Regime-dependent infection propagation fronts in an SIS model

We show the existence of traveling front solutions in a diffusive classical SIS epidemic model and the SIS model with a saturating incidence in the size of the susceptible population. We investigate the situation where both susceptible and infected populations move around at a comparable rate, but small compared to the spatial scale. In this case, we show that traveling front solutions exist for each fixed positive speed. In the regime where the infected population diffuses slower than the susceptible population, we show the existence of traveling wave solutions for each fixed positive speed and describe their structure and dependence on the wave speed as it is varied from 0 to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal infinity"> <mml:semantics> <mml:mi mathvariant="normal">∞</mml:mi> <mml:annotation encoding="application/x-tex">\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the regime where the infected population diffuses faster than the susceptible population, we derive a bound for the speeds of the fronts in this regime in which the infection propagates as a front. Moreover, for the classical SIS model we show that there is a case when the spread of the disease is governed by the Burgers-FKPP equation.