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<p>In this paper, we derive an exact analytical solution in parametric form for an infectious disease model (SIR) with migration and vaccination (SIRVN). Using derivatives and substitutions, we transform the SIRVN model into a nonlinear third-order differential equation, yielding a semi-analytical solution in parametric form. The model’s long-term oscillatory behavior approximates a Van der Pol-like equation with nonlinear damping. An analytic solution is derived using multi-scale expansion analysis and Laplace transform methods. Our comparison between the exact solution and data from the Jakarta outbreak shows a correlation of <i>R</i><sup>2</sup> = 0.99. We demonstrate that vaccination effectively reduces the peak of the outbreak, and the system’s asymptotic stability suggests a transition from pandemic to endemic in Jakarta. Lastly, the Van der Pol-like equation reveals that the model can explain the existence of multiple outbreak waves.</p>

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On the Exact Analytical Solution and Van Der Pol Like Equation of Infectious Diseases Model with the Time-Dependent Total Population

  • 期刊论文

  • 发表日期：

    2025年3月20日

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  • Zulfa Siti Zakia,
  • Intan Nurmala,
  • Mirda Prisma Wijayanto,
  • Albertus Sulaiman

摘要

In this paper, we derive an exact analytical solution in parametric form for an infectious disease model (SIR) with migration and vaccination (SIRVN). Using derivatives and substitutions, we transform the SIRVN model into a nonlinear third-order differential equation, yielding a semi-analytical solution in parametric form. The model’s long-term oscillatory behavior approximates a Van der Pol-like equation with nonlinear damping. An analytic solution is derived using multi-scale expansion analysis and Laplace transform methods. Our comparison between the exact solution and data from the Jakarta outbreak shows a correlation of R2 = 0.99. We demonstrate that vaccination effectively reduces the peak of the outbreak, and the system’s asymptotic stability suggests a transition from pandemic to endemic in Jakarta. Lastly, the Van der Pol-like equation reveals that the model can explain the existence of multiple outbreak waves.