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Applied Probability Insights into Nonlinear Epidemic Dynamics with Independent Jumps

Brahim Boukanjime, Mohamed El Fatini, Mohamed Maama

Abstract

This paper focuses on the analysis of a stochastic SAIRS-type epidemic model that explicitly incorporates the roles of asymptomatic and symptomatic infectious individuals in disease transmission dynamics. Asymptomatic carriers, often undetected due to the lack of symptoms, play a crucial role in the spread of many communicable diseases, including COVID-19. Our model also accounts for vaccination and considers the stochastic effects of environmental and population-level randomness using Lévy processes. We begin by demonstrating the existence and uniqueness of a global positive solution to the proposed stochastic system, ensuring the model's mathematical validity. Subsequently, we derive sufficient conditions under which the disease either becomes extinct or persists over time, depending on the parameters and initial conditions. The analysis highlights the influence of random perturbations, asymptomatic transmission, and vaccination strategies on disease dynamics. Finally, we conduct comprehensive numerical simulations to validate the theoretical findings and illustrate the behavior of the model under various scenarios of randomness and parameter settings. These results provide valuable insights into the stochastic dynamics of epidemic outbreaks and inform strategies for disease management and control.

Applied Probability Insights into Nonlinear Epidemic Dynamics with Independent Jumps

Abstract

This paper focuses on the analysis of a stochastic SAIRS-type epidemic model that explicitly incorporates the roles of asymptomatic and symptomatic infectious individuals in disease transmission dynamics. Asymptomatic carriers, often undetected due to the lack of symptoms, play a crucial role in the spread of many communicable diseases, including COVID-19. Our model also accounts for vaccination and considers the stochastic effects of environmental and population-level randomness using Lévy processes. We begin by demonstrating the existence and uniqueness of a global positive solution to the proposed stochastic system, ensuring the model's mathematical validity. Subsequently, we derive sufficient conditions under which the disease either becomes extinct or persists over time, depending on the parameters and initial conditions. The analysis highlights the influence of random perturbations, asymptomatic transmission, and vaccination strategies on disease dynamics. Finally, we conduct comprehensive numerical simulations to validate the theoretical findings and illustrate the behavior of the model under various scenarios of randomness and parameter settings. These results provide valuable insights into the stochastic dynamics of epidemic outbreaks and inform strategies for disease management and control.
Paper Structure (6 sections, 6 theorems, 45 equations, 11 figures, 1 table)

This paper contains 6 sections, 6 theorems, 45 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main theoretical results
  3. Stochastic extinction
  4. Persistence in mean
  5. Numerical simulation
  6. Conclusion

Key Result

Theorem 2.1

Under $(\textbf{A$1$})$ and $(\textbf{A$2$})$, for any given initial value $(S_0,A_0,I_0,R_0)$, there is a unique solution $(S(t),A(t),I(t),R(t))$ of model sys1 defined on $t\geqslant 0$, and will remain in $\mathbb{R}^4_+$ with probability one.

Figures (11)

  • Figure 1: Dynamics of the Susceptible (S) population over time.
  • Figure 2: Dynamics of the Asymptomatic (A) population over time.
  • Figure 3: Dynamics of the Symptomatic (I) population over time.
  • Figure 4: Dynamics of the Recovered (R) population over time.
  • Figure 5: 3D surface plot showing the evolution of all compartments (S, A, I, R) over time. This plot provides a comprehensive view of the transitions and interactions between compartments throughout the simulation period.
  • ...and 6 more figures

Theorems & Definitions (13)

  • Theorem 2.1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • Theorem 2.2
  • proof
  • Definition 1
  • Theorem 2.3
  • ...and 3 more