The immunities of computers against malware are actually heterogeneous depending upon the level of secure protection, which is largely determined by the security awareness of users. To understand the effects this can have on the propagation of malicious software, we herein develop a new compartmental model that takes into account heterogeneous immunization of the computer. In terms of protection levels, the traditional susceptible compartment is newly separated into two sub-compartments, weakly-protected and strongly-protected. Besides, our model includes two assumptions: a computer immediately possesses infectivity once it is infected by malware, and weakly-protected susceptible computers have a higher infection rate than strongly-protected susceptible computers. Appropriate Lyapunov functions are introduced to fully analyze the qualitative properties of this model. Specifically, it is proved that the malware-free equilibrium is globally asymptotically stable if the threshold is below unit, whereas the malware equilibrium is globally asymptotically stable if the threshold is above unit. Furthermore, the sensitivity of each parameter on the model's threshold is also analyzed. Through numerical simulations, a collection of effective measures for controlling malware spread is proposed, such as keeping as many systems strongly-protected as possible. These findings suggest that the security awareness of users should be taken both in the interpretation of malware parameters, as well as in the prediction of the evolution of future malware outbreaks.
Liu, W., Liu, C., Liu, X., Cui, S., & Huang, X. (2016). Modeling the spread of malware with the influence of heterogeneous immunization. Applied Mathematical Modelling, 40(4), 3141–3152. https://doi.org/10.1016/j.apm.2015.09.105