This work investigates the dynamics of positive classical solutions to a diffusive susceptible-exposed-infected-recovered-susceptible epidemic model with a mass-action incidence mechanism in spatially heterogeneous environments. Under minimal assumptions on the initial data, the global existence of classical solutions is established. Moreover, the eventual boundedness of these solutions is proved when either the spatial domain has dimension five or lower or the susceptible and exposed subpopulations share the same diffusion rate. Next, we define the basic reproduction number, $\mathcal{R}_0$, and demonstrate that the disease-free equilibrium is globally stable when $\mathcal{R}_0$ is sufficiently small. However, due to the complex interaction between population movement and spatial variation in transmission rates, we find that the disease may persist even when $\mathcal{R}_0$ is slightly less than one. In such cases, we show that the system admits at least two endemic equilibrium (EE) solutions, an outcome not observed under the frequency-dependent incidence mechanism. These results highlight the significant influence of the transmission mechanism on disease dynamics. Furthermore, we examine the spatial profiles of the EE solutions when diffusion rates are small. Our analysis suggests that limiting the movement of the susceptible population can significantly reduce disease prevalence, provided that the total population remains below a specific threshold. In contrast, restricting the movement of the infected, exposed, or recovered populations alone may not eradicate the disease. Overall, our findings provide important insights into the spatial dynamics of infectious diseases and may offer guidance for developing and implementing effective containment strategies.

A stochastic model for the spread of an SIR (susceptible $\to$ infective $\to$ removed) epidemic is considered. Infectives have independent and identically distributed infectivity profiles, which describe their infectiousness as a function of time since infection. The individual-to-individual infection rate depends also on the number of susceptibles present in the population. Exact results are derived for the distribution of statistics defined on the final outcome of the epidemic, including its final size. These are proved by using a generalisation of a Sellke construction to show that the distribution of the final outcome of the epidemic is the same as that of an associated discrete-time epidemic process, in which infectives are considered one at a time, and exploiting connection with death processes to analyse the final outcome of the latter. The results generalise easily to multipopulation epidemics.

We consider an infinite-dimension SIS model introduced by Delmas, Dronnier, and Zitt, with a more general incidence rate, and study its equilibria. Unsurprisingly, there exists at least one endemic equilibrium if and only if the basic reproduction number is larger than 1. When the pathogen transmission exhibits one way propagation, it is possible to observe different possible endemic equilibria. We characterize in a general setting all the equilibria, using a decomposition of the space into atoms, given by the transmission operator. We also prove that the proportion of infected individuals converges to an equilibrium, which is uniquely determined by the support of the initial condition. We extend those results to infinite-dimensional SIS models with reservoir or with immigration.

To investigate multiple effects of the interaction between V. cholerae and phage on cholera transmission, we propose a degenerate reaction-diffusion model with different dispersal rates, which incorporates a short-lived hyperinfectious (HI vibrios) state of V. cholerae and lower-infectious (LI vibrios) state of V. cholerae. Our main purpose is to investigate the existence and stability analysis of multi-class boundary steady states, which is much more complicated and challenging than the case when the boundary steady state is unique. In a spatially heterogeneous case, the basic reproduction number $\mathscr{R}_{0}$ is defined as the spectral radius of the sum of two linear operators associated with HI vibrios infection and LI vibrios infection. If $\mathscr{R}_{0}\leq 1$, the disease-free steady state is globally asymptotically stable. If $\mathscr{R}_{0}\gt 1$, the uniform persistence of phage-free model, as well as the existence of the phage-free steady state, are established. In a spatially homogeneous case, when $\ \;\widetilde{\!\!\!\mathscr{R}}_{0}\gt 1$, the global asymptotic stability of phage-free steady state and the uniform persistence of the phage-present model are discussed under some additional conditions. The mathematical approach here has wide applications in degenerate Partial Differential Equations.

Seasonal changes and cyclical human activities (such as periodic fishing bans, Wolbachia-based mosquito population control, and school term breaks) have significant impacts on population dynamics. We propose a general switching dynamical model to describe these periodic changes. The existence, uniqueness and stability of positive periodic solutions are thoroughly investigated. The results are stated in terms of an introduced threshold value. To demonstrate their practicability, the obtained results are applied to two biological situations.

