General Epidemic Models Research Articles

Network Reconstruction and Prediction of Epidemic Outbreaks for General Group-Based Compartmental Epidemic Models

The underlying core of most epidemic models is the graph that specifies the contacts between healthy and infected individuals. However, in the majority of applications, the contact network is unknown. To understand and predict an epidemic outbreak nonetheless, network reconstruction methods aim to estimate the contact network from viral state observations. This work considers general compartmental epidemic models (GEMF) in discrete time, which describe the viral spread between groups of individuals. The reconstruction of the network translates into a set of linear equations that is severely ill-conditioned. Counterintuitively, we show that the contact network cannot be reconstructed from one epidemic outbreak with any finite machine precision, although an accurate prediction of the epidemic outbreak is possible.

A note on the direction of the transcritical bifurcation in epidemic models

In classical epidemic models, it is common to observe that a disease-free equilibrium looses its stability for R0=1 and a transcritical bifurcation takes place. We analyze this aspect from the point of view of the mathematical structure of models, in order to assess which parts of the structure might be responsible of the direction of the transcritical bifurcation. We formulate a general criterion, which gives sufficient (resp. necessary) conditions for the occurrence of forward (resp. backward) bifurcations. The criterion, obtained as consequence of a well known analysis of the centre manifold for general epidemic models, is applied to several epidemic models taken from the literature.

Threshold dynamics for compartmental epidemic models with impulses

The basic reproductive number and its calculation for general impulsive compartmental epidemic models, with pulses on both the infected and the uninfected compartments, are established. Theoretical results show that the basic reproductive number serves as a threshold parameter: the disease dies out if the basic reproductive number is smaller than unity, and breaks out if it is larger than unity. The global dynamics of a viral dynamical model with impulsive immune response is analyzed to study how the vaccination strength and the vaccination interval affect the basic reproductive number and virus progression.

Message passing approach for general epidemic models

In most models of the spread of disease over contact networks it is assumed that the probabilities per unit time of disease transmission and recovery from disease are constant, implying exponential distributions of the time intervals for transmission and recovery. Time intervals for real diseases, however, have distributions that in most cases are far from exponential, which leads to disagreements, both qualitative and quantitative, with the models. In this paper, we study a generalized version of the susceptible-infected-recovered model of epidemic disease that allows for arbitrary distributions of transmission and recovery times. Standard differential equation approaches cannot be used for this generalized model, but we show that the problem can be reformulated as a time-dependent message passing calculation on the appropriate contact network. The calculation is exact on trees (i.e., loopless networks) or locally treelike networks (such as random graphs) in the large system size limit. On non-tree-like networks we show that the calculation gives a rigorous bound on the size of disease outbreaks. We demonstrate the method with applications to two specific models and the results compare favorably with numerical simulations.

General compartmental epidemic models

The age of infection approach introduced by Kermack and Mckendrick in 1927 gives a unified way of describing and analyzing a variety of epidemic models, including models with multiple stages, treatment, and heterogeneous mixing. The author gives a description of the main results for such models, emphasizing the use of the final size relation to estimate the size of the epidemic.

Subthreshold and superthreshold coexistence of pathogen variants: The impact of host age-structure

It is well known that in the most general epidemic models with multiple pathogen variants a competitive exclusion principle is valid, such that the variant with the highest reproduction number eliminates the rest. Mechanisms such as super-infection, coinfection, and cross-immunity can lead to pathogen polymorphism where multiple strains coexist. It is also known that variability of infectivity with host age can destabilize the endemic equilibrium and cause oscillations. In this article we show that the hosts’ chronological age can itself lead to coexistence of microparasites in the most basic model where competitive exclusion will occur without the age structure. Moreover, the host age-structure leads to multiple subthreshold dominance equilibria, and both weakly and strongly subthreshold coexistence. We find that the two pathogens cannot cooperate to persist subthreshold if neither one of them can persist subthreshold by itself. If, however, one of them can persist subthreshold by itself, it can cause the two pathogens to coexist in a strongly subthreshold equilibrium. The second strain that persists subthreshold through the mediation of the first always has a lower virulence. Our results show that age structure in infectivity can permit the coexistence of competing pathogens when the incidence is of proportionate mixing type (frequency-dependent transmission) and at least one of the strains is virulent.

Compound Poisson limits for household epidemics

We consider epidemics in populations that are partitioned into small groups known as households. Whilst infectious, a typical infective makes global and local contact with individuals chosen independently and uniformly from the whole population or their own household, as appropriate. Previously, the classical Poisson approximation for the number of survivors of a severe epidemic has been extended to the household model. However, in the current work we exploit a Sellke-type construction of the epidemic process, which enables the derivation of sufficient conditions for the existence of a compound Poisson limit theorem for the survivors of the epidemic. The results are specialised to the Reed-Frost and general stochastic epidemic models.

Compound Poisson limits for household epidemics

We consider epidemics in populations that are partitioned into small groups known as households. Whilst infectious, a typical infective makes global and local contact with individuals chosen independently and uniformly from the whole population or their own household, as appropriate. Previously, the classical Poisson approximation for the number of survivors of a severe epidemic has been extended to the household model. However, in the current work we exploit a Sellke-type construction of the epidemic process, which enables the derivation of sufficient conditions for the existence of a compound Poisson limit theorem for the survivors of the epidemic. The results are specialised to the Reed-Frost and general stochastic epidemic models.

Optimization over Time. Dynamic Programming and Stochastic Control. Volume 1

"Optimization over Time. Dynamic Programming and Stochastic Control. Volume 1." Journal of the Operational Research Society, 34(3), p. 255