Gamma-distributed antivirus cleaning delays in an SVEIQRS smartphone virus model

Abstract

This paper revisits the delayed SVEIQRS smartphone-virus model of Duan and Ke \cite{DuanKe2022} by replacing the deterministic antivirus-cleaning delay with weak and strong gamma distributed delays. A fixed delay assumes that all infected and quarantined devices complete cleaning after the same waiting time, whereas a distributed delay allows cleaning times to vary across devices, antivirus versions, network conditions and user behaviour. The weak kernel describes broadly dispersed memory, while the strong kernel describes a more concentrated cleaning-time distribution with the same mean.  The virus-existing equilibrium and the basic reproduction number coincide with those of the fixed-delay model, because normalized kernels preserve constant histories. The local stability mechanism is nevertheless different: the exponential factors in the fixed-delay characteristic equation are replaced by rational transfer functions, or equivalently by characteristic polynomials of higher degree. Using the parameter set and equilibrium of the reference model, the fixed-delay system undergoes the reported Hopf stability loss, while the weak and strong gamma formulations remain locally stable over the investigated range of mean cleaning times. The numerical simulations show that distributed cleaning memory desynchronizes the delayed feedback, damps the oscillations of the infected, quarantined and recovered compartments, and provides a stabilizing alternative to a synchronized point delay.