We analyse a Markovian SIR epidemic model where individuals either recover naturally or are diagnosed, leading to isolation and potential contact tracing. Our focus is on digital contact tracing via a tracing app, considering both its standalone use and its combination with manual tracing. We prove that as the population size n grows large, the epidemic process converges to a limiting process, which, unlike with typical epidemic models, is not a branching process due to dependencies created by contact tracing. However, by grouping to-be-traced individuals into macro-individuals, we derive a multi-type branching process interpretation, allowing computation of the reproduction number R. This is then converted to an individual reproduction number $R^\mathrm{(ind)}$, which, in contrast to R, decays monotonically with the fraction of app-users, while both share the same threshold at 1. Finally, we compare digital (only) contact tracing and manual (only) contact tracing, proving that the critical fraction of app-users, $\pi_{\mathrm{c}}$, required for $R=1$ is higher than the critical fraction manually contact-traced, $p_{\mathrm{c}}$, for manual tracing.

We study a variant of the classical Markovian logistic SIS epidemic model on a complete graph, which has the additional feature that healthy individuals can become infected without contacting an infected member of the population. This additional ‘self-infection’ is used to model situations where there is an unknown source of infection or an external disease reservoir, such as an animal carrier population. In contrast to the classical logistic SIS epidemic model, the version with self-infection has a non-degenerate stationary distribution, and we derive precise asymptotics for the time to converge to stationarity (mixing time) as the population size becomes large. It turns out that the chain exhibits the cutoff phenomenon, which is a sharp transition in time from one to zero of the total variation distance to stationarity. We obtain the exact leading constant for the cutoff time and show that the window size is of constant (optimal) order. While this result is interesting in its own right, an additional contribution of this work is that the proof illustrates a recently formalised methodology of Barbour, Brightwell and Luczak (2022), ‘Long-term concentration of measure and cut-off’, Stochastic Processes and their Applications 152, 378–423, which can be used to show cutoff via a combination of concentration-of-measure inequalities for the trajectory of the chain and coupling techniques.

In this paper, we consider an optimal distributed control problem for a reaction-diffusion-based SIR epidemic model with human behavioural effects. We develop a model wherein non-pharmaceutical intervention methods are implemented, but a portion of the population does not comply with them, and this non-compliance affects the spread of the disease. Drawing from social contagion theory, our model allows for the spread of non-compliance parallel to the spread of the disease. The quantities of interest for control are the reduction in infection rate among the compliant population, the rate of spread of non-compliance and the rate at which non-compliant individuals become compliant after, e.g., receiving more or better information about the underlying disease. We prove the existence of global-in-time solutions for fixed controls and study the regularity properties of the resulting control-to-state map. The existence of optimal control is then established in an abstract framework for a fairly general class of objective functions. Necessary first–order optimality conditions are obtained via a Lagrangian-based stationarity system. We conclude with a discussion regarding minimisation of the size of infected and non-compliant populations and present simulations with various parameters values to demonstrate the behaviour of the model.

This paper is concerned with the growth rate of susceptible–infectious–recovered epidemics with general infectious period distribution on random intersection graphs. This type of graph is characterised by the presence of cliques (fully connected subgraphs). We study epidemics on random intersection graphs with a mixed Poisson degree distribution and show that in the limit of large population sizes the number of infected individuals grows exponentially during the early phase of the epidemic, as is generally the case for epidemics on asymptotically unclustered networks. The Malthusian parameter is shown to satisfy a variant of the classical Euler–Lotka equation. To obtain these results we construct a coupling of the epidemic process and a continuous-time multitype branching process, where the type of an individual is (essentially) given by the length of its infectious period. Asymptotic results are then obtained via an embedded single-type Crump–Mode–Jagers branching process.

The study of many population growth models is complicated by only partial observation of the underlying stochastic process driving the model. For example, in an epidemic outbreak we might know when individuals show symptoms to a disease and are removed, but not when individuals are infected. Motivated by the above example and the long-established approximation of epidemic processes by branching processes, we explore the number of individuals alive in a time-inhomogeneous branching process with a general phase-type lifetime distribution given only (partial) information on the times of deaths of individuals. Deaths are detected independently with a detection probability that can vary with time and type. We show that the number of individuals alive immediately after the kth detected death can be expressed as the mixture of random variables each of which consists of the sum of k independent zero-modified geometric distributions. Furthermore, in the case of an Erlang lifetime distribution, we derive an easy-to-compute mixture of negative binomial distributions as an approximation of the number of individuals alive immediately after the kth detected death.

