<p>This study investigates the impact of seasonal weather variations on malaria transmission by developing and analyzing both deterministic and stochastic non-autonomous models. We first show that the global dynamics of the deterministic model are governed by the basic reproduction number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation>: when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${{R}_{0}}<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the disease-free periodic solution is globally asymptotically stable; when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$R_0 > 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the disease is uniformly persistent in the population. Building on the same biological assumptions, we further formulate a time nonhomogeneous stochastic model and develop an analytical solution for the probability of disease extinction by applying multi-type branching process approximation and backward Kolmogorov differential equations. Our analytical solution reveals that the probability of disease extinction varies periodically with the initial time and depends on the initial number of infected individuals. Moreover, we derive analytical expressions for the mean and variance of the time to disease extinction. Additionally, this study investigates the malaria situation in Nigeria over the period 2018-2024. Our model demonstrates a strong fit with real-world data on both monthly cases and monthly cumulative cases during this period. Numerical simulations confirm the analytical results and illustrate the impact of the treatment rate on long-term disease dynamics and the probability of disease extinction. Further, we analyze two important indicators: the total infectivity time and the total treatment time. The theoretical analytical solutions are strongly supported by numerical simulations, which show close agreement between the branching process approximation and Monte Carlo estimates, thereby validating the reliability and predictive capability of the proposed stochastic modeling approach. These findings may offer useful insights for malaria control.</p>

Dynamical Analysis of Vector-Host Diseases Model in a Periodic Environment

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发表日期：

2026年6月27日

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Haijie Li,
Guirong Liu

摘要

This study investigates the impact of seasonal weather variations on malaria transmission by developing and analyzing both deterministic and stochastic non-autonomous models. We first show that the global dynamics of the deterministic model are governed by the basic reproduction number $R_0$ $R_{0}$ : when ${{R}_{0}}<1$ $R_{0} < 1$ , the disease-free periodic solution is globally asymptotically stable; when $R_0 > 1$ $R_{0} > 1$ , the disease is uniformly persistent in the population. Building on the same biological assumptions, we further formulate a time nonhomogeneous stochastic model and develop an analytical solution for the probability of disease extinction by applying multi-type branching process approximation and backward Kolmogorov differential equations. Our analytical solution reveals that the probability of disease extinction varies periodically with the initial time and depends on the initial number of infected individuals. Moreover, we derive analytical expressions for the mean and variance of the time to disease extinction. Additionally, this study investigates the malaria situation in Nigeria over the period 2018-2024. Our model demonstrates a strong fit with real-world data on both monthly cases and monthly cumulative cases during this period. Numerical simulations confirm the analytical results and illustrate the impact of the treatment rate on long-term disease dynamics and the probability of disease extinction. Further, we analyze two important indicators: the total infectivity time and the total treatment time. The theoretical analytical solutions are strongly supported by numerical simulations, which show close agreement between the branching process approximation and Monte Carlo estimates, thereby validating the reliability and predictive capability of the proposed stochastic modeling approach. These findings may offer useful insights for malaria control.