We study the problem of optimal control of the stochastic SIR model. Models of this type are used in mathematical epidemiology to capture the time evolution of highly infectious diseases such as COVID-19. Our approach relies on reformulating the Hamilton-Jacobi-Bellman equation as a stochastic minimum principle. This results in a system of forward backward stochastic differential equations, which is amenable to numerical solution via Monte Carlo simulations. We present a number of numerical solutions of the system under a variety of scenarios.
In recent years the role of epidemic models in informing public health policies has progressively grown. Models have become increasingly realistic and more complex, requiring the use of multiple data sources to estimate all quantities of interest. This review summarises the different types of stochastic epidemic models that use evidence synthesis and highlights current challenges.
To model the evolution of diseases with extended latency periods and the presence of asymptomatic patients like COVID-19, we define a simple discrete time stochastic SIR-type epidemic model. We include both latent periods as well as the presence of quarantine areas, to capture the evolutionary dynamics of such diseases.
Stochastic effects may cause fade-out of an infectious disease in a population immediately after an epidemic outbreak. We develop WKB theory to determine the most probable path of the system toward epidemic fade-out and to evaluate the fade-out probability. The most probable path is an instanton-like orbit in the phase space of the underlying Hamiltonian flow.
We present a generic epidemic model with stochastic parameters, in which the dynamics self-organize to a critical state with suppressed exponential growth. More precisely, the dynamics evolve into a quasi-steady-state, where the effective reproduction rate fluctuates close to the critical value one, as observed for different epidemics. The main assumptions underlying the model are that the rate at which each individual becomes infected changes stochastically in time with a heavy-tailed steady state. The critical regime is characterized by an extremely long duration of the epidemic. Its stabili…
We show how the standard field theoretical language based on creation and annihilation operators may be used for a straightforward derivation of closed master equations describing the population dynamics of multivariate stochastic epidemic models. In order to do that, we introduce an SIR-inspired stochastic model for hepatitis C virus epidemic, from which we obtain the time evolution of the mean number of susceptible, infected, recovered and chronically infected individuals in a population whose total size is allowed to change.
Epidemic models currently play a central role in our attempts to understand and control infectious diseases. Here, we derive a model for the diffusion limit of stochastic susceptible-infectious-removed (SIR) epidemic dynamics on a heterogeneous network. Using this, we consider analytically the early asymptotic exponential growth phase of such epidemics, showing how the higher order moments of the network degree distribution enter into the stochastic behaviour of the epidemic. We find that the first three moments of the network degree distribution are needed to specify the variance in disease p…
Stochastic epidemic models describe the dynamics of an epidemic as a disease spreads through a population. Typically, only a fraction of cases are observed at a set of discrete times. The absence of complete information about the time evolution of an epidemic gives rise to a complicated latent variable problem in which the state space size of the epidemic grows large as the population size increases. This makes analytically integrating over the missing data infeasible for populations of even moderate size. We present a data augmentation Markov chain Monte Carlo (MCMC) framework for Bayesian es…
We prove that, for Poisson transmission and recovery processes, the classic Susceptible $\to$ Infected $\to$ Recovered (SIR) epidemic model of Kermack and McKendrick provides, for any given time $t>0$, a strict lower bound on the expected number of suscpetibles and a strict upper bound on the expected number of recoveries in the general stochastic SIR epidemic. The proof is based on the recent message passing representation of SIR epidemics applied to a complete graph.
