| Citation: |
You Zhou, Beibei Zhang, Zhi Ling. DYNAMICAL BEHAVIOR OF THE FECAL-ORAL TRANSMISSION DISEASES MODEL ON A T-PERIODIC EVOLUTION DOMAIN[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 717-741. doi: 10.11948/20230025
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DYNAMICAL BEHAVIOR OF THE FECAL-ORAL TRANSMISSION DISEASES MODEL ON A T-PERIODIC EVOLUTION DOMAIN
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Abstract
We study the transmission dynamics of a fecal-oral diseases model on a $ T $-periodic evolution domain. We introduce the basic reproduction number $ R_0(\rho) $ as a threshold by some operator semigroup theory and give the relationship between it and that of the fixed domain, where $ \rho(t) $ is the domain evolution rate. By means of upper and lower solutions method, we investigate the existence, uniqueness and attractivity of endemic and disease-free equilibria respectively. Under certain conditions, there exists a unique global asymptotically stable positive periodic solution if $ R_0(\rho)>1 $. When $ R_0(\rho) \leq 1 $, the model possesses only zero solutions and is globally asymptotically stable. The final numerical simulations further verify our conclusions and illustrate the effect of the evolution rate. Based on the index $ \overline{\rho^{-2}} := \frac{1}{T}\int_{0}^{T}\frac{1}{\rho(t)^2}\mbox{d}t $, compared with the model on a fixed domain, we show that the transmission risk of the diseases increases if the index is lower than 1 and the risk decreases if the index is equal or greater than 1.
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References
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You Zhou, Beibei Zhang, Zhi Ling. DYNAMICAL BEHAVIOR OF THE FECAL-ORAL TRANSMISSION DISEASES MODEL ON A T-PERIODIC EVOLUTION DOMAIN[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 717-741. doi: 10.11948/20230025
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Figure 1.
$ a_{11} = 0.75 $ $ a_{12} = 2/3 $ $ a_{22} = 0.3 $ $ R_0(\rho) \geq \mathfrak{R_0} > 1 $ $ R_0(\rho) > 1 $ $ \rho(t) = \rho_1(t) $ $ \rho_2(t) $ $ \rho_3(t) $ -
Figure 2.
$ a_{11} = 0.3 $ $ a_{12} = 1/3 $ $ a_{22} = 0.95 $ $ \mathfrak{R_0} < 1 $ $ \rho(t) = \rho_1(t) $ $ \rho_2(t) $ $ \rho_3(t) $