| Citation: |
Shiyu Li, Yuanshun Tan, Xiaodan Sun, Yu Mu. AN EPIDEMIC MODEL COUPLED WITH ENVIRONMENTAL LEVEL: EXPLORE THE IMPACT OF DISEASE AWARENESS ON DIRECT AND INDIRECT TRANSMISSION[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1374-1397. doi: 10.11948/20240230
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AN EPIDEMIC MODEL COUPLED WITH ENVIRONMENTAL LEVEL: EXPLORE THE IMPACT OF DISEASE AWARENESS ON DIRECT AND INDIRECT TRANSMISSION
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Abstract
To study the impact of disease awareness on infectious diseases with direct and indirect transmission, we develop a mathematical model by coupling the transmission dynamics at the population level and the environmental level. The basic reproduction number $R_0$ of the coupled model is calculated, and the existence and stability of the disease-free and endemic equilibrium are analyzed in detail. By using center manifold theory, it is verified that the model undergoes backward bifurcation under certain conditions. Numerical simulations verify our theoretical results and indicate that enhancing disease awareness can help reduce both the risk of direct and indirect disease transmission. Interestingly, increasing disease awareness decreases the backward regime of the bifurcation curve, thereby the $R_0$ interval in which the endemic equilibrium and the disease-free equilibrium showed bistability becomes smaller, and the $R_0$ interval in which the disease-free equilibrium showed global stability becomes greater. If the disease cannot be eliminated, the number of infected persons at the steady state decreases with the increase in disease awareness. The findings have certain reference values for the development of effective non-pharmaceutical intervention policies.
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Keywords:
- Coupled epidemic model /
- disease awareness /
- backward bifurcation /
- global stability
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References
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Shiyu Li, Yuanshun Tan, Xiaodan Sun, Yu Mu. AN EPIDEMIC MODEL COUPLED WITH ENVIRONMENTAL LEVEL: EXPLORE THE IMPACT OF DISEASE AWARENESS ON DIRECT AND INDIRECT TRANSMISSION[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1374-1397. doi: 10.11948/20240230
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Figure 1.
Variation in the number of exposed and infected people and virus concentration in the environment under different values of
$ \sigma $ -
Figure 2.
The impact of
$ \mu $ $ \eta_i $ $ i=1, 2 $ $ R_0 $ -
Figure 3.
(a) shows that
$ E_0 $ $ R _ 0 ^ * <1 $ $ Q_1^* $ $ R_0>1 $ -
Figure 4.
Taking different initial values,
$ E_0 $ $ Q_1^* $ $ n<0 $ $ R_c<R_0<1 $ -
Figure 5.
Bifurcation diagram for different awareness impact factors. (a) shows the backward bifurcation diagram of system (2.1), where the dash curve represents unstable equilibrium while the solid curve represents stable equilibrium. (b) shows the influence of the value of different awareness impact factor
$ \sigma $