We study an epidemic patch model that describes the disease spread in population with variable latency due to the differences in immunologic tolerance between individuals. We focus on whether the disease can spread in space that leads to the emergence of epidemic wave, that is the travelling wave solution with constant speed. We first establish some properties of the linearized wave profile equations, which are helpful in obtaining the priori estimates of travelling waves and wave speeds. Then, applying the truncation method and limiting arguments, we can obtain threshold propagation dynamics of the epidemic model. Our result gives a complete characterization of the existence, nonexistence and minimal wave speed of travelling waves. To the best of our knowledge, this is the first time to characterize the propagation dynamics of epidemic patch model with variable latency, which contributes to the understanding of the transmission phenomenon of disease.

In this paper, we study the existence of travelling wave solutions and the spreading speed for the solutions of an age-structured epidemic model with nonlocal diffusion. Our proofs make use of the comparison principles both to construct suitable sub/super-solutions and to prove the regularity of travelling wave solutions.

In this paper, we consider reaction-diffusion epidemic models with mass action or standard incidence mechanism and study the impact of limiting population movement on disease transmissions. We set either the dispersal rate of the susceptible or infected people to zero and study the corresponding degenerate reaction-diffusion model. Our main approach to study the global dynamics of these models is to construct delicate Lyapunov functions. Our results show that the consequences of limiting the movement of susceptible or infected people depend on transmission mechanisms, model parameters and population size.

We examine several extensions to the basic Susceptible-Infected-Recovered model, which are able to induce recurrent outbreaks (the basic Susceptible-Infected-Recovered model by itself does not exhibit recurrent outbreaks). We first analyse how slow seasonal variations can destabilise the endemic equilibrium, leading to recurrent outbreaks. In the limit of slow immunity loss, we derive asymptotic thresholds that characterise this transition. In the outbreak regime, we use asymptotic matching to obtain a two-dimensional discrete map which describes outbreak times and strength. We then analyse the resulting map using linear stability and numerics. As the frequency of forcing is increased, the map exhibits a period-doubling route to chaos which alternates with periodic outbreaks of increasing frequency. Other extensions that can lead to recurrent outbreaks include the addition of noise, state-dependent variation and fine-graining of model classes.

We consider the propagation of a stochastic SIR-type epidemic in two connected populations: a relatively small local population of interest which is surrounded by a much larger external population. External infectives can temporarily enter the small population and contribute to the spread of the infection inside this population. The rules for entry of infectives into the small population as well as their length of stay are modeled by a general Markov queueing system. Our main objective is to determine the distribution of the total number of infections within both populations. To do this, the approach we propose consists of deriving a family of martingales for the joint epidemic processes and applying classical stopping time or convergence theorems. The study then focuses on several particular cases where the external infection is described by a linear branching process and the entry of external infectives obeys certain specific rules. Some of the results obtained are illustrated by numerical examples.

We introduce an extension to Kermack and McKendrick’s classic susceptible–infected–recovered (SIR) model in epidemiology, whose underlying mechanism of infection consists of individuals attending randomly generated social gatherings. This gives rise to a system of ordinary differential equations (ODEs) where the force of the infection term depends non-linearly on the proportion of infected individuals. Some specific instances yield models already studied in the literature, to which the present work provides a probabilistic foundation. The basic reproduction number is seen to depend quadratically on the average size of the gatherings, which may be helpful in understanding how restrictions on social gatherings affect the spread of the disease. We rigorously justify our model by showing that the system of ODEs is the mean-field limit of the jump Markov process corresponding to the evolution of the disease in a finite population.

