Random Modelling of Contagious Diseases

Modelling contagious diseases needs to include a mechanistic knowledge about contacts between hosts and pathogens as specific as possible, e.g., by incorporating in the model information about social networks through which the disease spreads. The unknown part concerning the contact mechanism can be modelled using a stochastic approach. For that purpose, we revisit SIR models by introducing first a microscopic stochastic version of the contacts between individuals of different populations (namely Susceptible, Infective and Recovering), then by adding a random perturbation in the vicinity of the endemic fixed point of the SIR model and eventually by introducing the definition of various types of random social networks. We propose as example of application to contagious diseases the HIV, and we show that a micro-simulation of individual based modelling (IBM) type can reproduce the current stable incidence of the HIV epidemic in a population of HIV-positive men having sex with men (MSM).     

Disease contact tracing in random and clustered networks

The efficacy of contact tracing, be it between individuals (e.g. sexually transmitted diseases or severe acute respiratory syndrome) or between groups of individuals (e.g. foot-and-mouth disease; FMD), is difficult to evaluate without precise knowledge of the underlying contact structure; i.e. who is connected to whom? Motivated by the 2001 FMD epidemic in the UK, we determine, using stochastic simulations and deterministic 'moment closure' models of disease transmission on networks of premises (nodes), network and disease properties that are important for contact tracing efficiency. For random networks with a high average number of connections per node, little clustering of connections and short latency periods, contact tracing is typically ineffective. In this case, isolation of infected nodes is the dominant factor in determining disease epidemic size and duration. If the latency period is longer and the average number of connections per node small, or if the network is spatially clustered, then the contact tracing performs better and an overall reduction in the proportion of nodes that are removed during an epidemic is observed.     

Growth rate and basic reproduction number for population models with a simple periodic factor

For continuous-time population models with a periodic factor which is sinusoidal, both the growth rate and the basic reproduction number are shown to be the largest roots of simple equations involving continued fractions. As an example, we reconsider an SEIS model with a fixed latent period, an exponentially distributed infectious period and a sinusoidal contact rate studied in Williams and Dye [B.G. Williams, C. Dye, Infectious disease persistence when transmission varies seasonally, Math. Biosci. 145 (1997) 77]. We show that apart from a few exceptional parameter values, the epidemic threshold depends not only on the mean contact rate, but also on the amplitude of fluctuations.     

Stochastic two-group models with transmission dependent on host infectivity or susceptibility

ABSTRACT

Stochastic epidemic models with two groups are formulated and applied to emerging and re-emerging infectious diseases. In recent emerging diseases, disease spread has been attributed to superspreaders, highly infectious individuals that infect a large number of susceptible individuals. In some re-emerging infectious diseases, disease spread is attributed to waning immunity in susceptible hosts. We apply a continuous-time Markov chain (CTMC) model to study disease emergence or re-emergence from different groups, where the transmission rates depend on either the infectious host or the susceptible host. Multitype branching processes approximate the dynamics of the CTMC model near the disease-free equilibrium and are used to estimate the probability of a minor or a major epidemic. It is shown that the probability of a major epidemic is greater if initiated by an individual from the superspreader group or by an individual from the highly susceptible group. The models are applied to Severe Acute Respiratory Syndrome and measles.     

Infectious disease in the Pleistocene: Old friends or old foes?

The impact of endemic and epidemic disease on humans has traditionally been seen as a comparatively recent historical phenomenon associated with the Neolithisation of human groups, an increase in population size led by sedentarism, and increasing contact with domesticated animals as well as species occupying opportunistic symbiotic and ectosymbiotic relationships with humans. The orthodox approach is that Neolithisation created the conditions for increasing population size able to support a reservoir of infectious disease sufficient to act as selective pressure. This orthodoxy is the result of an overly simplistic reliance on skeletal data assuming that no skeletal lesions equated to a healthy individual, underpinned by the assumption that hunter-gatherer groups were inherently healthy while agricultural groups acted as infectious disease reservoirs. The work of van Blerkom, Am. J. Phys. Anthropol., vol. suppl 37 (2003), Wolfe et al., Nature, vol. 447 (2007) and Houldcroft and Underdown, Am. J. Phys. Anthropol., vol. 160, (2016) has changed this landscape by arguing that humans and pathogens have long been fellow travelers. The package of infectious diseases experienced by our ancient ancestors may not be as dissimilar to modern infectious diseases as was once believed. The importance of DNA, from ancient and modern sources, to the study of the antiquity of infectious disease, and its role as a selective pressure cannot be overstated. Here we consider evidence of ancient epidemic and endemic infectious diseases with inferences from modern and ancient human and hominin DNA, and from circulating and extinct pathogen genomes. We argue that the pandemics of the past are a vital tool to unlock the weapons needed to fight pandemics of the future.     

