Travelling waves for an epidemic model with non-smooth treatment rates

Abstract

We consider a susceptible–infected–removed (SIR) epidemic model with two types ofnonlinear treatment rates: (i) piecewise linear treatment rate with saturationeffect, (ii) piecewise constant treatment rate with a jump (Heaviside function).For case (i), we compute travelling front solutions whose profiles are heteroclinicorbits which connect either the disease-free state to an infective state or twoendemic states with each other. For case (ii), it is shown that the profile has thefollowing properties: the number of susceptibles is monotonically increasing and thenumber of infectives approaches zero at infinity, while their product converges toa constant. Numerical simulations are performed for all these cases. Abnormalbehaviour like travelling waves with non-monotonic profile or oscillations is observed.