Coalescent inference for infectious disease: meta-analysis of hepatitis C

Genetic analysis of pathogen genomes is a powerful approach to investigating the population dynamics and epidemic history of infectious diseases. However, the theoretical underpinnings of the most widely used, coalescent methods have been questioned, casting doubt on their interpretation. The aim of this study is to develop robust population genetic inference for compartmental models in epidemiology. Using a general approach based on the theory of metapopulations, we derive coalescent models under susceptible–infectious (SI), susceptible–infectious–susceptible (SIS) and susceptible–infectious–recovered (SIR) dynamics. We show that exponential and logistic growth models are equivalent to SI and SIS models, respectively, when co-infection is negligible. Implementing SI, SIS and SIR models in BEAST, we conduct a meta-analysis of hepatitis C epidemics, and show that we can directly estimate the basic reproductive number (R₀) and prevalence under SIR dynamics. We find that differences in genetic diversity between epidemics can be explained by differences in underlying epidemiology (age of the epidemic and local population density) and viral subtype. Model comparison reveals SIR dynamics in three globally restricted epidemics, but most are better fit by the simpler SI dynamics. In summary, metapopulation models provide a general and practical framework for integrating epidemiology and population genetics for the purposes of joint inference.     

How well can the exponential-growth coalescent approximate constant-rate birth–death population dynamics?

One of the central objectives in the field of phylodynamics is the quantification of population dynamic processes using genetic sequence data or in some cases phenotypic data. Phylodynamics has been successfully applied to many different processes, such as the spread of infectious diseases, within-host evolution of a pathogen, macroevolution and even language evolution. Phylodynamic analysis requires a probability distribution on phylogenetic trees spanned by the genetic data. Because such a probability distribution is not available for many common stochastic population dynamic processes, coalescent-based approximations assuming deterministic population size changes are widely employed. Key to many population dynamic models, in particular epidemiological models, is a period of exponential population growth during the initial phase. Here, we show that the coalescent does not well approximate stochastic exponential population growth, which is typically modelled by a birth–death process. We demonstrate that introducing demographic stochasticity into the population size function of the coalescent improves the approximation for values of R₀ close to 1, but substantial differences remain for large R₀. In addition, the computational advantage of using an approximation over exact models vanishes when introducing such demographic stochasticity. These results highlight that we need to increase efforts to develop phylodynamic tools that correctly account for the stochasticity of population dynamic models for inference.     

Inference of Epidemiological Dynamics Based on Simulated Phylogenies Using Birth-Death and Coalescent Models

Quantifying epidemiological dynamics is crucial for understanding and forecasting the spread of an epidemic. The coalescent and the birth-death model are used interchangeably to infer epidemiological parameters from the genealogical relationships of the pathogen population under study, which in turn are inferred from the pathogen genetic sequencing data. To compare the performance of these widely applied models, we performed a simulation study. We simulated phylogenetic trees under the constant rate birth-death model and the coalescent model with a deterministic exponentially growing infected population. For each tree, we re-estimated the epidemiological parameters using both a birth-death and a coalescent based method, implemented as an MCMC procedure in BEAST v2.0. In our analyses that estimate the growth rate of an epidemic based on simulated birth-death trees, the point estimates such as the maximum a posteriori/maximum likelihood estimates are not very different. However, the estimates of uncertainty are very different. The birth-death model had a higher coverage than the coalescent model, i.e. contained the true value in the highest posterior density (HPD) interval more often (2–13% vs. 31–75% error). The coverage of the coalescent decreases with decreasing basic reproductive ratio and increasing sampling probability of infecteds. We hypothesize that the biases in the coalescent are due to the assumption of deterministic rather than stochastic population size changes. Both methods performed reasonably well when analyzing trees simulated under the coalescent. The methods can also identify other key epidemiological parameters as long as one of the parameters is fixed to its true value. In summary, when using genetic data to estimate epidemic dynamics, our results suggest that the birth-death method will be less sensitive to population fluctuations of early outbreaks than the coalescent method that assumes a deterministic exponentially growing infected population.     

