Modeling the simple epidemic with deterministic differential equations and random initial conditions

In a simple epidemic the only transition in the population is from susceptible to infected and the total population size is fixed for all time. This paper investigates the effect of random initial conditions on the deterministic model for the simple epidemic. By assuming a Beta distribution on the initial proportion of susceptibles, we define a distribution that describes the proportion of susceptibles in a population at any time during an epidemic. The mean and variance for this distribution are derived as hypergeometric functions, and the behavior of these functions is investigated. Lastly, we define a distribution to describe the time until a given proportion of the population remains susceptible. A method for finding the quantiles of this distribution is developed and used to make confidence statements regarding the time until a given proportion of the population is susceptible.     

An SEI model for sarcoptic mange among chamois

We consider a simple model to study the dynamics of sarcoptic mange in a population of chamois. The epidemiological patterns observed during an epidemic in Italy are reconstructed and key parameters of the model are estimated from field data. In particular, we calculate the basic reproductive ratio R (0), a threshold value for chamois density for the occurrence of an epidemic and the speed of propagation of the epidemic wave. The model is then used to obtain indications on the effect of culling as a possible control measure in a closed population and extended to analyse the spatial diffusion of the epidemic. Our results are in agreement with mange epidemiology and observations, and suggest that intervention could be efficacious in reducing the impact of an epidemic.     

An SEI model for sarcoptic mange among chamois

We consider a simple model to study the dynamics of sarcoptic mange in a population of chamois. The epidemiological patterns observed during an epidemic in Italy are reconstructed and key parameters of the model are estimated from field data. In particular, we calculate the basic reproductive ratio R ₀, a threshold value for chamois density for the occurrence of an epidemic and the speed of propagation of the epidemic wave. The model is then used to obtain indications on the effect of culling as a possible control measure in a closed population and extended to analyse the spatial diffusion of the epidemic. Our results are in agreement with mange epidemiology and observations, and suggest that intervention could be efficacious in reducing the impact of an epidemic.     

An evolutionary epidemiological mechanism, with applications to type a influenza

In this paper I develop a model that describes an evolutionary epidemiological mechanism and apply this model to the epidemiology of type A influenza. This evolutionary epidemiological model differs from the classical nonevolutionary epidemiological model which has been applied to diseases like measles, rubella, and whooping cough in having a novel mechanism which causes susceptible individuals to be introduced into the host population. In the nonevolutionary model, susceptibles are continually introduced into the host population by demographic processes: most hosts that die are immune, while newborn hosts are susceptible. In this evolutionary model, the susceptible class is continually replenished because the pathogen changes genetically, and hence immunologically, from one epidemic to the next, causing previously immune hosts to become susceptible. I derive formulae which describe how the equilibrium number of infected hosts, the interepidemic period, and the probability that a host will become reinfected depend on the rate of amino acid substitution in the pathogen, m, a parameter describing the effect of these substitutions on host immunity, gamma, as well as the host population size, N, and the recovery rate, r. To apply the model to influenza, I show how the nondimensional parameter epsilon = m gamma N/r2 may be estimated from four types of data. The methods are applied to several data sets, and I conclude that epsilon much less than 1; sampling variation and inconsistencies between the various data sets do not permit epsilon to be estimated more precisely. The evolutionary epidemiological model has no threshold host population size, in contrast to the nonevolutionary model.     

Simple Epidemic Models with Positive Latent Period

This paper deals with two types of simple epidemic models, namely, deterministic and stochastic wherein the latent period is assumed to be positive. In the deterministic epidemic model, the distributions of susceptibles, inactive infectives, active infectives and that of epidemic curve which gives the rate at which new infections take place have been obtained. The expression for the expected time of the entire epidemic has been derived. Also the partial differential equation for the moment generating function of the proportion of susceptibles in the population is established. In the end, we have studied a stochastic approach of the system.     

