Heterogeneity in multiple transmission pathways: modelling the spread of cholera and other waterborne disease in networks with a common water source

Many factors influencing disease transmission vary throughout and across populations. For diseases spread through multiple transmission pathways, sources of variation may affect each transmission pathway differently. In this paper we consider a disease that can be spread via direct and indirect transmission, such as the waterborne disease cholera. Specifically, we consider a system of multiple patches with direct transmission occurring entirely within patch and indirect transmission via a single shared water source. We investigate the effect of heterogeneity in dual transmission pathways on the spread of the disease. We first present a 2-patch model for which we examine the effect of variation in each pathway separately and propose a measure of heterogeneity that incorporates both transmission mechanisms and is predictive of R₀. We also explore how heterogeneity affects the final outbreak size and the efficacy of intervention measures. We conclude by extending several results to a more general n-patch setting.     

Historical Epidemiology of the Second Cholera Pandemic: Relevance to Present Day Disease Dynamics

Despite nearly two centuries of study, the fundamental transmission dynamic properties of cholera remain incompletely characterized. We used historical time-series data on the spread of cholera in twelve European and North American cities during the second cholera pandemic, as reported in Amariah Brigham’s 1832 A Treatise on Epidemic Cholera, to parameterize simple mathematical models of cholera transmission. Richards growth models were used to derive estimates of the basic reproductive number (R₀) (median: 16.0, range: 1.9 to 550.9) and the proportion of unrecognized cases (mean: 96.3%, SD: 0.04%). Heterogeneity in model-generated R₀ estimates was consistent with variability in cholera dynamics described by contemporary investigators and may represent differences in the nature of cholera spread. While subject to limitations associated with measurement and the absence of microbiological diagnosis, historical epidemic data are a potentially rich source for understanding pathogen dynamics in the absence of control measures, particularly when used in conjunction with simple and readily interpretable mathematical models.     

Dynamical behavior of a hepatitis B virus transmission model with vaccination

Hepatitis B virus (HBV) infection is a globally health problem. In 2005, the WHO Western Pacific Regional Office set a goal of reducing chronic HBV infection rate to less than 2% among children five years of age by 2012, as an interim milestone towards the final goal of less than 1%. Many countries made some plans (such as free HBV vaccination program for all neonates in China now) to control the transmission HBV. We develop a model to explore the impact of vaccination and other controlling measures of HBV infection. The model has simple dynamical behavior which has a globally asymptotically stable disease-free equilibrium when the basic reproduction number R₀≤1, and a globally asymptotically stable endemic equilibrium when R₀>1. Numerical simulation results show that the vaccination is a very effective measure to control the infection and they also give some useful comments on controlling the transmission of HBV.     

Qualitative analysis of models with different treatment protocols to prevent antibiotic resistance

This paper is concerned with the qualitative analysis of two models [S. Bonhoeffer, M. Lipsitch, B.R. Levin, Evaluating treatment protocols to prevent antibiotic resistance, Proc. Natl. Acad. Sci. USA 94 (1997) 12106] for different treatment protocols to prevent antibiotic resistance. Detailed qualitative analysis about the local or global stability of the equilibria of both models is carried out in term of the basic reproduction number R₀. For the model with a single antibiotic therapy, we show that if R₀ < 1, then the disease-free equilibrium is globally asymptotically stable; if R₀ > 1, then the disease-endemic equilibrium is globally asymptotically stable. For the model with multiple antibiotic therapies, stabilities of various equilibria are analyzed and combining treatment is shown better than cycling treatment. Numerical simulations are performed to show that the dynamical properties depend intimately upon the parameters.     

The Interaction between Seasonality and Pulsed Interventions against Malaria in Their Effects on the Reproduction Number

The basic reproduction number (R ₀) is an important quantity summarising the dynamics of an infectious disease, as it quantifies how much effort is needed to control transmission. The relative change in R ₀ due to an intervention is referred to as the effect size. However malaria and other diseases are often highly seasonal and some interventions have time-varying effects, meaning that simple reproduction number formulae cannot be used. Methods have recently been developed for calculating R ₀ for diseases with seasonally varying transmission. I extend those methods to calculate the effect size of repeated rounds of mass drug administration, indoor residual spraying and other interventions against Plasmodium falciparum malaria in seasonal settings in Africa. I show that if an intervention reduces transmission from one host to another by a constant factor, then its effect size is the same in a seasonal as in a non-seasonal setting. The optimal time of year for drug administration is in the low season, whereas the best time for indoor residual spraying or a vaccine which reduces infection rates is just before the high season. In general, the impact of time-varying interventions increases with increasing seasonality, if carried out at the optimal time of year. The effect of combinations of interventions that act at different stages of the transmission cycle is roughly the product of the separate effects. However for individual time-varying interventions, it is necessary to use methods such as those developed here rather than inserting the average efficacy into a simple formula.     