Phylodynamics of Infectious Disease Epidemics

We present a formalism for unifying the inference of population size from genetic sequences and mathematical models of infectious disease in populations. Virus phylogenies have been used in many recent studies to infer properties of epidemics. These approaches rely on coalescent models that may not be appropriate for infectious diseases. We account for phylogenetic patterns of viruses in susceptible–infected (SI), susceptible–infected–susceptible (SIS), and susceptible–infected–recovered (SIR) models of infectious disease, and our approach may be a viable alternative to demographic models used to reconstruct epidemic dynamics. The method allows epidemiological parameters, such as the reproductive number, to be estimated directly from viral sequence data. We also describe patterns of phylogenetic clustering that are often construed as arising from a short chain of transmissions. Our model reproduces the moments of the distribution of phylogenetic cluster sizes and may therefore serve as a null hypothesis for cluster sizes under simple epidemiological models. We examine a small cross-sectional sample of human immunodeficiency (HIV)-1 sequences collected in the United States and compare our results to standard estimates of effective population size. Estimated prevalence is consistent with estimates of effective population size and the known history of the HIV epidemic. While our model accurately estimates prevalence during exponential growth, we find that periods of decline are harder to identify.COALESCENT theory has found wide applications for inference of viral phylogenies (Nee et al. 1996; Rosenberg and Nordborg 2002; Drummond et al. 2005) and estimation of epidemic prevalence (Yusim et al. 2001; Robbins et al. 2003; Wilson et al. 2005), yet there have been few attempts to formally integrate coalescent theory with standard epidemiological models (Pybus et al. 2001; Goodreau 2006). While epidemiological models such as susceptible–infected–recovered (SIR) consider the dynamics of an entire population going forward in time, the coalescent theory operates on a small sample of an infected subpopulation and models the merging of lineages backward in time until a common ancestor has been reached. The original coalescent theory was based on a population of constant size with discrete generations (Kingman 1982a,b). Numerous extensions have been made for populations with overlapping generations in continuous time, exponential or logistic growth (Griffiths and Tavare 1994), and stochastically varying size (Kaj and Krone 2003). However, infectious disease epidemics are a special case of a variable size population, often characterized by early explosive growth followed by decline that leads to extinction or an endemic steady state.If superinfection is rare and if mutation is fast relative to epidemic growth, each lineage in a phylogenetic tree corresponds to a single infected individual with its own unique viral population. An infection event viewed in reverse time is equivalent to the coalescence of two lineages and every transmission of the virus between hosts can generate a new branch in the phylogeny of consensus viral isolates from infected individuals. Recently diverged sequences should represent transmissions in the recent past, and branches close to the root of a tree should represent transmissions from long ago. Consequently, branching patterns provide information about the frequency of transmissions over time (Wilson et al. 2005). The correspondence between transmission and phylogenetic branching is easiest to detect for viruses such as human immunodeficiency virus (HIV) and hepatitis C virus that have a high mutation rate relative to dispersal. Underlying SIR dynamics also apply to other pathogens, although in some cases it may be more difficult to reconstruct the transmission history.We examined the properties of viral phylogenies generated by the most common epidemiological models based on ordinary differential equations (ODEs). We are able to fit epidemiological models to a reconstructed phylogeny for sampled viral sequence data and make inferences regarding the size of the corresponding infected population. Our solution takes the form of an ODE analogous to those used to track epidemic prevalence and thereby provides a convenient link between commonly used epidemiological models and phylodynamics. Virtually all coalescent theory to date has been expressed in terms of integer-valued stochastic processes. Our motivation for using differential equations to describe the coalescent process is a desire to formalize a link with standard epidemiological models that are also expressed in terms of differential equations.We use our method to calculate the distribution of coalescent times for samples of viral sequences, fit SIR models to a viral phylogeny, and calculate median time to the most recent common ancestor (MRCA) of the sample. Our method also provides equations that describe the time evolution of the cluster size distribution (CSD)—the distribution of the number of descendants of a lineage over time. Clusters of related virus are often interpreted as epidemiologically linked. For example, clusters of acute HIV infections may represent short transmission chains between high-risk individuals (Yerly et al. 2001; Hue et al. 2005; Pao et al. 2005; Goodreau 2006; Brenner et al. 2007; Drumright and Frost 2008; Lewis et al. 2008). Because our model reproduces the moments of the cluster size distribution, it can be used to predict the level of clustering as a function of epidemiological conditions. The moments could be directly compared to empirical values or they could be used to reconstruct the entire CSD, whereupon standard statistical tests could be used for comparing distributions.Although our equations describe the macroscopic properties of the population distribution of cluster sizes, we generalize our method to the case of a small cross-sectional sample of sequences. This allows us to develop a likelihood-based approach to fitting SIR models to observed sequences.By considering variable degrees of incidence and the size of the infected population, our solution sheds light on the relationship between coalescent rates and epidemic dynamics. Coalescent rates are low near peak prevalence, but higher when there is a large ratio of incidence to prevalence. This can occur early on, when the epidemic is entering its expansion phase, as well as late if the epidemic has multiple periods of growth.     

