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A consistent SIR model on time scales with exact solution

Márcia Lemos-Silva, Sandra Vaz, Delfim F. M. Torres

TL;DR

For the new SIR model, an explicit solution is obtained, the asymptotic stability of the extinction and disease-free equilibria are proved, and some necessary conditions for the monotonic behavior of the infected population are deduced.

Abstract

We propose a new dynamic SIR model that, in contrast with the available model on time scales, is biological relevant. For the new SIR model we obtain an explicit solution, we prove the asymptotic stability of the extinction and disease-free equilibria, and deduce some necessary conditions for the monotonic behavior of the infected population. The new results are illustrated with several examples in the discrete, continuous, and quantum settings.

A consistent SIR model on time scales with exact solution

TL;DR

For the new SIR model, an explicit solution is obtained, the asymptotic stability of the extinction and disease-free equilibria are proved, and some necessary conditions for the monotonic behavior of the infected population are deduced.

Abstract

We propose a new dynamic SIR model that, in contrast with the available model on time scales, is biological relevant. For the new SIR model we obtain an explicit solution, we prove the asymptotic stability of the extinction and disease-free equilibria, and deduce some necessary conditions for the monotonic behavior of the infected population. The new results are illustrated with several examples in the discrete, continuous, and quantum settings.

Paper Structure

This paper contains 4 sections, 20 theorems, 71 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Time-scale fundamentals
  3. Main Results
  4. Conclusion

Key Result

Theorem 1

Let $p \in \mathcal{R}$ and $t_0 \in \mathbb{T}$. Then the regressive IVP problem of the form has the exponential function as its unique solution, denoted by $e_p(\cdot, t_0)$.

Figures (5)

  • Figure 1: Dynamics of infected population with constant coefficients $b = 1.5$ and $c = 0.1$ for the time-scale models \ref{['dynamic:sir']} and sir:ts with $\mathbb{T} = \mathbb{Z}$ versus the classical continuous SIR model \ref{['sir:continuous']}.
  • Figure 2: Dynamics of infected population with time-dependent coefficients \ref{['eq:bt']} and \ref{['eq:ct']} for the time-scale models \ref{['dynamic:sir']} and sir:ts with $\mathbb{T} = \mathbb{Z}$ versus the classical continuous SIR model \ref{['sir:continuous']}.
  • Figure 3: Dynamics of Example \ref{['ex:1']} with $x_0 = 0.6$, $y_0 = 0.4$ and $m = -1$.
  • Figure 4: Dynamics of Example \ref{['ex:2']} with $x_0 = 0.6$, $y_0 = 0.4$ and $q = 1.1$.
  • Figure 5: Illustration of Theorem \ref{['thm:9']}: dynamics of the infected population $y(t)$ of Example \ref{['ex:03']} with $x_0 = 0.8$, $y_0 = 0.2$, $b = 0.15$ and $c = 0.1$.

Theorems & Definitions (38)

  • Definition 1: See book:ts
  • Definition 2: See book:ts
  • Definition 3: See book:ts
  • Definition 4: See book:ts
  • Definition 5: See book:ts
  • Theorem 1: See book:ts
  • Theorem 2: See book:ts
  • Theorem 3: See book:ts
  • Theorem 4: See book:ts
  • Theorem 5: Variation of constants advanced:book:ts
  • ...and 28 more