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.     

The effect of transmission route on plant virus epidemic development and disease control

A model for indirect vector transmission and epidemic development of plant viruses is extended to consider direct transmission through vector mating. A basic reproduction number is derived which is the sum of the R₀ values specific for three transmission routes. We analyse the model to determine the effect of direct transmission on plant disease control directed against indirect transmission. Increasing the rate of horizontal sexual transmission means that vector control rate or indirect transmission rate must be increased/decreased substantially to maintain R₀ at a value less than 1. By contrast, proportionately increasing the probability of transovarial transmission has little effect. Expressions are derived for the steady-state values of the viruliferous vector population. There is clear advantage for an insect virus in indirect transmission to plants, especially where the sexual and transovarial transmission rates are low; however information on virulence-transmissibility relationships is required to explain the evolution of a plant virus from an insect virus.     

An SIS patch model with variable transmission coefficients

In this paper, an SIS patch model with non-constant transmission coefficients is formulated to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches. The basic reproduction number R₀ is determined. It is shown that the disease-free equilibrium is globally asymptotically stable if R₀?1, and the disease is uniformly persistent and there exists at least one endemic equilibrium if R₀>1. In particular, when the disease is non-fatal and the travel rates of susceptible and infectious individuals in each patch are the same, the endemic equilibrium is unique and is globally asymptotically stable as R₀>1. Numerical calculations are performed to illustrate some results for the case with two patches.     

Separate roles of the latent and infectious periods in shaping the relation between the basic reproduction number and the intrinsic growth rate of infectious disease outbreaks

Two approximations are commonly used to describe the spread of an infectious disease at its early phase: (i) the branching processes based on the generation concept and (ii) the exponential growth over calendar time. The former is characterized by a mean parameter: the reproduction number R₀. The latter is characterized by a growth rate ρ, also known as the Malthusian number. It is common to use empirically observed ρ to assess R₀ using formulae derived either when both the latent and infectious periods follow exponential distributions or assuming both are fixed non-random quantities. This paper first points out that most of these formulae are special cases when the latent and infectious periods are gamma distributed, given by a closed-form solution in Anderson and Watson [1980. On the spread of a disease with gamma distributed latent and infectious periods. Biometrika 67 (1), 191-198]. A more general result will be then established which takes the result in Anderson and Watson [1980. On the spread of a disease with gamma distributed latent and infectious periods. Biometrika 67 (1), 191-198] as its special case. Three aspects separately shape the relationship between ρ and R₀. They are: (i) the intensity of infectious contacts as a counting process; (ii) the distribution of the latent period and (iii) the distribution of the infectious period. This article also distinguishes the generation time from the transmission interval. It shows that whereas the distribution of the generation time can be derived by the latent and infectious period distributions, the distribution of the transmission interval is also determined by the intensity of infectious contacts as a counting process and hence by R₀. Some syntheses among R₀, ρ and the average transmission interval are discussed. Numerical examples and simulation results are supplied to support the theoretical arguments.     

Multiple Transmission Pathways and Disease Dynamics in a Waterborne Pathogen Model

Multiple transmission pathways exist for many waterborne diseases, including cholera, Giardia, Cryptosporidium, and Campylobacter. Theoretical work exploring the effects of multiple transmission pathways on disease dynamics is incomplete. Here, we consider a simple ODE model that extends the classical SIR framework by adding a compartment (W) that tracks pathogen concentration in the water. Infected individuals shed pathogen into the water compartment, and new infections arise both through exposure to contaminated water, as well as by the classical SIR person–person transmission pathway. We compute the basic reproductive number (ℛ₀), epidemic growth rate, and final outbreak size for the resulting “SIWR” model, and examine how these fundamental quantities depend upon the transmission parameters for the different pathways. We prove that the endemic disease equilibrium for the SIWR model is globally stable. We identify the pathogen decay rate in the water compartment as a key parameter determining when the distinction between the different transmission routes in the SIWR model is important. When the decay rate is slow, using an SIR model rather than the SIWR model can lead to under-estimates of the basic reproductive number and over-estimates of the infectious period.     