Phylodynamic Inference of Bacterial Outbreak Parameters Using Nanopore Sequencing

Nanopore sequencing and phylodynamic modeling have been used to reconstruct the transmission dynamics of viral epidemics, but their application to bacterial pathogens has remained challenging. Cost-effective bacterial genome sequencing and variant calling on nanopore platforms would greatly enhance surveillance and outbreak response in communities without access to sequencing infrastructure. Here, we adapt random forest models for single nucleotide polymorphism (SNP) polishing developed by Sanderson and colleagues (2020. High precision Neisseria gonorrhoeae variant and antimicrobial resistance calling from metagenomic nanopore sequencing. Genome Res. 30(9):1354–1363) to estimate divergence and effective reproduction numbers (R_e) of two methicillin-resistant Staphylococcus aureus (MRSA) outbreaks from remote communities in Far North Queensland and Papua New Guinea (PNG; n = 159). Successive barcoded panels of S. aureus isolates (2 × 12 per MinION) sequenced at low coverage (>5× to 10×) provided sufficient data to accurately infer genotypes with high recall when compared with Illumina references. Random forest models achieved high resolution on ST93 outbreak sequence types (>90% accuracy and precision) and enabled phylodynamic inference of epidemiological parameters using birth–death skyline models. Our method reproduced phylogenetic topology, origin of the outbreaks, and indications of epidemic growth (R_e > 1). Nextflow pipelines implement SNP polisher training, evaluation, and outbreak alignments, enabling reconstruction of within-lineage transmission dynamics for infection control of bacterial disease outbreaks on portable nanopore platforms. Our study shows that nanopore technology can be used for bacterial outbreak reconstruction at competitive costs, providing opportunities for infection control in hospitals and communities without access to sequencing infrastructure, such as in remote northern Australia and PNG.     

Physiological analysis on oscillatory behavior of glucose-insulin regulation by model with delays

Despite temporally forced transmission driving many infectious diseases, analytical insight into its role when combined with stochastic disease processes and non-linear transmission has received little attention. During disease outbreaks, however, the absence of saturation effects early on in well-mixed populations mean that epidemic models may be linearised and we can calculate outbreak properties, including the effects of temporal forcing on fade-out, disease emergence and system dynamics, via analysis of the associated master equations. The approach is illustrated for the unforced and forced SIR and SEIR epidemic models. We demonstrate that in unforced models, initial conditions (and any uncertainty therein) play a stronger role in driving outbreak properties than the basic reproduction number R₀, while the same properties are highly sensitive to small amplitude temporal forcing, particularly when R₀ is small. Although illustrated for the SIR and SEIR models, the master equation framework may be applied to more realistic models, although analytical intractability scales rapidly with increasing system dimensionality. One application of these methods is obtaining a better understanding of the rate at which vector-borne and waterborne infectious diseases invade new regions given variability in environmental drivers, a particularly important question when addressing potential shifts in the global distribution and intensity of infectious diseases under climate change.     

