Adaptive Susceptibility and Heterogeneity in Contagion Models on Networks

Contagious processes, such as spread of infectious diseases, social behaviors, or computer viruses, affect biological, social, and technological systems. Epidemic models for large populations and finite populations on networks have been used to understand and control both transient and steady-state behaviors. Typically it is assumed that after recovery from an infection, every agent will either return to its original susceptible state or acquire full immunity to reinfection. We study the network SIRI (Susceptible-Infected-Recovered-Infected) model, an epidemic model for the spread of contagious processes on a network of heterogeneous agents that can adapt their susceptibility to reinfection. The model generalizes existing models to accommodate realistic conditions in which agents acquire partial or compromised immunity after first exposure to an infection. We prove necessary and sufficient conditions on model parameters and network structure that distinguish four dynamic regimes: infection-free, epidemic, endemic, and bistable. For the bistable regime, which is not accounted for in traditional models, we show how there can be a rapid resurgent epidemic after what looks like convergence to an infection-free population. We use the model and its predictive capability to show how control strategies can be designed to mitigate problematic contagious behaviors.Comment: 14 pages, 5 figure

Active Control and Sustained Oscillations in actSIS Epidemic Dynamics

An actively controlled Susceptible-Infected-Susceptible (actSIS) contagion model is presented for studying epidemic dynamics with continuous-time feedback control of infection rates. Our work is inspired by the observation that epidemics can be controlled through decentralized disease-control strategies such as quarantining, sheltering in place, social distancing, etc., where individuals actively modify their contact rates with others in response to observations of infection levels in the population. Accounting for a time lag in observations and categorizing individuals into distinct sub-populations based on their risk profiles, we show that the actSIS model manifests qualitatively different features as compared with the SIS model. In a homogeneous population of risk-averters, the endemic equilibrium is always reduced, although the transient infection level can exhibit overshoot or undershoot. In a homogeneous population of risk-tolerating individuals, the system exhibits bistability, which can also lead to reduced infection. For a heterogeneous population comprised of risk-tolerators and risk-averters, we prove conditions on model parameters for the existence of a Hopf bifurcation and sustained oscillations in the infected population

SIS Epidemic Model under Mobility on Multi-layer Networks

We study the influence of heterogeneous mobility patterns in a population on the SIS epidemic model. In particular, we consider a patchy environment in which each patch comprises individuals belonging the different classes, e.g., individuals in different socio-economic strata. We model the mobility of individuals of each class across different patches through an associated Continuous Time Markov Chain (CTMC). The topology of these multiple CTMCs constitute the multi-layer network of mobility. At each time, individuals move in the multi-layer network of spatially-distributed patches according to their CTMC and subsequently interact with the local individuals in the patch according to an SIS epidemic model. We derive a deterministic continuum limit model describing these mobility-epidemic interactions. We establish the existence of a Disease-Free Equilibrium (DFE) and an Endemic Equilibrium (EE) under different parameter regimes and establish their (almost) global asymptotic stability using Lyapunov techniques. We derive simple sufficient conditions that highlight the influence of the multi-layer network on the stability of DFE. Finally, we numerically illustrate that the derived model provides a good approximation to the stochastic model with a finite population and also demonstrate the influence of the multi-layer network structure on the transient performance.Comment: Extended version of the paper to appear at ACC 2020 and is an extension to the arXiv paper- On Epidemic Spreading under Mobility on Networks (arXiv:1909.02647v2) to Multi-layer settin