X. Chen, R. Wang, M. Tang, S. Cai, H. E. Stanley, L. A. Braunstein, New J. Phys. 20, 013007 (2018)
Although suppressing the spread of a disease is usually achieved by investing in public resources, in the real world only a small percentage of the population have access to government assistance when there is an outbreak, and most must rely on resources from family or friends.Westudy the dynamics of disease spreading in social-contact multiplex networks when the recovery of infected nodes depends on resources from healthy neighbors in the social layer.Weinvestigate how degree heterogeneity affects the spreading dynamics. Using theoretical analysis and simulations we find that degree heterogeneity promotes disease spreading. The phase transition of the infected density is hybrid and increases smoothly from zero to a finite small value at the first invasion threshold and then suddenly jumps at the second invasion threshold.Wealso find a hysteresis loop in the transition of the infected density.We further investigate how an overlap in the edges between two layers affects the spreading dynamics.We find that when the amount of overlap is smaller than a critical value the phase transition is hybrid and there is a hysteresis loop, otherwise the phase transition is continuous and the hysteresis loop vanishes. In addition, the edge overlap allows an epidemic outbreak when the transmission rate is below the first invasion threshold, but suppresses any explosive transition when the transmission rate is above the first invasion threshold.