Oscillatory phenomena in a model of infectious diseases

Oscillations of the number of cases around an average endemic level are common in several infectious diseases. In this paper we study simple deterministic models, where the oscillations arise either solely from periodically varying contact rates or from the combined effect of large initial perturbation, small periodic variation of the contact rate, and the destabilizing nature of infectious and latent periods when described as time delays. The main results are: (a) For a model with a periodically varying contact rate and a recovery rate, a threshold amplitude of variation is found by numerical and analytic methods at which 2-year subharmonic resonance appears. (b) Approximate analytic relationships are derived for the amplitude and phase of the forced 1-year oscillations below this threshold and for the 2-year oscillations above it—in terms of the reproduction rate of the infection. (c) Similar calculations are performed when the recovery rate is replaced by a fixed infectious period represented by a pure time delay. The threshold amplitude of variation in the contact rate is found here to be smaller than in the recovery rate model. (d) A model with a fixed infectious period and a constant contact rate is considered. The nontrivial steady state is shown to be locally stable for the parameter range of interest. However, the ratio of the imaginary to real parts of the eigenvalues in the characteristic equation is increased as compared to the corresponding model with a recovery rate. (e) For the model with a fixed infectious period and a constant contact rate an approximation method indicates consistency in a certain range of contact rates with the existence of an unstable periodic solution about the locally stable steady state. The actual existence of such a solution is not verified. The interpretation is that the destabilizing effect of the introduction of a pure delay into the model becomes more significant as the distance in the variables space from the endemic steady state is increased. (f) For a fixed infectious period and very small subthreshold variation in the contact rate, two different types of solutions are found numerically: yearly small-amplitude oscillations about an endemic average and large-amplitude oscillations of a subharmonic period. The pattern seen depends on the initial conditions. For a sufficiently large initial deviation from the endemic level even very small seasonal variations lead to regular recurrent outbreaks of the disease. The effect of latent periods and of changing the form of the interaction are also considered.     

An SIRVS epidemic model with pulse vaccination strategy

The aim of this paper is to analyze an SIRVS epidemic model in which pulse vaccination strategy (PVS) is included. We are interested in finding the basic reproductive number of the model which determine whether or not the disease dies out. The global attractivity of the disease-free periodic solution (DFPS for short) is obtained when the basic reproductive number is less than unity. The disease is permanent when the basic reproductive number is greater than unity, i.e., the epidemic will turn out to endemic. Our results indicate that the disease will go to extinction when the vaccination rate reaches some critical value.     

The relationship between real-time and discrete-generation models of epidemic spread

Many important results in stochastic epidemic modelling are based on the Reed-Frost model or on other similar models that are characterised by unrealistic temporal dynamics. Nevertheless, they can be extended to many other more realistic models thanks to an argument first provided by Ludwig [Final size distributions for epidemics, Math. Biosci. 23 (1975) 33-46], that states that, for a disease leading to permanent immunity after recovery, under suitable conditions, a continuous-time infectious process has the same final size distribution as another more tractable discrete-generation contact process; in other words, the temporal dynamics of the epidemic can be neglected without affecting the final size distribution. Despite the importance of such an argument, its presence behind many results is often not clearly stated or hidden in references to previous results. In this paper, we reanalyse Ludwig’s result, highlighting some of the conditions under which it does not hold and providing a general framework to examine the differences between the continuous-time and the discrete-generation process.     

Network resonance can be generated independently at distinct levels of neuronal organization

Resonance is defined as maximal response of a system to periodic inputs in a limited frequency band. Resonance may serve to optimize inter-neuronal communication, and has been observed at multiple levels of neuronal organization. However, it is unknown how neuronal resonance observed at the network level is generated and how network resonance depends on the properties of the network building blocks. Here, we first develop a metric for quantifying spike timing resonance in the presence of background noise, extending the notion of spiking resonance for in vivo experiments. Using conductance-based models, we find that network resonance can be inherited from resonances at other levels of organization, or be intrinsically generated by combining mechanisms across distinct levels. Resonance of membrane potential fluctuations, postsynaptic potentials, and single neuron spiking can each be generated independently of resonance at any other level and be propagated to the network level. At all levels of organization, interactions between processes that give rise to low- and high-pass filters generate the observed resonance. Intrinsic network resonance can be generated by the combination of filters belonging to different levels of organization. Inhibition-induced network resonance can emerge by inheritance from resonance of membrane potential fluctuations, and be sharpened by presynaptic high-pass filtering. Our results demonstrate a multiplicity of qualitatively different mechanisms that can generate resonance in neuronal systems, and provide analysis tools and a conceptual framework for the mechanistic investigation of network resonance in terms of circuit components, across levels of neuronal organization.     