Stochastic models for virus and immune system dynamics

New stochastic models are developed for the dynamics of a viral infection and an immune response during the early stages of infection. The stochastic models are derived based on the dynamics of deterministic models. The simplest deterministic model is a well-known system of ordinary differential equations which consists of three populations: uninfected cells, actively infected cells, and virus particles. This basic model is extended to include some factors of the immune response related to Human Immunodeficiency Virus-1 (HIV-1) infection. For the deterministic models, the basic reproduction number, R₀, is calculated and it is shown that if R₀<1, the disease-free equilibrium is locally asymptotically stable and is globally asymptotically stable in some special cases. The new stochastic models are systems of stochastic differential equations (SDEs) and continuous-time Markov chain (CTMC) models that account for the variability in cellular reproduction and death, the infection process, the immune system activation, and viral reproduction. Two viral release strategies are considered: budding and bursting. The CTMC model is used to estimate the probability of virus extinction during the early stages of infection. Numerical simulations are carried out using parameter values applicable to HIV-1 dynamics. The stochastic models provide new insights, distinct from the basic deterministic models. For the case R₀>1, the deterministic models predict the viral infection persists in the host. But for the stochastic models, there is a positive probability of viral extinction. It is shown that the probability of a successful invasion depends on the initial viral dose, whether the immune system is activated, and whether the release strategy is bursting or budding.     

Inference for nonlinear epidemiological models using genealogies and time series

Phylodynamics - the field aiming to quantitatively integrate the ecological and evolutionary dynamics of rapidly evolving populations like those of RNA viruses - increasingly relies upon coalescent approaches to infer past population dynamics from reconstructed genealogies. As sequence data have become more abundant, these approaches are beginning to be used on populations undergoing rapid and rather complex dynamics. In such cases, the simple demographic models that current phylodynamic methods employ can be limiting. First, these models are not ideal for yielding biological insight into the processes that drive the dynamics of the populations of interest. Second, these models differ in form from mechanistic and often stochastic population dynamic models that are currently widely used when fitting models to time series data. As such, their use does not allow for both genealogical data and time series data to be considered in tandem when conducting inference. Here, we present a flexible statistical framework for phylodynamic inference that goes beyond these current limitations. The framework we present employs a recently developed method known as particle MCMC to fit stochastic, nonlinear mechanistic models for complex population dynamics to gene genealogies and time series data in a Bayesian framework. We demonstrate our approach using a nonlinear Susceptible-Infected-Recovered (SIR) model for the transmission dynamics of an infectious disease and show through simulations that it provides accurate estimates of past disease dynamics and key epidemiological parameters from genealogies with or without accompanying time series data.     

A Class of Pairwise Models for Epidemic Dynamics on Weighted Networks

In this paper, we study the SIS (susceptible–infected–susceptible) and SIR (susceptible–infected–removed) epidemic models on undirected, weighted networks by deriving pairwise-type approximate models coupled with individual-based network simulation. Two different types of theoretical/synthetic weighted network models are considered. Both start from non-weighted networks with fixed topology followed by the allocation of link weights in either (i) random or (ii) fixed/deterministic way. The pairwise models are formulated for a general discrete distribution of weights, and these models are then used in conjunction with stochastic network simulations to evaluate the impact of different weight distributions on epidemic thresholds and dynamics in general. For the SIR model, the basic reproductive ratio R ₀ is computed, and we show that (i) for both network models R ₀ is maximised if all weights are equal, and (ii) when the two models are ‘equally-matched’, the networks with a random weight distribution give rise to a higher R ₀ value. The models with different weight distributions are also used to explore the agreement between the pairwise and simulation models for different parameter combinations.     

Analytic approximation of spatial epidemic models of foot and mouth disease

The effect of spatial heterogeneity in epidemic models has improved with computational advances, yet far less progress has been made in developing analytical tools for understanding such systems. Here, we develop two classes of second-order moment closure methods for approximating the dynamics of a stochastic spatial model of the spread of foot and mouth disease. We consider the performance of such ‘pseudo-spatial’ models as a function of R₀, the locality in disease transmission, farm distribution and geographically-targeted control when an arbitrary number of spatial kernels are incorporated. One advantage of mapping complex spatial models onto simpler deterministic approximations lies in the ability to potentially obtain a better analytical understanding of disease dynamics and the effects of control. We exploit this tractability by deriving analytical results in the invasion stages of an FMD outbreak, highlighting key principles underlying epidemic spread on contact networks and the effect of spatial correlations.     

