The Effect of Limited Medical Resource, Socio-Economic Processes and Their Time Scales on Epidemic Models

Abstract This paper investigates an SIS version of the Poletti model that incorporates treatments under limited medical resources. Specifically, we examine how various types of bifurcations in the model depend on three key factors: socio-economic processes (SEP), medical resources (MR), and the time scales (TS) associated with SEP and the epidemic. We treat the basic reproduction number under normal behavior ( $$R_{0n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> ) as the bifurcation parameter. The model exhibits its simplest bifurcation-where the infected population initially grows and eventually declines as $$R_{0n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mn>0</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> increases-when MR are adequate, the epidemic’s TS is significantly shorter than that of SEP, or the SEP factor is low. The most complex bifurcation scenario includes a degenerate Hopf bifurcation, a discontinuous Hopf-like bifurcation, and disappearance of stable and unstable limit cycles in a saddle-node bifurcation, ultimately leading the system to settle into a low-prevalence equilibrium. This complexity arises under the following conditions: (1) the two TS are approximately equal, (2) the SEP factor is high, and (3) MR are inadequate. Intermediate bifurcations between the simplest and most complex cases may occur, depending on the specific combination of the three aforementioned factors.