Exact solution of generalized cooperative susceptible-infected-removed (SIR) dynamics

In this paper, we introduce a general framework for coinfection as cooperative susceptible-infected-removed (SIR) dynamics. We first solve the SIR model analytically for two symmetric cooperative contagions [L. Chen et al., Europhys. Lett. 104, 50001 (2013)] and then generalize and solve the model exactly in the symmetric scenarios for three and more cooperative contagions. We calculate the transition points and order parameters, i.e., the total number of infected hosts. We show that the behavior of the system does not change qualitatively with the inclusion of more diseases. We also show analytically that there is a saddle-node-like bifurcation for two cooperative SIR dynamics and that the transition is hybrid. Moreover, we investigate where the symmetric solution is stable for initial fluctuations. We finally explore sets of parameters which give rise to asymmetric cases, namely, the asymmetric cases of primary and secondary infection rates of one pathogen with respect to another. This setting can lead to fewer infected hosts, a higher epidemic threshold, and also continuous transitions. These results open the road to a better understanding of disease ecology.

Figure 1

Flow chart of a two-disease coinfection with A, B. The capital letters A and B represent infective states, while the lower-case letters a and b stand for recovered ones. $X$ is defined as the fraction of agents that can transfer disease A (or B) and $P$ is defined as the fraction of agents who have experienced only one of the diseases.

Figure 2

Phase transitions from disease-free to epidemic regime of two coupled SIR [11]. The order parameter $R=1-S_{\infty }$ is plotted against $\alpha$ for $\epsilon =0.005$. The curves correspond to different levels of cooperation, $c$.

Figure 3

Phase portraits of the dynamical system in two-disease coinfection for $\alpha =0.8$ and $c=1$ (top) and $c=10$ (bottom). Note that not all the points in the above phase portrait are physical and thus we should confine the portrait to the area where $p+s\le 1$.

Figure 4

$X$ plotted against $\tau$ for $\epsilon =0.01$ and $c=10$. The curve corresponds to different rates for a primary infection $\alpha$. The number of fixed points, where $X\left({\tau }_{\infty }\right)=0$, changes as we increase $\alpha$. Comparing the curves, one observes a saddle-node-like bifurcation in the system.

Figure 5

Comparison of the analytical approximation with the numerical result data for $c=15$ and $\epsilon =0.005$. We have intentionally taken a relatively large value for $\epsilon$. For smaller values of $\epsilon$ the difference of the two curves was not observable.

Figure 6

Potential $\delta V$ plotted against $\tau$ for $\epsilon =0.001$ and $c=30$. The curves correspond to different rates for the primary infection $\alpha =0.62,0.85,0.95$. The inset graph shows a larger view for very small values of $\tau$. It is clear that for $\alpha =0.62$ there is a minimum near $\tau =0$ which vanishes for large values of $\alpha$.

Figure 7

Schematic of three-disease coinfection with A, B, C symmetry and restrictions on the infection rates as discussed in the text. Capital letters A, B, and C represent infective states, and lower-case letters a, b, and c stand for recovered ones. $X$ is defined as the fraction of agents that can transfer the disease A (or B or C) and $P_0$, $P_1$, and $P_2$ are defined as the fraction of agents who have experienced none, only one, or only two of the diseases, respectively. All the edges have arrows with the same direction as the first and the last ones shown, but for simplification are not drawn.

Figure 8

Asymmetric initial conditions. The real parts of the eigenvalues of the matrix $G$ for two cases, which show if the symmetric solution is an attractive fixed point. (a) Below the transition point with $c=15$, $\epsilon ={10}^{-4}$, and $\alpha =0.9$. (b) Above the transition point with the same values for $c$ and $\epsilon$ but with $\alpha =0.97$. In both graphs, $X$ is also sketched. For the top graph, the real part of the eigenvalues is negative for $\tau <{\tau }_{\infty }$. For the case above the transition point, at the beginning, the real part of the eigenvalues is positive; however, in most parts of the dynamics it is negative. Note that $X$ is plotted 400 (a) and 4 (b) times larger, in order to be seen.

Figure 9

The order parameter of the asymmetric system as a function of ${\alpha }_B$ (the smaller infection rate) for $k=1,2,...,6$, where $({\alpha }_A,{\beta }_A)=k({\alpha }_B,{\beta }_B)$ and $c=10$, $\epsilon ={10}^{-3}$. The discontinuity becomes smaller as $k$ increases and vanishes at $k\simeq 4$.

Figure 10

Time evolution of the variables $S$, $P_A$, $P_B$, $100X_A$, and $100X_B$ as functions of time for a system with $\epsilon =0.001$, ${\alpha }_B=0.38$, $c=10$, and $k=3$. Note that ${\alpha }_A=k{\alpha }_B=1.11>1$. There are five different and disjoint peaks for infecting populations.