Modeling and Statistical Analysis of the Spatio-Temporal Patterns of Seasonal Influenza in Israel

Background

Seasonal influenza outbreaks are a serious burden for public health worldwide and cause morbidity to millions of people each year. In the temperate zone influenza is predominantly seasonal, with epidemics occurring every winter, but the severity of the outbreaks vary substantially between years. In this study we used a highly detailed database, which gave us both temporal and spatial information of influenza dynamics in Israel in the years 1998–2009. We use a discrete-time stochastic epidemic SIR model to find estimates and credible confidence intervals of key epidemiological parameters.

Findings

Despite the biological complexity of the disease we found that a simple SIR-type model can be fitted successfully to the seasonal influenza data. This was true at both the national levels and at the scale of single cities.The effective reproductive number R_e varies between the different years both nationally and among Israeli cities. However, we did not find differences in R_e between different Israeli cities within a year. R _e was positively correlated to the strength of the spatial synchronization in Israel. For those years in which the disease was more “infectious”, then outbreaks in different cities tended to occur with smaller time lags. Our spatial analysis demonstrates that both the timing and the strength of the outbreak within a year are highly synchronized between the Israeli cities. We extend the spatial analysis to demonstrate the existence of high synchrony between Israeli and French influenza outbreaks.

Conclusions

The data analysis combined with mathematical modeling provided a better understanding of the spatio-temporal and synchronization dynamics of influenza in Israel and between Israel and France. Altogether, we show that despite major differences in demography and weather conditions intra-annual influenza epidemics are tightly synchronized in both their timing and magnitude, while they may vary greatly between years. The predominance of a similar main strain of influenza, combined with population mixing serve to enhance local and global influenza synchronization within an influenza season.     

A comparison of a deterministic and stochastic model for Hepatitis C with an isolation stage

We formulate a deterministic epidemic model for the spread of Hepatitis C containing an acute, chronic and isolation class and analyse the effects of the isolation class on the transmission dynamics of the disease. We calculate the basic reproduction number R₀ and show that for R₀≤1, the disease-free equilibrium is globally asymptotically stable. In addition, it is shown that for a special case when R₀>1, the endemic equilibrium is locally asymptotically stable. Furthermore, an analogous stochastic epidemic model for Hepatitis C is formulated using a continuous time Markov chain. Numerical simulations are used to estimate the mean, variance and probability distributions of the discrete random variables and these are compared to the steady-state solutions of the deterministic model. Finally, the expected time to disease extinction is estimated for the stochastic model and the impact of isolation on the time to extinction is explored.     

Fundamental Identifiability Limits in Molecular Epidemiology

Viral phylogenies provide crucial information on the spread of infectious diseases, and many studies fit mathematical models to phylogenetic data to estimate epidemiological parameters such as the effective reproduction ratio (R_e) over time. Such phylodynamic inferences often complement or even substitute for conventional surveillance data, particularly when sampling is poor or delayed. It remains generally unknown, however, how robust phylodynamic epidemiological inferences are, especially when there is uncertainty regarding pathogen prevalence and sampling intensity. Here, we use recently developed mathematical techniques to fully characterize the information that can possibly be extracted from serially collected viral phylogenetic data, in the context of the commonly used birth-death-sampling model. We show that for any candidate epidemiological scenario, there exists a myriad of alternative, markedly different, and yet plausible “congruent” scenarios that cannot be distinguished using phylogenetic data alone, no matter how large the data set. In the absence of strong constraints or rate priors across the entire study period, neither maximum-likelihood fitting nor Bayesian inference can reliably reconstruct the true epidemiological dynamics from phylogenetic data alone; rather, estimators can only converge to the “congruence class” of the true dynamics. We propose concrete and feasible strategies for making more robust epidemiological inferences from viral phylogenetic data.     

Estimated epidemiologic parameters and morbidity associated with pandemic H1N1 influenza

Background

In the face of an influenza pandemic, accurate estimates of epidemiologic parameters are required to help guide decision-making. We sought to estimate epidemiologic parameters for pandemic H1N1 influenza using data from initial reports of laboratory-confirmed cases.