Modeling influenza seasonality in the tropics and subtropics

Climate drivers such as humidity and temperature may play a key role in influenza seasonal transmission dynamics. Such a relationship has been well defined for temperate regions. However, to date no models capable of capturing the diverse seasonal pattern in tropical and subtropical climates exist. In addition, multiple influenza viruses could cocirculate and shape epidemic dynamics. Here we construct seven mechanistic epidemic models to test the effect of two major climate drivers (humidity and temperature) and multi-strain co-circulation on influenza transmission in Hong Kong, an influenza epidemic center located in the subtropics. Based on model fit to long-term influenza surveillance data from 1998 to 2018, we found that a simple model incorporating the effect of both humidity and temperature best recreated the influenza epidemic patterns observed in Hong Kong. The model quantifies a bimodal effect of absolute humidity on influenza transmission where both low and very high humidity levels facilitate transmission quadratically; the model also quantifies the monotonic but nonlinear relationship with temperature. In addition, model results suggest that, at the population level, a shorter immunity period can approximate the co-circulation of influenza virus (sub)types. The basic reproductive number R₀ estimated by the best-fit model is also consistent with laboratory influenza survival and transmission studies under various combinations of humidity and temperature levels. Overall, our study has developed a simple mechanistic model capable of quantifying the impact of climate drivers on influenza transmission in (sub)tropical regions. This model can be applied to improve influenza forecasting in the (sub)tropics in the future.     

Network epidemic models with two levels of mixing

The study of epidemics on social networks has attracted considerable attention recently. In this paper, we consider a stochastic SIR (susceptible-->infective-->removed) model for the spread of an epidemic on a finite network, having an arbitrary but specified degree distribution, in which individuals also make casual contacts, i.e. with people chosen uniformly from the population. The behaviour of the model as the network size tends to infinity is investigated. In particular, the basic reproduction number R(0), that governs whether or not an epidemic with few initial infectives can become established is determined, as are the probability that an epidemic becomes established and the proportion of the population who are ultimately infected by such an epidemic. For the case when the infectious period is constant and all individuals in the network have the same degree, the asymptotic variance and a central limit theorem for the size of an epidemic that becomes established are obtained. Letting the rate at which individuals make casual contacts decrease to zero yields, heuristically, corresponding results for the model without casual contacts, i.e. for the standard SIR network epidemic model. A deterministic model that approximates the spread of an epidemic that becomes established in a large population is also derived. The theory is illustrated by numerical studies, which demonstrate that the asymptotic approximations work well, even for only moderately sized networks, and that the degree distribution and the inclusion of casual contacts can each have a major impact on the outcome of an epidemic.     

Optimal intervention for epidemic models with general infection and removal rate functions

 In this paper, known results on optimal intervention policies for the general stochastic epidemic model are extended to epidemic models with more general infection and removal rate functions. We consider first policies allowing for the isolation of any number of infectives from the susceptible population at any time, secondly policies allowing for the immunisation of the entire susceptible population at any time, and finally policies allowing for either of these interventions. In each case the costs of infection, isolation and immunisation are assumed to have a particular, rather simple, form. Sufficient conditions are given on the infection and removal rate functions of the model for the optimal policies to take the same simple form as in the case of the general stochastic epidemic model. More general costs are briefly discussed, and some numerical examples given. Finally, we discuss possible directions for further work. Received: 16 February 1999     

Times on trees, and the age of an allele

In this paper we consider the genealogy of a random sample of n chromosomes from a panmictic population which has evolved with constant size N over many generations. We address two related problems. First we describe how genealogical information may be usefully partitioned into information on the events (mutations and coalescences) which occur in the genealogy, and the times between these events. We show that the distribution of the times given information on the events is particularly simple and describe how this can considerably reduce the computational burden when performing inference for these times. Second we investigate the effect on the genealogy of conditioning on a single mutation having occurred during the ancestry of the sample. In particular we use results from the first part of the paper to derive explicit formulae for the density of the age of a mutant allele, conditional on its frequency in either a sample or the population.     

