE
  11. The control reproduction number
  12. Local stability of the disease-free equilibrium point
  13. Global stability of the disease-free equilibrium
  14. Existence of endemic equilibrium points
  15. Numerical scheme and simulations
  16. ...and 11 more sections

Key Result

Lemma 1

(Local stability of the DFE) The stationary point $\mathcal{E}_0$ of system Covid19_model-reduce is locally asymptotically stable (LAS) if $\mathcal{R}_{c}<1$, and unstable otherwise.

Figures (30)

  • Figure 1: Cumulative reported cases in Germany versus model fitting. $t=0$ stands for December 31, 2020 and $t=59$ stands for February 28, 2021.
  • Figure 2: Long-term Covid-19 prediction.
  • Figure 3: Cumulative reported cases in Cameroon versus model fitting. $t=0$ stands for April 2021 and $t=12$ stands for April 2022.
  • Figure 4: Initial distribution of model state variables where there is one southern peak.
  • Figure 5: Solution after $t_f=250$ days.
  • ...and 25 more figures

Theorems & Definitions (11)

  • Lemma 1
  • Theorem 1
  • proof
  • Proposition 1
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • proof
  • Remark 1
  • ...and 1 more