Congruent epidemic models for unstructured and structured populations: analytical reconstruction of a 2003 SARS outbreak

Both the threat of bioterrorism and the natural emergence of contagious diseases underscore the importance of quantitatively understanding disease transmission in structured human populations. Over the last few years, researchers have advanced the mathematical theory of scale-free networks and used such theoretical advancements in pilot epidemic models. Scale-free contact networks are particularly interesting in the realm of mathematical epidemiology, primarily because these networks may allow meaningfully structured populations to be incorporated in epidemic models at moderate or intermediate levels of complexity. Moreover, a scale-free contact network with node degree correlation is in accord with the well-known preferred mixing concept. The present author describes a semi-empirical and deterministic epidemic modeling approach that (a) focuses on time-varying rates of disease transmission in both unstructured and structured populations and (b) employs probability density functions to characterize disease progression and outbreak controls. Given an epidemic curve for a historical outbreak, this modeling approach calls for Monte Carlo calculations (that define the average new infection rate) and solutions to integro-differential equations (that describe outbreak dynamics in an aggregate population or across all network connectivity classes). Numerical results are obtained for the 2003 SARS outbreak in Taiwan and the dynamical implications of time-varying transmission rates and scale-free contact networks are discussed in some detail.     

Stochastic epidemic models: A survey

This paper is a survey paper on stochastic epidemic models. A simple stochastic epidemic model is defined and exact and asymptotic (relying on a large community) properties are presented. The purpose of modelling is illustrated by studying effects of vaccination and also in terms of inference procedures for important parameters, such as the basic reproduction number and the critical vaccination coverage. Several generalizations towards realism, e.g. multitype and household epidemic models, are also presented, as is a model for endemic diseases.     

Contact switching as a control strategy for epidemic outbreaks

We study the effects of switching social contacts as a strategy to control epidemic outbreaks. Connections between susceptible and infective individuals can be broken by either individual, and then reconnected to a randomly chosen member of the population. It is assumed that the reconnecting individual has no previous information on the epidemiological condition of the new contact. We show that reconnection can completely suppress the disease, both by continuous and discontinuous transitions between the endemic and the infection-free states. For diseases with an asymptomatic phase, we analyze the conditions for the suppression of the disease, and show that—even when these conditions are not met—the increase of the endemic infection level is usually rather small. We conclude that, within some simple epidemiological models, contact switching is a quite robust and effective control strategy. This suggests that it may also be an efficient method in more complex situations.     

Comparison of deterministic and stochastic SIS and SIR models in discrete time

The dynamics of deterministic and stochastic discrete-time epidemic models are analyzed and compared. The discrete-time stochastic models are Markov chains, approximations to the continuous-time models. Models of SIS and SIR type with constant population size and general force of infection are analyzed, then a more general SIS model with variable population size is analyzed. In the deterministic models, the value of the basic reproductive number R0 determines persistence or extinction of the disease. If R0 < 1, the disease is eliminated, whereas if R0 > 1, the disease persists in the population. Since all stochastic models considered in this paper have finite state spaces with at least one absorbing state, ultimate disease extinction is certain regardless of the value of R0. However, in some cases, the time until disease extinction may be very long. In these cases, if the probability distribution is conditioned on non-extinction, then when R0 > 1, there exists a quasi-stationary probability distribution whose mean agrees with deterministic endemic equilibrium. The expected duration of the epidemic is investigated numerically.     

Modeling the impact of random screening and contact tracing in reducing the spread of HIV

Mathematical models can help predict the effectiveness of control measures on the spread of HIV and other sexually transmitted diseases (STDs) by reducing the uncertainty in assessing the impact of intervention strategies such as random screening and contact tracing. Even though contact tracing is one of the most effective methods used for controlling treatable STDs, it is still a controversial strategy for controlling HIV because of cost and confidentiality issues. To help estimate the effectiveness of these control measures, we formulate two models with random screening and contact tracing based on the differential infectivity (DI) model and the staged-progression (SP) model. We derive formulas for the reproductive numbers and the endemic equilibria and compare the impact that random screening and contact tracing have in slowing the epidemic in the two models. In the DI model the infected population is divided into groups according to their infectiousness, and HIV is largely spread by a small, highly infectious, group of superspreaders. In this model contact tracing is an effective approach to identifying the superspreaders and has a large effect in slowing the epidemic. In the SP model every infected individual goes through a series of infection stages and the virus is primarily spread by individuals in an initial highly infectious stage or in the late stages of the disease. In this model random screening is more effective than for the DI model, and contact tracing is less effective. Thus the effectiveness of the intervention strategy strongly depends on the underlying etiology of the disease transmission.     

