On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections

Abstract

A Markov-modulated framework is presented to incorporate correlated inter-event times into the stochastic susceptible-infectious-recovered (SIR) epidemic model for a closed finite community. The resulting process allows us to deal with non-exponential distributional assumptions on the contact process between the compartment of infectives and the compartment of susceptible individuals, and the recovery process of infected individuals, but keeping the dimensionality of the underlying Markov chain model tractable. The variability between SIR-models with distinct level of correlation is discussed in terms of extinction times, the final size of the epidemic, and the basic reproduction number, which is defined here as a random variable rather than an expected value.1. Introduction. The SIR-model is a well-studied epidemic model that, together with its generalizations, has been widely applied to infectious diseases such as measles, chickenpox, or mumps, among other situations where infection confers (typically lifelong) immunity; a good discussion about SIR-models analyzed from deterministic and stochastic perspectives can be found in the lucid texts by Allen [1, 2], Andersson and Britton [8], Bailey [16] and Keeling and Rohani [36]. The SIR-model is first analyzed in depth by Kermack and McKendrick [37] in 1927 in order to study the evolution of a disease in a closed community of finite size where, at time t, individuals are classified into three categories: S(t) susceptibles, I(t) infectives, and R(t) removed individuals. The general description in [37] is related to an homogeneously mixed population (i.e., any infective can infect any susceptible with equal probability), where the infection and recovery rates of a given infective depend on the total time that this individual has been infected for. The analytical

Perturbation analysis in finite LD‐QBD processes and applications to epidemic models

“…This block‐bidiagonal form in sections 3 and 4 in the work of Neuts and Li permits recursive procedures for computing the final size of the epidemic and the maximum size distribution in a similar manner to that in the work of Amador et al, where the joint distribution of the maximum number of infectives during an outbreak and the random time to reach this maximum number is also derived in terms of Laplace–Stieltjes transforms. For S I R epidemic models with Markov‐modulated events, we refer the reader to the work of Almaraz and Gómez‐Corral; more concretely, LD‐QBD processes are used in the work of Almaraz and Gómez‐Corral to derive the probability distributions of the length of an outbreak, the final size of the epidemic, and the number of secondary cases. When these models are used for representing epidemics in reality, statistical estimation techniques, such as Bayesian approaches (for instance, approximate Bayesian computation and Monte Carlo Markov chain methods), are usually implemented in order to estimate parameters of these models from clinical data; see, for example, the work of Kypraios et al In the statistical setting, it is therefore important to evaluate the impact that a small perturbation of the underlying parameters may have in the dynamics of the epidemic model, so that one can identify parameters that the model is most sensitive to, allowing for potentially devoting more computational and statistical efforts in estimating those parameter values.…”

Agent-Based Markov Modeling for Improved COVID-19 Mitigation Policies

“…Again, a Markov process is not assumed and probabilities are not updated over time. Another line of previous work (Li et al, 2018;Lefèvre & Simon, 2019;Marwa et al, 2018;Almaraz & Gómez-Corral, 2018;Britton & Pardoux, 2019) models epidemic progression as a Markov process. However, such models assume full observability regarding the susceptible, infected, and recovered sub-groups.…”

A Stochastic SVIR Model with Imperfect Vaccine and External Source of Infection

“…The SIR model, and a number of different variations, has been widely analysed both for homogeneous [6,7] and heterogeneous populations [8]. In these systems, of particular interest can be specific summary statistics that characterize an outbreak, such as the size of the outbreak [9], its length [10,11] or the reproduction number [12].…”