Sufficient conditions for existence of positive periodic solution of a generalized nonresident computer virus model

TL;DR

This paper establishes conditions under which a generalized nonresident computer virus model, with time-dependent rates, guarantees at least one positive periodic solution using topological degree theory.

Contribution

It introduces a biologically inspired, time-dependent computer virus model and proves the existence of positive periodic solutions using topological degree methods.

Findings

Existence of at least one positive periodic solution is proven.

The model incorporates time-dependent rates for infection and recovery.

Mathematical proof uses topological degree and operator reformulation.

Abstract

In this paper, we introduce a nonresident computer virus model and prove the existence of at least one positive periodic solution. The proposed model is based on a biological approach and is obtained by considering that all rates (rates that the computers are disconnected from the Internet, the rate that the computers are cured, etc) are time dependent real functions. Assuming that the initial condition is a positive vector and the coefficients are positive $\omega-$periodic and applying the topological degree arguments we deduce that generalized nonresident computer virus model has at least one positive $\omega-$periodic solution. The proof consists of two big parts. First, an appropriate change of variable which conserves the periodicity property and implies the positive behavior. Second, a reformulation of transformed system as an operator equation which is analyzed by applying the continuation theorem of the coincidence degree theory.