Pathways to explosive transitions in interacting contagion dynamics

Technical Summary: Pathways to explosive transitions in interacting contagion dynamics

Problem Statement
Complex systems often exhibit collective behaviors that cannot be understood by studying isolated components. In the context of contagion dynamics, a critical gap exists in understanding how dynamical interdependencies between concomitant processes alter intrinsic equilibria and bifurcations. Specifically, while isolated epidemic dynamics typically undergo second-order (continuous) transitions, and social dynamics often exhibit first-order (explosive) transitions due to complex contagion mechanisms, the pathways through which their interaction generates or suppresses explosive transitions remain to be fully characterized. The emergence of such explosive transitions compromises system stability, as small perturbations can lead to vastly different dynamical outcomes. This work addresses the mechanisms driving these transitions when two spreading processes interact, ranging from mutual cooperation to negative feedback loops.

Methodology
The authors propose a general mean-field framework coupling two simultaneous spreading processes, denoted as dynamics A and dynamics B, within a closed population of $N$ individuals. The model extends the standard Susceptible-Infected-Susceptible (SIS) compartmental model to four states: Susceptible ($S$), infected by A only ($A$), infected by B only ($B$), and co-infected ($AB$).

The dynamics are governed by a set of three differential equations describing the evolution of the fractions of the population in each infected state ($\rho_A, \rho_B, \rho_{AB}$). The model incorporates:

1. Recovery: A uniform recovery rate $\mu = 1$ for both processes.
2. Interaction Rates: The infection rates are modulated by coupling parameters $\epsilon_{AB}$ and $\epsilon_{BA}$.
   • $\epsilon_{ij} = 1$: Non-interacting processes.
   • $\epsilon_{ij} > 1$: Cooperative/Excitatory interaction (presence of $i$ enhances $j$).
   • $\epsilon_{ij} < 1$: Competitive/Inhibitory interaction (presence of $i$ suppresses $j$).
3. Contagion Nature:
   • Dynamics B is modeled as a standard epidemic with a constant contagion rate $\lambda_B$, exhibiting a second-order transition at $\lambda_B = 1$.
   • Dynamics A is modeled with a tunable synergistic contagion rate $\tilde{\lambda}_A = \lambda_A e^{\gamma(1-\rho_{AT})}$, where $\rho_{AT}$ is the total fraction infected by A. The parameter $\gamma$ controls the nature of the transition: $\gamma = 0$ corresponds to standard epidemic dynamics, while $\gamma < 0$ introduces group pressure effects characteristic of social contagions, capable of inducing explosive (first-order) transitions below a critical $\gamma_c$.

The study analyzes the system's stationary states and bifurcation points analytically and numerically. Numerical simulations employ forward and backward continuations of control parameters to identify bistability and hysteresis cycles, which are signatures of first-order transitions.

Key Contributions and Results

1. Mutual Cooperation of Epidemics ($\gamma=0, \epsilon > 1$):
   The model reproduces established literature results where two standard epidemic processes mutually enhance each other. This mutual cooperation transforms the typical second-order transition of isolated epidemics into a first-order (explosive) transition, creating a bistable region where the system can jump abruptly from a disease-free state to an endemic state. Conversely, mutual competition ($\epsilon < 1$) leads to the extinction of the less transmissible pathogen but does not alter the continuous nature of the transitions.

2. Unidirectional Coupling (Social Dynamics Driving Epidemics):
   When a social contagion (Dynamics A, $\gamma < 0$) unidirectionally drives an epidemic (Dynamics B, $\epsilon_{BA}=1, \epsilon_{AB} > 1$), the nature of the transition in B depends on the interplay between social pressure ($\gamma$) and coupling strength ($\epsilon_{AB}$).

   • If Dynamics A is not inherently explosive, the transition in B remains continuous.
   • If Dynamics A is inherently explosive, it can induce an explosive transition in B, but only if the coupling strength exceeds a critical threshold $\epsilon^c_{AB}$.
   • The authors derive analytical expressions for the critical infectivity $\lambda^c_A$ and the critical coupling strength $\epsilon^c_{AB}$ that determine the onset of discontinuous transitions in the driven epidemic.

3. Negative Feedback Loops (Suppression of Explosiveness):
   The study explores scenarios where Dynamics A enhances its inhibitor B (negative feedback, $\epsilon_{AB} > 1, \epsilon_{BA} < 1$).

   • The presence of an active inhibitor B can suppress the explosive nature of Dynamics A. As the coupling strength increases, the hysteresis cycles associated with the first-order transition in A shrink and eventually vanish, restoring a smooth (second-order) transition.
   • This suppression is most effective when the inhibitor is in an active state and the coupling strength is sufficiently high. This highlights a mechanism for increasing system robustness against external perturbations.

4. Mutual Competition with Complex Contagion:
   In scenarios involving mutually inhibitory interactions between two processes where at least one exhibits complex contagion (synergistic rates), the system can still exhibit discontinuous transitions. Specifically, mutually exclusive competing pathogens with synergistic transmission rates can display explosive transitions, a phenomenon distinct from the smooth extinction observed in standard competing epidemics.

Significance
The paper provides a general perspective on the pathways leading to explosive phenomena in interacting contagion processes. It demonstrates that explosiveness is not solely an intrinsic property of a single process but can emerge, be transmitted, or be suppressed through the nature of the interdependencies between simultaneous dynamics.

The authors claim that their framework unifies previous findings regarding mutual cooperation and unidirectional driving while revealing a new pathway for stability: negative feedback loops can effectively "switch off" first-order transitions, thereby enhancing system robustness. This finding aligns with observations in gene regulatory networks where negative feedback ensures stability. The work underscores the necessity of including interdependencies when modeling the simultaneous propagation of contagion processes, as these interactions fundamentally alter the dynamical states and transition nature compared to isolated systems. The study is limited to well-mixed populations, with the authors noting that exploring these dynamics on complex contact networks remains future work.