Heterogeneous recovery rates against SIS epidemics in directed networks

The nodes in communication networks are possibly and most likely equipped with different recovery resources, which allow them to recover from a virus with different rates. In this paper, we aim to understand know how to allocate the limited recovery resources to efficiently prevent the spreading of epidemics. We study the susceptible-infected-susceptible (SIS) epidemic model on directed scale-free networks. In the classic SIS model, a susceptible node can be infected by an infected neighbor with the infection rate β and an infected node can be recovered to be susceptible again with the recovery rate δ. In the steady state a fraction y
_∞
of nodes are infected, which shows how severely the network is infected. Instead of considering the same recovery rate for all the nodes in the SIS model, we propose to allocate the recovery rate δ
_ί
for node i according to its indegree and outdegree-δ
_i
~ k
^{α in}
_{i, in}
k
^{α out}
_{i, out}
given the finite average recovery rate (δ) representing the limited recovery resources over the whole network. We consider directed networks with the same power law indegree and outdegree distribution, but different values of the directionality ξ (the proportion of unidirectional links) and linear correlation coefficients ρ between the indegree and outdegree. We find that, by tuning the two scaling exponents α
_in
and α
_out
, we can always reduce the infection fraction y
_∞
thus reducing the extent of infections, comparing to the homogeneous recovery rates allocation. Moreover, we can find our optimal strategy via the optimal choice of the exponent α
_in
and α
_out
. Our optimal strategy indicates that when the recovery resources are sufficient, more resources should be allocated to the nodes with a larger indegree or outdegree, but when the recovery resource is very limited, only the nodes with a larger outdegree should be equipped with more resources. We also find that our optimal strategy works better when the recovery resources are sufficient but not yet able to make the epidemic die out, and when the indegree outdegree correlation is small.