Threshold theorem is probably the most important development of mathematical epidemic modelling. Unfortunately, some models may not behave according to the threshold. In this paper, we will focus on the final outcome of SIR model with demography. The behaviour of the model approached by deteministic and stochastic models will be introduced, mainly using simulations. Furthermore, we will also investigate the dynamic of susceptibles in population in absence of infective. We have successfully showed that both deterministic and stochastic models performed similar results when $R_0 \leq 1$. That is…
Emerging epidemics and local infection clusters are initially prone to stochastic effects that can substantially impact the epidemic trajectory. While numerous studies are devoted to the deterministic regime of an established epidemic, mathematical descriptions of the initial phase of epidemic growth are comparatively rarer. Here, we review existing mathematical results on the epidemic size over time, and derive new results to elucidate the early dynamics of an infection cluster started by a single infected individual. We show that the initial growth of epidemics that eventually take off is ac…
Developing robust, quantitative methods to optimize resource allocations in response to epidemics has the potential to save lives and minimize health care costs. In this paper, we develop and apply a computationally efficient algorithm that enables us to calculate the complete probability distribution for the final epidemic size in a stochastic Susceptible-Infected-Recovered (SIR) model. Based on these results, we determine the optimal allocations of a limited quantity of vaccine between two non-interacting populations. We compare the stochastic solution to results obtained for the traditional…
Studies about epidemic modelling have been conducted since before 19th century. Both deterministic and stochastiic model were used to capture the dynamic of infection in the population. The purpose of this project is to investigate the behaviour of the models when we set the basic reproduction number, $R_0$. This quantity is defined as the expected number of contacts made by a typical infective to susceptibles in the population. According to the epidemic threshold theory, when $R_0 \leq 1$, minor epidemic occurs with probability one in both approaches, but when $R_0 > 1$, the deterministic and…
We propose a stochastic SICA epidemic model for HIV transmission, described by stochastic ordinary differential equations, and discuss its perturbation by environmental white noise. Existence and uniqueness of the global positive solution to the stochastic HIV system is proven, and conditions under which extinction and persistence in mean hold, are given. The theoretical results are illustrated via numerical simulations.
We propose a stochastic SIR model, specified as a system of stochastic differential equations, to analyse the data of the Italian COVID-19 epidemic, taking also into account the under-detection of infected and recovered individuals in the population. We find that a correct assessment of the amount of under-detection is important to obtain reliable estimates of the critical model parameters. Moreover, a single SIR model over the whole epidemic period is unable to correctly describe the behaviour of the pandemic. Then, the adaptation of the model in every time-interval between relevant governmen…
Consider a uniformly mixing population which grows as a super-critical linear birth and death process. At some time an infectious disease (of SIR or SEIR type) is introduced by one individual being infected from outside. It is shown that three different scenarios may occur: 1) an epidemic never takes off, 2) an epidemic gets going and grows but at a slower rate than the community thus still being negligible in terms of population fractions, or 3) an epidemic takes off and grows quicker than the community eventually leading to an endemic equilibrium. Depending on the parameter values, either sc…
To better describe the spread of a disease, we extend a discrete time stochastic SIR-type epidemic model of Tuckwell and Williams. We assume the dependence on time of the number of daily encounters and include a parameter to represent a possible quarantine of the infectious individuals. We provide an analytic description of this Markovian model and investigate its dynamics. Both a diffusion approximation and the basic reproduction number are derived. Through several simulations, we show how the evolution of a disease is affected by the distribution of the number of daily encounters and its dep…
We extend a technique of approximation of the long-term behavior of a supercritical stochastic epidemic model, using the WKB approximation and a Hamiltonian phase space, to the subcritical case. The limiting behavior of the model and approximation are qualitatively different in the subcritical case, requiring a novel analysis of the limiting behavior of the Hamiltonian system away from its deterministic subsystem. This yields a novel, general technique of approximation of the quasistationary distribution of stochastic epidemic and birth-death models, and may lead to techniques for analysis of …
We developed a model and a software package for stochastic simulations of transmission of COVID-19 and other similar infectious diseases, that takes into account contact network structures and geographical distribution of population density, detailed up to a level of location of individuals. Our analysis framework includes a surrogate model optimization process for quick fitting of the model's parameters to the observed epidemic curves for cases, hospitalizations and deaths. This set of instruments (the model, the simulation code, and the optimizer) is a useful tool for policymakers and epidem…
A feature often observed in epidemiological networks is significant heterogeneity in degree. A popular modelling approach to this has been to consider large populations with highly heterogeneous discrete contact rates. This paper defines an individual-level non-Markovian stochastic process that converges on standard ODE models of such populations in the appropriate asymptotic limit. A generalised Sellke construction is derived for this model, and this is then used to consider final outcomes in the case where heterogeneity follows a truncated Zipf distribution.
