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Integrable discretizations of the SIR model

Yuta Tanaka, Ken-ichi Maruno

Abstract

Structure-preserving discretizations of the SIR model are presented by focusing on the hodograph transformation and the conditions for integrability for their discrete SIR models are given. For those integrable discrete SIR models, we derive their exact solutions as well as conserved quantities. If we choose the parameter appropriately for one of our proposed discrete SIR models, it conserves the conserved quantities of the SIR model. We also investigate an ultradiscretizable discrete SIR model.

Integrable discretizations of the SIR model

Abstract

Structure-preserving discretizations of the SIR model are presented by focusing on the hodograph transformation and the conditions for integrability for their discrete SIR models are given. For those integrable discrete SIR models, we derive their exact solutions as well as conserved quantities. If we choose the parameter appropriately for one of our proposed discrete SIR models, it conserves the conserved quantities of the SIR model. We also investigate an ultradiscretizable discrete SIR model.
Paper Structure (9 sections, 146 equations, 5 figures)

This paper contains 9 sections, 146 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Construction of conserved quantities and exact solutions of the SIR model by the hodograph transformation
  3. Integrable discretizations of the SIR model
  4. The discrete SIR-1 model
  5. The discrete SIR-2 model
  6. The generalized discrete SIR model
  7. The nonautonomous generalized discrete SIR model
  8. Integrability of an ultradiscretizable SIR model
  9. Conclusions

Figures (5)

  • Figure 1: The graphs of the exact solution to the initial value problem for the SIR model. The parameters and initial values are $\beta=0.0005$, $\gamma=0.1$, $S(0)=997$, $I(0)=3$, $R(0)=0$. The horizontal axis in the left panel is $t$, the horizontal axis in the right panel is $\tau$.
  • Figure 2: The graphs of an exact solution to the initial value problem for the dSIR1 model. The parameters and initial values are $\beta=0.0005, \gamma=0.1, S(0)=997, I(0)=3, R(0)=0$, $\epsilon_k=0.5$. The horizontal axis in the left panel is $t$, the horizontal axis in the right panel is $\tau$.
  • Figure 3: The graphs of an exact solution to the initial value problem for the dSIR2 model. The parameters and initial values are $\beta=0.0005, \gamma=0.1, S(0)=997, I(0)=3, R(0)=0$, $\epsilon_k=0.5$. The horizontal axis in the left panel is $t$, the horizontal axis in the right panel is $\tau$.
  • Figure 4: The graph of an exact solution to the initial value problem for the gdSIR model with $p=0$ (the dSIR1 model), $p=0.5$, $p=1$ (the dSIR2 model). The parameters and initial values are $\beta=0.0005, \gamma=0.1, S(0)=997, I(0)=3, R(0)=0$, $\epsilon_k=500$. The horizontal axis is $\tau$.
  • Figure 5: The graph of a numerical solution to the initial value problem for the gdSIR model with the best value $p=0.50021\cdots$ . The parameters and initial values are $\beta=0.0005, \gamma=0.1, S(0)=997, I(0)=3, R(0)=0$, $\epsilon_k=0.5$.