Analysis of a general SIS model with infective vectors on the complex networks

A general SIS model with infective vectors on complex networks is studied in this paper. In particular, the model considers the linear combination of three possible routes of disease propagation between infected and susceptible individuals as well as two possible transmission types which describe how the susceptible vectors attack the infected individuals. A new technique based on the basic reproduction matrix is introduced to obtain the following results. First, necessary and sufficient conditions are obtained for the global stability of the model through a unified approach. As a result, we are able to produce the exact basic reproduction number and the precise epidemic thresholds with respect to three spreading strengths, the curing strength or the immunization strength all at once. Second, the monotonicity of the basic reproduction number and the above mentioned epidemic thresholds with respect to all other parameters can be rigorously characterized. Finally, we are able to compare the effectiveness of various immunization strategies under the assumption that the number of persons getting vaccinated is the same for all strategies. In particular, we prove that in the scale-free networks, both targeted and acquaintance immunizations are more effective than uniform and active immunizations and that active immunization is the least effective strategy among those four. We are also able to determine how the vaccine should be used at minimum to control the outbreak of the disease.