Using a stochastic Susceptible-Infected-Removed (SIR) meta-population model of disease transmission, we present analytical calculations and numerical simulations dissecting the interplay between stochasticity and the division of a population into mutually independent sub-populations. We show that subdivision activates two stochastic effects---extinction and desynchronization---diminishing the overall impact of the outbreak, even when the total population has already left the stochastic regime and the basic reproduction number is not altered by the subdivision. Both effects are quantitatively c…
We study a stochastic process describing the onset of spreading dynamics of an epidemic in a population composed by individuals of three classes: susceptible (S), infected (I), and recovered (R). The stochastic process is defined by local rules and involves the following cyclic process: S$\to$I$\to$R$\to$S (SIRS). The open process S$\to$I$\to$R (SIR) is studied as a particular case of the SIRS process. The epidemic process is analyzed at different levels of description: by a stochastic lattice gas model and by a birth and death process. By means of Monte Carlo simulations and dynamical mean-fi…
A simple, but ``classical``, stochastic model for epidemic spread in a finite, but large, population is studied. The progress of the epidemic can be divided into three different phases that requires different tools to analyse. Initially the process is approximated by a branching process. It is discussed for how long time this approximation is valid. When a non-negligible proportion of the population is already infected the process can be studied using differential equations. In a final phase the spread will fade out. The results are used to investigate what happens if two strains of infectious…
The primary goal of this research is to investigate the impact of delay on the dynamics of the Susceptible-Exposed-Infected-Recovered-Death and Susceptible (SEIRDS) model, to which we add a stochastic term to account for uncertainty in COVID-19 parameter estimations. We run two models, one deterministic and one stochastic, and show that their solutions exist and are unique. We also numerically investigate the impact of immunity loss on the emerging time of a new wave, as well as the necessary condition for the extinction and persistence of the disease.
An adaptive network model using SIS epidemic propagation with link-type dependent link activation and deletion is considered. Bifurcation analysis of the pairwise ODE approximation and the network-based stochastic simulation is carried out, showing that three typical behaviours may occur; namely, oscillations can be observed besides disease-free or endemic steady states. The oscillatory behaviour in the stochastic simulations is studied using Fourier analysis, as well as through analysing the exact master equations of the stochastic model. A compact pairwise approximation for the dynamic netwo…
We study the stochastic susceptible-infected-recovered (SIR) model with time-dependent forcing using analytic techniques which allow us to disentangle the interaction of stochasticity and external forcing. The model is formulated as a continuous time Markov process, which is decomposed into a deterministic dynamics together with stochastic corrections, by using an expansion in inverse system size. The forcing induces a limit cycle in the deterministic dynamics, and a complete analysis of the fluctuations about this time-dependent solution is given. This analysis is applied when the limit cycle…
Recently, emerging epidemics like COVID-19 and its variants require predictive mathematical models to implement suitable responses in order to limit their negative and profound impact on society. The SIR (Susceptible-InfectedRemoved) system is a straightforward mathematical formulation to model the dissemination of many infectious diseases. The present paper reports novel theoretical and analytical results for a perturbed version of an SIR model with Gamma-distributed delay. Notably, our epidemic model is represented by Itô-L'evy stochastic differential equations in order to simulate sudden an…
Stochastic epidemic models on networks are inherently high-dimensional and the resulting exact models are intractable numerically even for modest network sizes. Mean-field models provide an alternative but can only capture average quantities, thus offering little or no information about variability in the outcome of the exact process. In this paper we conjecture and numerically prove that it is possible to construct PDE-limits of the exact stochastic SIS epidemics on regular and Erdős-Rényi networks. To do this we first approximate the exact stochastic process at population level by a Birth-an…
Traditionally, the identification of parameters in the formulation and solution of inverse problems considers that models, variables and mathematical parameters are free of uncertainties. This aspect simplifies the estimation process, but does not consider the influence of relatively small changes in the design variables in terms of the objective function. In this work, the SIDR (Susceptible, Infected, Dead and Recovered) model is used to simulate the dynamic behavior of the novel coronavirus disease (COVID-19), and its parameters are estimated by formulating a robust inverse problem, that is,…