Modeling epidemic spread with awareness and heterogeneous transmission rates in networks

During an epidemic outbreak in a human population, susceptibility to infection can be reduced by raising awareness of the disease. In this paper, we investigate the effects of three forms of awareness (i.e., contact, local, and global) on the spread of a disease in a random network. Connectivity-correlated transmission rates are assumed. By using the mean-field theory and numerical simulation, we show that both local and contact awareness can raise the epidemic thresholds while the global awareness cannot, which mirrors the recent results of Wu et al. The obtained results point out that individual behaviors in the presence of an infectious disease has a great influence on the epidemic dynamics. Our method enriches mean-field analysis in epidemic models.     

Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases

It is well known that behavioral changes in contact patterns may significantly affect the spread of an epidemic outbreak. Here we focus on simple endemic models for recurrent epidemics, by modelling the social contact rate as a function of the available information on the present and the past disease prevalence. We show that social behavior change alone may trigger sustained oscillations. This indicates that human behavior might be a critical explaining factor of oscillations in time-series of endemic diseases. Finally, we briefly show how the inclusion of seasonal variations in contacts may imply chaos.     

Small amplitude,long period outbreaks in seasonally driven epidemics

It is now documented that childhood diseases such as measles, mumps, and chickenpox exhibit a wide range of recurrent behavior (periodic as well as chaotic) in large population centers in the first world. Mathematical models used in the past (such as the SEIR model with seasonal forcing) have been able to predict the onset of both periodic and chaotic sustained epidemics using parameters of childhood diseases. Although these models possess stable solutions which appear to have the correct frequency content, the corresponding outbreaks require extremely large populations to support the epidemic. This paper shows that by relaxing the assumption of uniformity in the supply of susceptibles, simple models predict stable long period oscillatory epidemics having small amplitude. Both coupled and single population models are considered.     

On the probability of extinction in a periodic environment

For a certain class of multi-type branching processes in a continuous-time periodic environment, we show that the extinction probability is equal to (resp. less than) 1 if the basic reproduction number $R_0$ is less than (resp. bigger than) 1. The proof uses results concerning the asymptotic behavior of cooperative systems of differential equations. In epidemiology the extinction probability may be used as a time-periodic measure of the epidemic risk. As an example we consider a linearized SEIR epidemic model and data from the recent measles epidemic in France. Discrete-time models with potential applications in conservation biology are also discussed.     

The influence of immune cross-reaction on phase structure in resonant solutions of a multi-strain seasonal SIR model

Resonance in seasonally forced SIR epidemiological models may lead to stable solutions in which the epidemic period is an integer multiple of the forcing period. We examine the influence of immune cross-protection and cross-enhancement on the epidemic phase relationship of resonance solutions in an annually forced two-strain SIR model. Solutions with epidemics of the two strains in-phase commonly occur for wide ranges of cross-reaction intensity. Solutions with epidemics out-of-phase are less common and limited to narrow ranges of cross-reaction intensity. This is broadly as predicted by the two natural periods of the system. The natural period corresponding to out-of-phase solutions is sensitive to changes in the cross-reaction parameter but the natural period corresponding to in-phase solutions is constant. Bifurcation analysis indicates that the stability of in-phase orbits is controlled by pitchfork and period doubling bifurcations while out-of-phase orbits may also be influenced by Andronov-Hopf bifurcations. In order to develop an intuitive understanding of the epidemiological factors governing the occurrence of different solutions we consider how the susceptible, infected and removed components of the system must interact to form a stable solution. This shows that the impact of cross-reaction is moderated by in-phase structures but amplified by out-of-phase structures. Although the average infection rate over long time periods is not affected by phase structure, this analysis indicates that in-phase epidemic patterns are likely to be more consistent and thus allow more effective health care management.     

A Comparison of Methods for Calculating the Basic Reproductive Number for Periodic Epidemic Systems

When using mathematics to study epidemics, oftentimes the goal is to determine when an infection can invade and persist within a population. The most common way to do so uses threshold quantities called reproductive numbers. An infection’s basic reproductive number (BRN), typically denoted \(R_0\), measures the infection’s initial ability to reproduce in a naive population and is tied mathematically to the stability of the disease-free equilibrium. Next-generation methods have long been used to derive \(R_0\) for autonomous continuous-time systems; however, many diseases exhibit seasonal behavior. Incorporating seasonality into models may improve accuracy in important ways, but the resulting non-autonomous systems are much more difficult to analyze. In the literature, two principal methods have been used to derive BRNs for periodic epidemic models. One, based on time-averages, is simple to apply but does not always describe the correct threshold behavior. The other, based on linear operator theory, is more general but also more complicated, and no detailed explanations of the necessary computations have yet been laid out. This paper reconciles the two methods by laying out an explicit procedure for the second and then identifying conditions (and some important classes of models) under which the two methods agree. This allows the use of the simpler method, which yields interpretable closed-form expressions, when appropriate, and illustrates in detail the simplest possible case where they disagree. Results show that seasonality alone cannot affect disease persistence, but must act in conjunction with non-hierarchical heterogeneity in the infected population, in order to do so.