Multiple Lattice Model for Influenza Spreading

Behavioral differences among age classes, together with the natural tendency of individuals to prefer contacts with individuals of similar age, naturally point to the existence of a community structure in the population network, in which each community can be identified with a different age class. Data on age-dependent contact patterns also reveal how relevant is the role of the population age structure in shaping the spreading of an infectious disease. In the present paper we propose a simple model for epidemic spreading, in which a contact network with an intrinsic community structure is coupled with a simple stochastic SIR model for the epidemic spreading. The population is divided in 4 different age-communities and hosted on a multiple lattice, each community occupying a specific age-lattice. Individuals are allowed to move freely to nearest neighbor empty sites on the age-lattice. Different communities are connected with each other by means of inter-lattices edges, with a different number of external links connecting different age class populations. The parameters of the contact network model are fixed by requiring the simulated data to fully reproduce the contact patterns matrices of the Polymod survey. The paper shows that adopting a topology which better implements the age-class community structure of the population, one gets a better agreement between experimental contact patterns and simulated data, and this also improves the accordance between simulated and experimental data on the epidemic spreading.     

Uncovering epidemiological dynamics in heterogeneous host populations using phylogenetic methods

Host population structure has a major influence on epidemiological dynamics. However, in particular for sexually transmitted diseases, quantitative data on population contact structure are hard to obtain. Here, we introduce a new method that quantifies host population structure based on phylogenetic trees, which are obtained from pathogen genetic sequence data. Our method is based on a maximum-likelihood framework and uses a multi-type branching process, under which each host is assigned to a type (subpopulation). In a simulation study, we show that our method produces accurate parameter estimates for phylogenetic trees in which each tip is assigned to a type, as well for phylogenetic trees in which the type of the tip is unknown. We apply the method to a Latvian HIV-1 dataset, quantifying the impact of the intravenous drug user epidemic on the heterosexual epidemic (known tip states), and identifying superspreader dynamics within the men-having-sex-with-men epidemic (unknown tip states).     

Semi-empirical power-law scaling of new infection rate to model epidemic dynamics with inhomogeneous mixing

The expected number of new infections per day per infectious person during an epidemic has been found to exhibit power-law scaling with respect to the susceptible fraction of the population. This is in contrast to the linear scaling assumed in traditional epidemiologic modeling. Based on simulated epidemic dynamics in synthetic populations representing Los Angeles, Chicago, and Portland, we find city-dependent scaling exponents in the range of 1.7-2.06. This scaling arises from variations in the strength, duration, and number of contacts per person. Implementation of power-law scaling of the new infection rate is quite simple for SIR, SEIR, and histogram-based epidemic models. Treatment of the effects of the social contact structure through this power-law formulation leads to significantly lower predictions of final epidemic size than the traditional linear formulation.     

The propagation of a cultural or biological trait by neutral genetic drift in a subdivided population

We study fixation probabilities and times as a consequence of neutral genetic drift in subdivided populations, motivated by a model of the cultural evolutionary process of language change that is described by the same mathematics as the biological process. We focus on the growth of fixation times with the number of subpopulations, and variation of fixation probabilities and times with initial distributions of mutants. A general formula for the fixation probability for arbitrary initial condition is derived by extending a duality relation between forwards- and backwards-time properties of the model from a panmictic to a subdivided population. From this we obtain new formulae(formally exact in the limit of extremely weak migration) for the mean fixation time from an arbitrary initial condition for Wright's island model, presenting two cases as examples. For more general models of population subdivision, formulae are introduced for an arbitrary number of mutants that are randomly located, and a single mutant whose position is known. These formulae contain parameters that typically have to be obtained numerically, a procedure we follow for two contrasting clustered models. These data suggest that variation of fixation time with the initial condition is slight, but depends strongly on the nature of subdivision. In particular, we demonstrate conditions under which the fixation time remains finite even in the limit of an infinite number of demes. In many cases-except this last where fixation in a finite time is seen--the time to fixation is shown to be in precise agreement with predictions from formulae for the asymptotic effective population size.     

Contributions to the mathematical theory of epidemics—I

(1) A mathematical investigation has been made of the progress of an epidemic in a homogeneous population. It has been assumed that complete immunity is conferred by a single attack, and that an individual is not infective at the moment at which he receives infection. With these reservations the problem has been investigated in its most general aspects, and the following conclusions have been arrived at. (2) In general a threshold density of population is found to exist, which depends upon the infectivity, recovery and death rates peculiar to the epidemic. No epidemic can occur if the population density is below this threshold value. (3) Small increases of the infectivity rate may lead to large epidemics; also, if the population density slightly exceeds its threshold value the effect of an epidemic will be to reduce the density as far below the threshold value as initially it was above it. (4) An epidemic, in general, comes to an end, before the susceptible population has been exhausted. (5) Similar results are indicated for the case in which transmission is through an intermediate host.     