We consider an SIR (susceptible $\to$ infective $\to$ recovered) epidemic in a closed population of size n, in which infection spreads via mixing events, comprising individuals chosen uniformly at random from the population, which occur at the points of a Poisson process. This contrasts sharply with most epidemic models, in which infection is spread purely by pairwise interaction. A sequence of epidemic processes, indexed by n, and an approximating branching process are constructed on a common probability space via embedded random walks. We show that under suitable conditions the process of infectives in the epidemic process converges almost surely to the branching process. This leads to a threshold theorem for the epidemic process, where a major outbreak is defined as one that infects at least $\log n$ individuals. We show further that there exists $\delta \gt 0$, depending on the model parameters, such that the probability that a major outbreak has size at least $\delta n$ tends to one as $n \to \infty$.

Motivated by applications to COVID dynamics, we describe a model of a branching process in a random environment $\{Z_n\}$ whose characteristics change when crossing upper and lower thresholds. This introduces a cyclical path behavior involving periods of increase and decrease leading to supercritical and subcritical regimes. Even though the process is not Markov, we identify subsequences at random time points $\{(\tau_j, \nu_j)\}$—specifically the values of the process at crossing times, viz. $\{(Z_{\tau_j}, Z_{\nu_j})\}$—along which the process retains the Markov structure. Under mild moment and regularity conditions, we establish that the subsequences possess a regenerative structure and prove that the limiting normal distributions of the growth rates of the process in supercritical and subcritical regimes decouple. For this reason, we establish limit theorems concerning the length of supercritical and subcritical regimes and the proportion of time the process spends in these regimes. As a byproduct of our analysis, we explicitly identify the limiting variances in terms of the functionals of the offspring distribution, threshold distribution, and environmental sequences.

Consider the following migration process based on a closed network of N queues with $K_N$ customers. Each station is a $\cdot$/M/$\infty$ queue with service (or migration) rate $\mu$. Upon departure, a customer is routed independently and uniformly at random to another station. In addition to migration, these customers are subject to a susceptible–infected–susceptible (SIS) dynamics. That is, customers are in one of two states: I for infected, or S for susceptible. Customers can swap their state either from I to S or from S to I only in stations. More precisely, at any station, each susceptible customer becomes infected with the instantaneous rate $\alpha Y$ if there are Y infected customers in the station, whereas each infected customer recovers and becomes susceptible with rate $\beta$. We let N tend to infinity and assume that $\lim_{N\to \infty} K_N/N= \eta $, where $\eta$ is a positive constant representing the customer density. The main problem of interest concerns the set of parameters of such a system for which there exists a stationary regime where the epidemic survives in the limiting system. The latter limit will be referred to as the thermodynamic limit. We use coupling and stochastic monotonicity arguments to establish key properties of the associated Markov processes, which in turn allow us to give the structure of the phase transition diagram of this thermodynamic limit with respect to $\eta$. The analysis of the Kolmogorov equations of this SIS model reduces to that of a wave-type PDE for which we have found no explicit solution. This plain SIS model is one among several companion stochastic processes that exhibit both random migration and contagion. Two of them are discussed in the present paper as they provide variants to the plain SIS model as well as some bounds and approximations. These two variants are the departure-on-change-of-state (DOCS) model and the averaged-infection-rate (AIR) model, which both admit closed-form solutions. The AIR system is a classical mean-field model where the infection mechanism based on the actual population of infected customers is replaced by a mechanism based on some empirical average of the number of infected customers in all stations. The latter admits a product-form solution. DOCS features accelerated migration in that each change of SIS state implies an immediate departure. This model leads to another wave-type PDE that admits a closed-form solution. In this text, the main focus is on the closed stochastic networks and their limits. The open systems consisting of a single station with Poisson input are instrumental in the analysis of the thermodynamic limits and are also of independent interest. This class of SIS dynamics has incarnations in virtually all queueing networks of the literature.

We consider a stochastic SIR (susceptible $\rightarrow$ infective $\rightarrow$ removed) model in which the infectious periods are modulated by a collection of independent and identically distributed Feller processes. Each infected individual is associated with one of these processes, the trajectories of which determine the duration of his infectious period, his contamination rate, and his type of removal (e.g. death or immunization). We use a martingale approach to derive the distribution of the final epidemic size and severity for this model and provide some general examples. Next, we focus on a single infected individual facing a given number of susceptibles, and we determine the distribution of his outcome (number of contaminations, severity, type of removal). Using a discrete-time formulation of the model, we show that this distribution also provides us with an alternative, more stable method to compute the final epidemic outcome distribution.