Epidemic models with inhomogeneous populations have been used to study major outbreaks and recently Britton and Lindenstrand \cite{BL} described the case when latency and infectivity have independent gamma distributions. They found that variability in these random variables had opposite effects on the epidemic growth rate. That rate increased with greater variability in latency but decreased with greater variability in infectivity. Here we extend their result by using the McKay bivariate gamma distribution for the joint distribution of latency and infectivity, recovering the above effects of v…
We consider a stochastic Susceptible-Exposed-Infected-Recovered (SEIR) epidemiological model with a contact rate that fluctuates seasonally. Through the use of a nonlinear, stochastic projection, we are able to analytically determine the lower dimensional manifold on which the deterministic and stochastic dynamics correctly interact. Our method produces a low dimensional stochastic model that captures the same timing of disease outbreak and the same amplitude and phase of recurrent behavior seen in the high dimensional model. Given seasonal epidemic data consisting of the number of infectious …
Some recent works reveal that there are models of differential equations for the mean and variance of infected individuals that reproduce the SIS epidemic model at some point. This stochastic SIS epidemic model can be interpreted as a Hamiltonian system, therefore we wondered if it could be geometrically handled through the theory of Lie--Hamilton systems, and this happened to be the case. The primordial result is that we are able to obtain a general solution for the stochastic/ SIS-epidemic model (with fluctuations) in form of a nonlinear superposition rule that includes particular stochastic…
Phylodynamics is a set of population genetics tools that aim at reconstructing demographic history of a population based on molecular sequences of individuals sampled from the population of interest. One important task in phylodynamics is to estimate changes in (effective) population size. When applied to infectious disease sequences such estimation of population size trajectories can provide information about changes in the number of infections. To model changes in the number of infected individuals, current phylodynamic methods use non-parametric approaches, parametric approaches, and stocha…
The interplay of seasonality, the system's nonlinearities and intrinsic stochasticity is studied for a seasonally forced susceptible-exposed-infective-recovered stochastic model. The model is explored in the parameter region that corresponds to childhood infectious diseases such as measles. The power spectrum of the stochastic fluctuations around the attractors of the deterministic system that describes the model in the thermodynamic limit is computed analytically and validated by stochastic simulations for large system sizes. Size effects are studied through additional simulations. Other effe…
Adaptive-network models are typically studied using deterministic differential equations which approximately describe their dynamics. In simulations, however, the discrete nature of the network gives rise to intrinsic noise which can radically alter the system's behaviour. In this article we develop a method to predict the effects of stochasticity in adaptive networks by making use of a pair-based proxy model. The technique is developed in the context of an epidemiological model of a disease spreading over an adaptive network of infectious contact. Our analysis reveals that in this model the s…
We present a new Bayesian inference method for compartmental models that takes into account the intrinsic stochasticity of the process. We show how to formulate a SIR-type Markov jump process as the solution of a stochastic differential equation with respect to a Poisson Random Measure (PRM), and how to simulate the process trajectory deterministically from a parameter value and a PRM realisation. This forms the basis of our Data Augmented MCMC, which consists in augmenting parameter space with the unobserved PRM value. The resulting simple Metropolis-Hastings sampler acts as an efficient simu…
Dynamics on networks is considered from the perspective of Markov stochastic processes. We partially describe the state of the system through network motifs and infer any missing data using the available information. This versatile approach is especially well adapted for modelling spreading processes and/or population dynamics. In particular, the generality of our systematic framework and the fact that its assumptions are explicitly stated suggests that it could be used as a common ground for comparing existing epidemics models too complex for direct comparison, such as agent-based computer si…
We study the spread of susceptible-infected-recovered (SIR) infectious diseases where an individual's infectiousness and probability of recovery depend on his/her "age" of infection. We focus first on early outbreak stages when stochastic effects dominate and show that epidemics tend to happen faster than deterministic calculations predict. If an outbreak is sufficiently large, stochastic effects are negligible and we modify the standard ordinary differential equation (ODE) model to accommodate age-of-infection effects. We avoid the use of partial differential equations which typically appear …
