Mutually cooperative epidemics on power-law networks

The spread of an infectious disease can, in some cases, promote the propagation of other pathogens favoring violent outbreaks, which cause a discontinuous transition to an endemic state. The topology of the contact network plays a crucial role in these cooperative dynamics. We consider a susceptible-infected-removed-type model with two mutually cooperative pathogens: An individual already infected with one disease has an increased probability of getting infected by the other. We present a heterogeneous mean-field theoretical approach to the coinfection dynamics on generic uncorrelated power-law degree-distributed networks and validate its results by means of numerical simulations. We show that, when the second moment of the degree distribution is finite, the epidemic transition is continuous for low cooperativity, while it is discontinuous when cooperativity is sufficiently high. For scale-free networks, i.e., topologies with diverging second moment, the transition is instead always continuous. In this way we clarify the effect of heterogeneity and system size on the nature of the transition, and we validate the physical interpretation about the origin of the discontinuity.

Figure 1

The six panels represent the total density ${\rho }_{ab}$ of population recovered from both A and B in the final state plotted versus $p$ on power-law networks with $\gamma =3,4,5,6,7,8$, built by applying the uncorrelated configuration model. The system size is $N={10}^5$, and $q=0.99$. The results are averaged over a number $N_r$ of independent realizations, that is, 200 for $\gamma =3$ to 5, and is increased to 2000 for $\gamma =6$ to 8.

Figure 2

Total density ${\rho }_{ab}$ of population recovered from both A and B in the final state, plotted versus $p$ on power-law networks of different sizes ($N={10}^3,{10}^4,{10}^5,{10}^6$, respectively), with $\gamma =7$. The parameter $q$ is 0.99. The results are averaged over $N_r=200,500,2000,2000$ independent realizations in panels (a), (b), (c), and (d), respectively.

Figure 3

(a) Mass distribution of doubly infected clusters at the threshold $p_c=〈k〉/(〈k^2〉-〈k〉)=0.43388$. The number of realizations is at least $N_r={10}^5$. The solid straight line represents the decay $n^{-1.5}$. (b) Probability $P_{ab}$ of forming a giant cluster (with size larger than $0.05N$) as a function of $p$ and system sizes ranging (top to bottom) from $N={10}^4$ to $N={10}^6$. The number of realizations is $N_r=2000$. (c) Plot of the size of the giant cluster as a function of $N$ for $p=p_c$, showing a linear growth. All data are for $\gamma =7$ and $q=0.99$.

Figure 4

Large red symbols represent the final fraction ${\rho }_{ab}$ of doubly infected nodes on power-law networks of size $N={10}^5$, with $\gamma =7$. They are plotted vs $p$ for various fixed values of the cooperativity $W=\beta /\alpha$. The small blue symbols are the results of the numerical integration of Eq. (5). The plotted values are computed as ${\rho }_{ab}={\sum }_k[R_k\left(\infty \right)-Y_k\left(\infty \right)]$. The vertical black line is $〈k〉/(〈k^2〉-〈k〉)$.

Figure 5

The value of the minimum cooperativity $W_c$ needed for a discontinuous transition, as predicted by HMF, as a function of $\gamma$.