Methods

We obtained data on laboratory-confirmed cases of pandemic H1N1 influenza reported in the province of Ontario, Canada, with dates of symptom onset between Apr. 13 and June 20, 2009. Incubation periods and duration of symptoms were estimated and fit to parametric distributions. We used competing-risk models to estimate risk of hospital admission and case-fatality rates. We used a Markov Chain Monte Carlo model to simulate disease transmission.

Results

The median incubation period was 4 days and the duration of symptoms was 7 days. Recovery was faster among patients less than 18 years old than among older patients (hazard ratio 1.23, 95% confidence interval 1.06–1.44). The risk of hospital admission was 4.5% (95% CI 3.8%–5.2%) and the case-fatality rate was 0.3% (95% CI 0.1%–0.5%). The risk of hospital admission was highest among patients less than 1 year old and those 65 years or older. Adults more than 50 years old comprised 7% of cases but accounted for 7 of 10 initial deaths (odds ratio 28.6, 95% confidence interval 7.3–111.2). From the simulation models, we estimated the following values (and 95% credible intervals): a mean basic reproductive number (R₀, the number of new cases created by a single primary case in a susceptible population) of 1.31 (1.25–1.38), a mean latent period of 2.62 (2.28–3.12) days and a mean duration of infectiousness of 3.38 (2.06–4.69) days. From these values we estimated a serial interval (the average time from onset of infectiousness in a case to the onset of infectiousness in a person infected by that case) of 4–5 days.

Interpretation

The low estimates for R₀ indicate that effective mitigation strategies may reduce the final epidemic impact of pandemic H1N1 influenza.The emergence and global spread of pandemic H1N1 influenza led the World Health Organization to declare a pandemic on June 11, 2009. As the pandemic spreads, countries will need to make decisions about strategies to mitigate and control disease in the face of uncertainty.For novel infectious diseases, accurate estimates of epidemiologic parameters can help guide decision-making. A key parameter for any new disease is the basic reproductive number (R₀), defined as the average number of new cases created by a single primary case in a susceptible population. R₀ affects the growth rate of an epidemic and the final number of infected people. It also informs the optimal choice of control strategies. Other key parameters that affect use of resources, disease burden and societal costs during a pandemic are duration of illness, rate of hospital admission and case-fatality rate. Early in an epidemic, the case-fatality rate may be underestimated because of the temporal lag between onset of infection and death; the delay between initial identification of a new case and death may lead to an apparent increase in deaths several weeks into an epidemic that is an artifact of the natural history of the disease.We used data from initial reports of laboratory-confirmed pandemic H1N1 influenza to estimate epidemiologic parameters for pandemic H1N1 influenza. The parameters included R₀, incubation period and duration of illness. We also estimated risk of hospital admission and case-fatality rates, which can be used to estimate the burden of illness likely to be associated with this disease.     

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.     

Phylodynamic Inference for Structured Epidemiological Models

Coalescent theory is routinely used to estimate past population dynamics and demographic parameters from genealogies. While early work in coalescent theory only considered simple demographic models, advances in theory have allowed for increasingly complex demographic scenarios to be considered. The success of this approach has lead to coalescent-based inference methods being applied to populations with rapidly changing population dynamics, including pathogens like RNA viruses. However, fitting epidemiological models to genealogies via coalescent models remains a challenging task, because pathogen populations often exhibit complex, nonlinear dynamics and are structured by multiple factors. Moreover, it often becomes necessary to consider stochastic variation in population dynamics when fitting such complex models to real data. Using recently developed structured coalescent models that accommodate complex population dynamics and population structure, we develop a statistical framework for fitting stochastic epidemiological models to genealogies. By combining particle filtering methods with Bayesian Markov chain Monte Carlo methods, we are able to fit a wide class of stochastic, nonlinear epidemiological models with different forms of population structure to genealogies. We demonstrate our framework using two structured epidemiological models: a model with disease progression between multiple stages of infection and a two-population model reflecting spatial structure. We apply the multi-stage model to HIV genealogies and show that the proposed method can be used to estimate the stage-specific transmission rates and prevalence of HIV. Finally, using the two-population model we explore how much information about population structure is contained in genealogies and what sample sizes are necessary to reliably infer parameters like migration rates.     

