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On quasi-linear reaction diffusion systems arising from compartmental SEIR models

Juan Yang, Jeff Morgan, Bao Quoc Tang

Abstract

The global existence and boundedness of solutions to quasi-linear reaction-diffusion systems are investigated. The system arises from compartmental models describing the spread of infectious diseases proposed in [Viguerie et al, Appl. Math. Lett. (2021); Viguerie et al, Comput. Mech. (2020)], where the diffusion rate is assumed to depend on the total population, leading to quasilinear diffusion with possible degeneracy. The mathematical analysis of this model has been addressed recently in [Auricchio et al, Math. Method Appl. Sci. (2023] where it was essentially assumed that all sub-populations diffuse at the same rate, which yields a positive lower bound of the total population, thus removing the degeneracy. In this work, we remove this assumption completely and show the global existence and boundedness of solutions by exploiting a recently developed $L^p$-energy method. Our approach is applicable to a larger class of systems and is sufficiently robust to allow model variants and different boundary conditions.

On quasi-linear reaction diffusion systems arising from compartmental SEIR models

Abstract

The global existence and boundedness of solutions to quasi-linear reaction-diffusion systems are investigated. The system arises from compartmental models describing the spread of infectious diseases proposed in [Viguerie et al, Appl. Math. Lett. (2021); Viguerie et al, Comput. Mech. (2020)], where the diffusion rate is assumed to depend on the total population, leading to quasilinear diffusion with possible degeneracy. The mathematical analysis of this model has been addressed recently in [Auricchio et al, Math. Method Appl. Sci. (2023] where it was essentially assumed that all sub-populations diffuse at the same rate, which yields a positive lower bound of the total population, thus removing the degeneracy. In this work, we remove this assumption completely and show the global existence and boundedness of solutions by exploiting a recently developed -energy method. Our approach is applicable to a larger class of systems and is sufficiently robust to allow model variants and different boundary conditions.
Paper Structure (12 sections, 20 theorems, 240 equations)

This paper contains 12 sections, 20 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. Mathematical analysis of quasilinear SEIRD models
  3. Problem setting
  4. Main results
  5. Global existence of bounded weak solutions
  6. Approximate system
  7. Uniform-in-$\varepsilon$ estimate
  8. Passing to the limit - Global existence
  9. Uniform-in-time boundness
  10. Proof of Theorems \ref{['th2']}--\ref{['th5']}
  11. Applications
  12. Proof of Lemma \ref{['lem-L-infty-bound']}

Key Result

Theorem 1.1

Assume D:cond-1, D:cond-2, Q1, Q2, Q3, A1, A2, A3, A5 and A4 with Then for any nonnegative, bounded initial datum $u_0\in (L^{\infty}(\Omega))^m$, there exists a global weak solution to system with $u_i\in L^{\infty}_{loc}(0,\infty; L^{\infty}(\Omega))$ for all $i=1,\cdots,m.$ In particular, if $K_1<0$ or $K_1=K_2=0$, then the solution is bounded uniformly in time

Theorems & Definitions (44)

  • Theorem 1.1
  • Remark 1
  • Theorem 1.2
  • Remark 2
  • Theorem 1.3
  • Remark 3
  • Theorem 1.4
  • Remark 4
  • Theorem 1.5
  • Remark 5
  • ...and 34 more