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Dynamics of a Stochastic COVID-19 Epidemic Model with Jump-Diffusion

Almaz Tesfay, Tareq Saeed, Anwar Zeb, Daniel Tesfay, Anas Khalaf, James Brannan

Abstract

For a stochastic COVID-19 model with jump-diffusion, we prove the existence and uniqueness of the global positive solution. We also investigate some conditions for the extinction and persistence of the disease. We calculate the threshold of the stochastic epidemic system which determines the extinction or permanence of the disease at different intensities of the stochastic noises. This threshold is denoted by $ξ$ which depends on the white and jump noises. The effects of these noises on the dynamics of the model are studied. The numerical experiments show that the random perturbation introduced in the stochastic model suppresses disease outbreaks as compared to its deterministic counterpart. In other words, the impact of the noises on the extinction and persistence is high. When the noise is large or small, our numerical findings show that the COVID-19 vanishes from the population if $ξ<1;$ whereas the epidemic can't go out of control if $ξ>1.$ From this, we observe that white noise and jump noise have a significant effect on the spread of COVID-19 infection, i.e., we can conclude that the stochastic model is more realistic than the deterministic one. Finally, to illustrate this phenomenon, we put some numerical simulations.

Dynamics of a Stochastic COVID-19 Epidemic Model with Jump-Diffusion

Abstract

For a stochastic COVID-19 model with jump-diffusion, we prove the existence and uniqueness of the global positive solution. We also investigate some conditions for the extinction and persistence of the disease. We calculate the threshold of the stochastic epidemic system which determines the extinction or permanence of the disease at different intensities of the stochastic noises. This threshold is denoted by which depends on the white and jump noises. The effects of these noises on the dynamics of the model are studied. The numerical experiments show that the random perturbation introduced in the stochastic model suppresses disease outbreaks as compared to its deterministic counterpart. In other words, the impact of the noises on the extinction and persistence is high. When the noise is large or small, our numerical findings show that the COVID-19 vanishes from the population if whereas the epidemic can't go out of control if From this, we observe that white noise and jump noise have a significant effect on the spread of COVID-19 infection, i.e., we can conclude that the stochastic model is more realistic than the deterministic one. Finally, to illustrate this phenomenon, we put some numerical simulations.

Paper Structure

This paper contains 10 sections, 6 theorems, 60 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dynamical analysis of deterministic COVID-19 model
  4. Dynamics of the stochastic COVID-19 system
  5. Existence and uniqueness of the solution
  6. Extinction and persistence of the disease
  7. Extinction of the disease
  8. Persistence of the disease
  9. Discussion and numerical experiments
  10. Conclusion

Key Result

Lemma 1

(The one dimensional It$\hat{o}$ formula). Here we will give It$\hat{o}$ formula for the following $n$-dimensional stochastic differential equation (SDE) with jump noise applebaum2009levy where $G:\mathbb{R}_+\times \mathbb{R}^n\rightarrow\mathbb{R}^n$, $F:\mathbb{R}_+\times \mathbb{R}^n\rightarrow\mathbb{R}^n\times\mathbb{R}^d$, $H:\mathbb{R}_+\times \mathbb{R}^n\times\mathbb{R}^n\rightarrow\ma

Figures (4)

  • Figure 1: Sample path of $\frac{dI}{dt}$ (a) when $\xi_0=1.0286$ and $\xi_0=0.0103$. (b) The phaseline of $dI_t/dt$ at different values of $\nu$. (c) When the reproduction number $\xi_0 <1.$
  • Figure 2: The numerical results of model (\ref{['CV-BM-LM']}). (a) The graph of the susceptible. (b) The graph of the infected people.(c) The graph of the recoveblack people. Parameters $S_0=70,\, I_0=50,\, R_0=20,\, \Lambda=0.0072,\, \beta=0.002,\,\nu=0.001,\,\sigma=0.01,\,\gamma=0.02,\,\lambda_j=0.047$, $\epsilon_j(y)=0.004, \,\, j=1,2,3,\,\, \xi=0.9760 <1.$
  • Figure 3: The numerical simulation of model (\ref{['CV-BM-LM']}). (a) The graph of the susceptible. (b) The graph of the infected people.(c) The graph of the recoveblack people from COVID-19. Parameters $S_0=70,\, I_0=50,\, R_0=20,\, \Lambda=0.0072,\, \beta=0.002,\,\nu=0.001,\,\sigma=0.01,\,\gamma=0.02,\,\lambda_1=0.047,\, \lambda_2=0.019,\,\, \lambda_3=0.047,\,\epsilon_j(y)=0.004, \, j=1,2,3,\,\, \xi=1.02 > 1$.
  • Figure 4: This Figure shows the numerical simulation of the stochastic COVID-19 model (\ref{['CV-BM-LM']}) with $S_0=70,\, I_0=50,\, R_0=20,\, \Lambda=0.0072,\, \beta=0.002,\,\nu=0.001,\,\sigma=0.01,\,\gamma=0.02,\,\, \xi=0.9284 <1$, $j=1,2,3$.

Theorems & Definitions (16)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • Proof 1
  • Theorem 1
  • Proof 2
  • Theorem 2
  • Proof 3
  • Remark 2
  • Definition 1
  • ...and 6 more