Malaysian Journal of Mathematical Sciences, June 2026, Vol. 20, No. 2
Traveling Waves in the SIS Generalized Epidemic Model With Infection-Age Structure
Lafta, A. H., Helal, M. M., Mohsen, A. A.
Corresponding Email: aamuhseen@gmail.com
Received date: 10 September 2025
Accepted date: 9 November 2025
Abstract:
In this work, traveling wave solutions (TWS) of a diffusive SIS epidemic model incorporating infection age structure and a general nonlinear incidence function are investigated. Through analysis of the associated differential system, the basic reproduction number \(R_0\) is identified as the fundamental threshold parameter: when \(R_0\leq 1\), disease invasion is prevented and no nontrivial wave emerges, whereas for \(R_0>1\), the dynamics are characterized by a strictly positive minimal wave speed \(c^*>0\). By employing spectral methods together with rigorously constructed upper and lower solutions, it is demonstrated that monotone waves connecting the disease free and endemic equilibria exist if and only if the propagation speed satisfies \(c \geq c^*\). Moreover, such waves are shown not to occur for 0 < c < \(c^*\). Numerical simulations, formulated through phase plane trajectories and traveling wave profiles, are presented to validate the theoretical prediction of a sharp threshold at \(c = c^*\).
Keywords: minimal wave speed (MWS); upper and lower solutions; age structured; basic reproduction number.
