Extreme outbreaks in non-Markovian epidemics on complex networks

Global Dynamics of a Non-Markovian Epidemic Model on Complex Networks

Abstract. In the early stages of a newly emerged or reemerged disease, there is a rapid increase in new infections, which can potentially lead to a healthcare crisis due to constraints of available medical resources. It is important to note that the recovery period of these emerging diseases typically follows a Gamma distribution rather than being narrowly centered around the mean. In this study, we propose a susceptible-infectious-recovered (SIR) network model that incorporates a general recovery rate and a saturation treatment function. We establish the well-posedness and global stability of the disease-free equilibrium in the model by employing semigroup theory and the standard comparison principle, respectively. From an epidemiological perspective, when the delayed treatment effect exceeds a significant threshold, a phenomenon known as backward bifurcation emerges near the disease-free equilibrium. This assertion is supported by an updated version of the Lyapunov–Schmidt approach. Additionally, we conduct numerical simulations to investigate how network topology and non-Markovian processes affect the patterns of disease transmission.

Applied Mathematical Modelling

Turing instability and bistable epidemic spread on complex networks: Effects of topology and initial seeding.

A reaction-diffusion framework on complex networks is developed to investigate how local bistability, dispersal, and network topology jointly determine infection dynamics, revealing how local nonlinearities and network heterogeneity interact to shape epidemic transitions.

Evaluating the Impact of Epidemic Control via State-Dependent Markovian Switching Modeling

We develop an exact finite-population stochastic framework for SIR epidemics evolving under Markovian switching between intervention regimes. The epidemic state is augmented by a finite phase component, allowing transmission, recovery, and direct immunity-acquisition rates to depend on the active regime. Phase-transition intensities may depend on the current epidemic state, so that policy escalation can react to the number of infectious individuals. Exploiting the monotonicity of the susceptible compartment, we derive level-wise recursions for the joint Laplace--Stieltjes transform and probability generating function of the extinction time and the number of infections generated before extinction. These recursions yield the infection-count distribution, conditional extinction-time transforms, and mixed moments linking epidemic duration and infection burden, while replacing a large global linear system with small phase-level solves. The framework is illustrated using weekly mpox incidence data from Luxembourg. A baseline one-phase SIR model is calibrated by maximum likelihood under a Poisson observation model. The calibrated baseline is then used for conditional comparisons of fixed control regimes, early versus delayed strict intervention, vaccination-supported control, and state-dependent escalation. The results show how switching mechanisms affect both the total number of infected individuals and the extinction time, including their dispersion. Since the switching mechanisms are specified rather than estimated from the intervention history, the results are conditional model-based comparisons rather than estimates of the historical effects of interventions in Luxembourg.

Efficient stochastic epidemic simulation via the Sellke construction

An exact, event-driven implementation of the epidemic Sellke construction is developed that maintains infection and recovery events in priority queues and achieves computational performance comparable to the classical Gillespie algorithm while naturally accommodating non-Markovian infectious periods and complex infectiousness profiles.

Exact moment equivalence and structural nonidentifiability in nonlinear epidemics

Transmission through temporary groups does not necessarily preserve complete information about the group-size distribution in aggregate epidemic data. We show that, in finite-population SIS and SIR models, the group-size distribution enters the dynamics only through a finite set of moments selected by the order of the nonlinear transmission kernel. Consequently, markedly different distributions can generate identical stochastic dynamics when their relevant moments coincide. This equivalence extends to transient evolution, fluctuations, extinction-time statistics, and final outbreak sizes. In the deterministic limit, the first moment sets the invasion threshold, whereas the second controls the nature of the transition and the emergence of bistability and hysteresis. The two distributions become distinguishable only when a higher-order transmission mechanism activates their first unmatched moment; even a weak additional channel can shift the phase boundary and place the systems in different dynamical regimes. Solutions of the master equation and stochastic simulations support these analytical predictions. These results establish an intrinsic limit on epidemic inference: a single aggregate dynamical protocol can identify only an equivalence class of group-size distributions, rather than uniquely reconstructing the full distribution.