Existing compartmental models in epidemiology are limited in terms of optimizing the resource allocation to control an epidemic outbreak under disease growth uncertainty. In this study, we address this core limitation by presenting a multi-stage stochastic programming compartmental model, which integrates the uncertain disease progression and resource allocation to control an infectious disease outbreak. The proposed multi-stage stochastic program involves various disease growth scenarios and optimizes the distribution of treatment centers and resources while minimizing the total expected numb…
This paper is concerned with Crump-Mode-Jagers branching processes, describing spread of an epidemic depending on the proportion of the population that is vaccinated. Births in the branching process are aborted independently with a time-dependent probability given by the fraction of the population vaccinated. Stochastic monotonicity and continuity results for a wide class of functions (e.g., extinction time and total number of births over all time) defined on such a branching process are proved using coupling arguments, leading to optimal vaccination schemes to control corresponding functions …
The spread of an epidemic process is considered in the context of a spatial SIR stochastic model that includes a parameter $0\le p\le 1$ that assigns weights $p$ and $1- p$ to global and local infective contacts respectively. The model was previously studied by other authors in different contexts. In this work we characterized the behavior of the system around the threshold for epidemic spreading. We first used a deterministic approximation of the stochastic model and checked the existence of a threshold value of $p$ for exponential epidemic spread. An analytical expression, which defines a fu…
For a recently derived pairwise model of network epidemics with non-Markovian recovery, we prove that under some mild technical conditions on the distribution of the infectious periods, smaller variance in the recovery time leads to higher reproduction number, and consequently to a larger epidemic outbreak, when the mean infectious period is fixed. We discuss how this result is related to various stochastic orderings of the distributions of infectious periods. The results are illustrated by a number of explicit stochastic simulations, suggesting that their validity goes beyond regular networks…
It is well recognized that population heterogeneity plays an important role in the spread of epidemics. While individual variations in social activity are often assumed to be persistent, i.e. constant in time, here we discuss the consequences of dynamic heterogeneity. By integrating the stochastic dynamics of social activity into traditional epidemiological models we demonstrate the emergence of a new long timescale governing the epidemic in broad agreement with empirical data. Our model captures multiple features of real-life epidemics such as COVID-19, including prolonged plateaus and multip…
Contacts between individuals serve as pathways where infections may propagate. These contact patterns can be represented by network structures. Static structures have been the common modeling paradigm but recent results suggest that temporal structures play different roles to regulate the spread of infections or infection-like dynamics. On temporal networks a vertex is active only at certain moments and inactive otherwise such that a contact is not continuously available. In several empirical networks, the time between two consecutive vertex-activation events typically follows heterogeneous ac…
Networks have become an important tool for infectious disease epidemiology. Most previous theoretical studies of transmission network models have either considered simple Markovian dynamics at the individual level, or have focused on the invasion threshold and final outcome of the epidemic. Here, we provide a general theory for early real-time behaviour of epidemics on large configuration model networks (i.e. static and locally unclustered), in particular focusing on the computation of the Malthusian parameter that describes the early exponential epidemic growth. Analytical, numerical and Mont…
Models of biological processes are often subject to different sources of noise. Developing an understanding of the combined effects of different types of uncertainty is an open challenge. In this paper, we study a variant of the susceptible-infective-recovered model of epidemic spread, which combines both agent-to-agent heterogeneity and intrinsic noise. We focus on epidemic cycles, driven by the stochasticity of infection and recovery events, and study in detail how heterogeneity in susceptibilities and propensities to pass on the disease affects these quasi-cycles. While the system can only …
We propose and study a shifted SICA epidemic model, extending the one of Silva and Torres (2017) to the stochastic setting driven by both Brownian motion processes and jump Lévy noise. Lévy noise perturbations are usually ignored by existing works of mathematical modelling in epidemiology, but its incorporation into the SICA epidemic model is worth to consider because of the presence of strong fluctuations in HIV/AIDS dynamics, often leading to the emergence of a number of discontinuities in the processes under investigation. Our work is organised as follows: (i) we begin by presenting our mod…
