Optimal weighted coefficient for heterogeneous epidemic spreading networks

In this paper, we study epidemic spreading on an arbitrary heterogeneous undirected weighted network, in which reinforcement effects among pathogens are considered to depict the characteristics of multiple infection based on a novel propagation model. In order to characterize the formation mechanism of contact weight between individuals, a new parameter defined as the weighted coefficient is introduced to coordinate each individual's own expected weights towards neighbors. A reasonable candidate for the expected weight relating to individual sensitivity and neighbor's infection level is also provided. Furthermore, an optimization problem is formulated to address a trade-off between individuals' infection levels and the adjustment of their weighted coefficients. The existence of an optimal solution is analyzed and the general expression of an optimal weighted coefficient is obtained. Numerical simulations performed by adopting the forward-backward sweep method are presented to illustrate the validity and efficiency of our theoretical results.