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.     

Accuracy of predicting the genetic risk of disease using a genome-wide approach

Background

The prediction of the genetic disease risk of an individual is a powerful public health tool. While predicting risk has been successful in diseases which follow simple Mendelian inheritance, it has proven challenging in complex diseases for which a large number of loci contribute to the genetic variance. The large numbers of single nucleotide polymorphisms now available provide new opportunities for predicting genetic risk of complex diseases with high accuracy.

Methodology/Principal Findings

We have derived simple deterministic formulae to predict the accuracy of predicted genetic risk from population or case control studies using a genome-wide approach and assuming a dichotomous disease phenotype with an underlying continuous liability. We show that the prediction equations are special cases of the more general problem of predicting the accuracy of estimates of genetic values of a continuous phenotype. Our predictive equations are responsive to all parameters that affect accuracy and they are independent of allele frequency and effect distributions. Deterministic prediction errors when tested by simulation were generally small. The common link among the expressions for accuracy is that they are best summarized as the product of the ratio of number of phenotypic records per number of risk loci and the observed heritability.

Conclusions/Significance

This study advances the understanding of the relative power of case control and population studies of disease. The predictions represent an upper bound of accuracy which may be achievable with improved effect estimation methods. The formulae derived will help researchers determine an appropriate sample size to attain a certain accuracy when predicting genetic risk.     

Persistence time of SIS infections in heterogeneous populations and networks

Journal of Mathematical Biology - For a susceptible–infectious–susceptible infection model in a heterogeneous population, we present simple formulae giving the leading-order asymptotic...     

Transition from growth to decay of an epidemic due to lockdown

We study the transition of an epidemic from growth phase to decay of the active infections in a population when lockdown health measures are introduced to reduce the probability of disease transmission. Although in the case of uniform lockdown, a simple compartmental model would indicate instantaneous transition to decay of the epidemic, this is not the case when partially isolated active clusters remain with the potential to create a series of small outbreaks. We model this using the Gillespie stochastic simulation algorithm based on a connected set of stochastic susceptible-infected-removed/recovered networks representing the locked-down majority population (in which the reproduction number is less than 1) weakly coupled to a large set of small clusters in which the infection may propagate. We find that the presence of such active clusters can lead to slower than expected decay of the epidemic and significantly delayed onset of the decay phase. We study the relative contributions of these changes, caused by the active clusters within the population, to the additional total infected population. We also demonstrate that limiting the size of the inevitable active clusters can be efficient in reducing their impact on the overall size of the epidemic outbreak. The deceleration of the decay phase becomes apparent when the active clusters form at least 5% of the population.     

Optimal observation times in experimental epidemic processes

Summary .   This article describes a method for choosing observation times for stochastic processes to maximise the expected information about their parameters. Two commonly used models for epidemiological processes are considered: a simple death process and a susceptible-infected (SI) epidemic process with dual sources for infection spreading within and from outwith the population. The search for the optimal design uses Bayesian computational methods to explore the joint parameter-data-design space, combined with a method known as moment closure to approximate the likelihood to make the acceptance step efficient. For the processes considered, a small number of optimally chosen observations are shown to yield almost as much information as much more intensively observed schemes that are commonly used in epidemiological experiments. Analysis of the simple death process allows a comparison between the full Bayesian approach and locally optimal designs around a point estimate from the prior based on asymptotic results. The robustness of the approach to misspecified priors is demonstrated for the SI epidemic process, for which the computational intractability of the likelihood precludes locally optimal designs. We show that optimal designs derived by the Bayesian approach are similar for observational studies of a single epidemic and for studies involving replicated epidemics in independent subpopulations. Different optima result, however, when the objective is to maximise the gain in information based on informative and non-informative priors: this has implications when an experiment is designed to convince a naïve or sceptical observer rather than consolidate the belief of an informed observer. Some extensions to the methods, including the selection of information criteria and extension to other epidemic processes with transition probabilities, are briefly addressed.