<p>Improper household waste disposal generates several hazardous bacteria that contribute to the spread of deadly diseases in society. These bacteria interact with human populations directly through environmental exposure and indirectly via carrier populations like flies and rats. Controlling these bacterial populations has become a key focus for the scientific community. In this paper, we develop and study an SEIRS model to control bacteria generated from household waste through two interventions: impulsive vaccination of susceptible humans and bacterial disinfection of the environment at fixed periodic intervals. Both theoretical analysis and numerical simulations are used to assess the effectiveness of these strategies in controlling disease transmission. Our findings reveal that the SEIRS model produces a unique positive T-periodic solution, whose analysis depends on two critical thresholds, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40808_2024_2250_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$ R^* $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>R</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40808_2024_2250_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$ R_* $</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>R</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation>. The first threshold, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40808_2024_2250_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$ R^* $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>R</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>, determines whether the infection-free periodic solution is globally attractive, indicating disease eradication when <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40808_2024_2250_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$ R^* < 1 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>R</mi> <mo>∗</mo> </msup> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. The second threshold, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40808_2024_2250_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$ R_* $</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>R</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation>, signifies the system’s permanence, where the disease persists when <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40808_2024_2250_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$ R_* > 1 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mmultiscripts> <mi>R</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> <mrow /> </mmultiscripts> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Numerical simulations further support these theoretical results, demonstrating that high rates of impulsive vaccination and bacterial washout are essential for achieving global attractivity and effectively controlling the disease. In contrast, lower intervention rates lead to persistent infections. Additionally, reducing the interval between interventions is particularly effective for disease eradication when the disease’s latent period is short. The proposed hybrid control strategy is practical and cost-effective, offering valuable insights for policymakers to enhance control efforts, reduce disease transmission, and lower the societal costs of outbreaks.</p>

Modeling and control of infectious diseases from household waste using an SEIRS model with impulsive vaccination and bacterial disinfection

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发表日期：

2025年1月3日

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Kunwer Singh Mathur

摘要

Improper household waste disposal generates several hazardous bacteria that contribute to the spread of deadly diseases in society. These bacteria interact with human populations directly through environmental exposure and indirectly via carrier populations like flies and rats. Controlling these bacterial populations has become a key focus for the scientific community. In this paper, we develop and study an SEIRS model to control bacteria generated from household waste through two interventions: impulsive vaccination of susceptible humans and bacterial disinfection of the environment at fixed periodic intervals. Both theoretical analysis and numerical simulations are used to assess the effectiveness of these strategies in controlling disease transmission. Our findings reveal that the SEIRS model produces a unique positive T-periodic solution, whose analysis depends on two critical thresholds, $ R^* $ $R^{*}$ and $ R_* $ $R_{*}$ . The first threshold, $ R^* $ $R^{*}$ , determines whether the infection-free periodic solution is globally attractive, indicating disease eradication when $ R^* < 1 $ $R^{*} < 1$ . The second threshold, $ R_* $ $R_{*}$ , signifies the system’s permanence, where the disease persists when $ R_* > 1 $ $R_{*} > 1$ . Numerical simulations further support these theoretical results, demonstrating that high rates of impulsive vaccination and bacterial washout are essential for achieving global attractivity and effectively controlling the disease. In contrast, lower intervention rates lead to persistent infections. Additionally, reducing the interval between interventions is particularly effective for disease eradication when the disease’s latent period is short. The proposed hybrid control strategy is practical and cost-effective, offering valuable insights for policymakers to enhance control efforts, reduce disease transmission, and lower the societal costs of outbreaks.