<p>We study an extension of the stochastic SIS model generated under the assumption that the disease transmission coefficient in an infinitesimal time interval is <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\beta (t) + \sigma (t)\xi _t$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>σ</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>ξ</mi> <mi>t</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\beta (t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sigma (t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are bounded time-dependent functions and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\xi _t$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ξ</mi> <mi>t</mi> </msub> </math></EquationSource> </InlineEquation> is a white noise. We verify that there is a global positive solution to the stochastic differential equation (SIS-Itô-SDE) resultant for the infected and give the necessary conditions in their coefficients to achieve endemic persistence or disease extinction. Also, a numerical procedure to estimate the parameters of the respective SDE is proposed to make forecasts useful in the initial stages of an epidemic. The method begins with the discretization of infectives; our approach assumes that the SIS-Itô-SDE involves unknown transmission <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\beta (t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and diffusion <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\sigma (t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> rates, which we estimate. Specifically, we employ the maximum likelihood estimator (MLE) to gauge the transmission rate’s value, ensuring its strong consistency. For estimating the diffusion parameter, we utilize the quadratic variation of the data. To validate our methodology, we simulate using various datasets. Subsequently, using the empirical distributions of these same estimators and the Milstein method, the number of infected is predicted. To evaluate the effectiveness of this method, by using the RMSE metric, the forecast accuracy of the number of infected individuals reported in two studies is compared with real historical data.</p>

Stochastic perturbation of SIS model with bounded time-dependent parameters: formulation and numerical approximation

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发表日期：

2025年10月16日

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Yofre H. García,
Flor A. Gómez

摘要

We study an extension of the stochastic SIS model generated under the assumption that the disease transmission coefficient in an infinitesimal time interval is $\beta (t) + \sigma (t)\xi _t$ $β (t) + σ (t) ξ_{t}$ , where $\beta (t)$ $β (t)$ and $\sigma (t)$ $σ (t)$ are bounded time-dependent functions and $\xi _t$ $ξ_{t}$ is a white noise. We verify that there is a global positive solution to the stochastic differential equation (SIS-Itô-SDE) resultant for the infected and give the necessary conditions in their coefficients to achieve endemic persistence or disease extinction. Also, a numerical procedure to estimate the parameters of the respective SDE is proposed to make forecasts useful in the initial stages of an epidemic. The method begins with the discretization of infectives; our approach assumes that the SIS-Itô-SDE involves unknown transmission $\beta (t)$ $β (t)$ and diffusion $\sigma (t)$ $σ (t)$ rates, which we estimate. Specifically, we employ the maximum likelihood estimator (MLE) to gauge the transmission rate’s value, ensuring its strong consistency. For estimating the diffusion parameter, we utilize the quadratic variation of the data. To validate our methodology, we simulate using various datasets. Subsequently, using the empirical distributions of these same estimators and the Milstein method, the number of infected is predicted. To evaluate the effectiveness of this method, by using the RMSE metric, the forecast accuracy of the number of infected individuals reported in two studies is compared with real historical data.