Modelling HIV/AIDS in the presence of an HIV testing and screening campaign

Preventing and managing the HIV/AIDS epidemic in South Africa will dominate the next decade and beyond. Reduction of new HIV infections by implementing a comprehensive national HIV prevention programme at a sufficient scale to have real impact remains a priority. In this paper, a deterministic HIV/AIDS model that incorporates condom use, screening through HIV counseling and testing (HCT), regular testing and treatment as control strategies is proposed with the objective of quantifying the effectiveness of HCT in preventing new infections and predicting the long-term dynamics of the epidemic. It is found that a backward bifurcation occurs if the rate of screening is below a certain threshold, suggesting that the classical requirement for the basic reproduction number to be below unity though necessary, is not sufficient for disease control in this case. The global stabilities of the equilibria under certain conditions are determined in terms of the model reproduction number R₀. Numerical simulations are performed and the model is fitted to data on HIV prevalence in South Africa. The effects of changes in some key epidemiological parameters are investigated. Projections are made to predict the long-term dynamics of the disease. The epidemiological implications of such projections on public health planning and management are discussed.     

Contact rate calculation for a basic epidemic model

Mass-action epidemic models are the foundation of the majority of studies of disease dynamics in human and animal populations. Here, a kinetic model of mobile susceptible and infective individuals in a two-dimensional domain is introduced, and an examination of the contact process results in a mass-action-like term for the generation of new infectives. The conditions under which density dependent and frequency dependent transmission terms emerge are clarified. Moreover, this model suggests that epidemics in large mobile spatially distributed populations can be well described by homogeneously mixing mass-action models. The analysis generates an analytic formula for the contact rate (β) and the basic reproductive ratio (R₀) of an infectious pathogen, which contains a mixture of demographic and epidemiological parameters. The analytic results are compared with a simulation and are shown to give good agreement. The simulation permits the exploration of more realistic movement strategies and their consequent effect on epidemic dynamics.     

A generalized cholera model and epidemic–endemic analysis

The transmission of cholera involves both human-to-human and environment-to-human pathways that complicate its dynamics. In this paper, we present a new and unified deterministic model that incorporates a general incidence rate and a general formulation of the pathogen concentration to analyse the dynamics of cholera. Particularly, this work unifies many existing cholera models proposed by different authors. We conduct equilibrium analysis to carefully study the complex epidemic and endemic behaviour of the disease. Our results show that despite the incorporation of the environmental component, there exists a forward transcritical bifurcation at R ₀=1 for the combined human–environment epidemiological model under biologically reasonable conditions.     

A mechanistic and data-driven reconstruction of the time-varying reproduction number: Application to the COVID-19 epidemic

The effective reproduction number R_eff is a critical epidemiological parameter that characterizes the transmissibility of a pathogen. However, this parameter is difficult to estimate in the presence of silent transmission and/or significant temporal variation in case reporting. This variation can occur due to the lack of timely or appropriate testing, public health interventions and/or changes in human behavior during an epidemic. This is exactly the situation we are confronted with during this COVID-19 pandemic. In this work, we propose to estimate R_eff for the SARS-CoV-2 (the etiological agent of the COVID-19), based on a model of its propagation considering a time-varying transmission rate. This rate is modeled by a Brownian diffusion process embedded in a stochastic model. The model is then fitted by Bayesian inference (particle Markov Chain Monte Carlo method) using multiple well-documented hospital datasets from several regions in France and in Ireland. This mechanistic modeling framework enables us to reconstruct the temporal evolution of the transmission rate of the COVID-19 based only on the available data. Except for the specific model structure, it is non-specifically assumed that the transmission rate follows a basic stochastic process constrained by the observations. This approach allows us to follow both the course of the COVID-19 epidemic and the temporal evolution of its R_eff(t). Besides, it allows to assess and to interpret the evolution of transmission with respect to the mitigation strategies implemented to control the epidemic waves in France and in Ireland. We can thus estimate a reduction of more than 80% for the first wave in all the studied regions but a smaller reduction for the second wave when the epidemic was less active, around 45% in France but just 20% in Ireland. For the third wave in Ireland the reduction was again significant (>70%).