| Citation: |
Yaxin Zhou, Daqing Jiang. COMPARISON OF RESEARCH METHODS FOR DISEASE MODELS WITH TWO DIFFERENT RANDOM PERTURBATIONS UNDER THE INFLUENCE OF SANITATION AND AWARENESS[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1325-1354. doi: 10.11948/20240043
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COMPARISON OF RESEARCH METHODS FOR DISEASE MODELS WITH TWO DIFFERENT RANDOM PERTURBATIONS UNDER THE INFLUENCE OF SANITATION AND AWARENESS
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Abstract
Environmental hygiene and public awareness play a key role in controlling the spread of infectious diseases, and they are also very effective health intervention measures for the public. This paper studies the dynamical behaviors of a nonlinear mathematical model with health and publicity which controlled by investment budget. We compared and analyzed the research methods of two disease models generated under white noise and Ornstein-Uhlenbeck process perturbations. At first, we discuss the local stability of the endemic equilibrium by Lyapunov function method which avoids the tedious calculation process when studying the local stability of the positive solution of the models with dimensions greater than three. And then we conduct research on random models under the influence of white noise, we study the existence and uniqueness of positive solution. We get a critical value $ R^* $ which corresponding to the control reproduction number $ R_1 $ of the ordinary differential equation when we discuss the stationary distribution of the stochastic system. In addition, constructing a Lyapunov function is a method to obtain some sufficient conditions for the extinction of the disease. Finally, the numerical simulations illustrate our above theoretical results and several parameters have a significant impact on the model are pointed out. Specifically, we present the dynamic properties of the same model under Ornstein-Uhlenbeck process perturbations in the Appendix.
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References
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Yaxin Zhou, Daqing Jiang. COMPARISON OF RESEARCH METHODS FOR DISEASE MODELS WITH TWO DIFFERENT RANDOM PERTURBATIONS UNDER THE INFLUENCE OF SANITATION AND AWARENESS[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1325-1354. doi: 10.11948/20240043
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Figure 1.
The stationary distribution of system (2.3) with initial value
$ (S(0), I(0), B(0), M(0))=(0.5, 0.5, 0.5, 0.5) $ -
Figure 2.
The extinction of system (2.3) with initial value
$ (S(0), I(0), B(0), M(0))=(0.5, 0.5, 0.5, 0.5) $ -
Figure 3.
The trend of
$ R_{0}, \; R_{1}\; \text{and}\; R^{*} $ $ v\; \text{and}\; \alpha $ $ d=0.9, \; d=0.5\; \text{and}\; d=0.1 $ -
Figure 4.
The influence of system (2.3) under the different
$ \alpha $ $ I(t)\; \text{and}\; B(t) $ $ \alpha=0.1, \; \alpha=0.4, \; \alpha=0.7, \; \alpha=0.9, $ -
Figure 5.
The influence of system (2.3) under the different
$ d $ $ I(t)\; \text{and}\; B(t) $ $ d=0.1, \; d=0.4, \; d=0.7, \; d=0.9, $ -
Figure 6.
The influence of system (2.3) under the different
$ v $ $ I(t)\; \text{and}\; B(t) $ $ v=0.1, \; v=0.4, \; v=0.7, \; v=0.9, $