Effect of Synaptic Transmission on Viral Fitness in HIV Infection

HIV can spread through its target cell population either via cell-free transmission, or by cell-to-cell transmission, presumably through virological synapses. Synaptic transmission entails the transfer of tens to hundreds of viruses per synapse, a fraction of which successfully integrate into the target cell genome. It is currently not understood how synaptic transmission affects viral fitness. Using a mathematical model, we investigate how different synaptic transmission strategies, defined by the number of viruses passed per synapse, influence the basic reproductive ratio of the virus, R₀, and virus load. In the most basic scenario, the model suggests that R₀ is maximized if a single virus particle is transferred per synapse. R₀ decreases and the infection eventually cannot be maintained for larger numbers of transferred viruses, because multiple infection of the same cell wastes viruses that could otherwise enter uninfected cells. To explain the relatively large number of HIV copies transferred per synapse, we consider additional biological assumptions under which an intermediate number of viruses transferred per synapse could maximize R₀. These include an increased burst size in multiply infected cells, the saturation of anti-viral factors upon infection of cells, and rate limiting steps during the process of synapse formation.     

Contact-based transmission models in terrestrial gastropod populations infected with parasitic mites

Parasite transmission fundamentally affects the epidemiology of host-parasite systems, and is considered to be a key element in the epidemiological modelling of infectious diseases. Recent research has stressed the importance of detailed disease-specific variables involved in the transmission process. Riccardoella limacum is a hematophagous mite living in the mantle cavity of terrestrial gastropods. In this study, we experimentally examined whether the transmission success of R. limacum is affected by the contact frequency, parasite load and/or behaviour of the land snail Arianta arbustorum, a common host of R. limacum. In the experiment the transmission success was mainly affected by physical contacts among snails and slightly influenced by parasite intensity of the infected snail. Using these results we developed two different transmission models based on contact frequencies and transmission probability among host snails. As parameters for the models we used life-history data from three natural A. arbustorum populations with different population densities. Data on contact frequencies of video-recorded snail groups were used to fit the density response of the contact function, assuming either a linear relationship (model 1) or a second-degree polynomial relationship based on the ideal gas model of animal encounter (model 2). We calculated transmission coefficients (β), basic reproductive ratios (R₀) and host threshold population densities for parasite persistence in the three A. arbustorum populations. We found higher transmission coefficients (β) and larger R₀-values in model 1 than in model 2. Furthermore, the host population with the highest density showed larger R₀-values (16.47-22.59) compared to populations with intermediate (2.71-7.45) or low population density (0.75-4.10). Host threshold population density for parasite persistence ranged from 0.35 to 2.72 snails per m². Our results show that the integration of the disease-relevant biology of the organisms concerned may improve models of host-parasite dynamics.     

Understanding Spatio-Temporal Variability in the Reproduction Ratio of the Bluetongue (BTV-1) Epidemic in Southern Spain (Andalusia) in 2007 Using Epidemic Trees

Andalusia (Southern Spain) is considered one of the main routes of introduction of bluetongue virus (BTV) into Europe, evidenced by a devastating epidemic caused by BTV-1 in 2007. Understanding the pattern and the drivers of BTV-1 spread in Andalusia is critical for effective detection and control of future epidemics. A long-standing metric for quantifying the behaviour of infectious diseases is the case-reproduction ratio (R_t), defined as the average number of secondary cases arising from a single infected case at time t (for t>0). Here we apply a method using epidemic trees to estimate the between-herd case reproduction ratio directly from epidemic data allowing the spatial and temporal variability in transmission to be described. We then relate this variability to predictors describing the hosts, vectors and the environment to better understand why the epidemic spread more quickly in some regions or periods. The R_t value for the BTV-1 epidemic in Andalusia peaked in July at 4.6, at the start of the epidemic, then decreased to 2.2 by August, dropped below 1 by September (0.8), and by October it had decreased to 0.02. BTV spread was the consequence of both local transmission within established disease foci and BTV expansion to distant new areas (i.e. new foci), which resulted in a high variability in BTV transmission, not only among different areas, but particularly through time, which suggests that general control measures applied at broad spatial scales are unlikely to be effective. This high variability through time was probably due to the impact of temperature on BTV transmission, as evidenced by a reduction in the value of R_t by 0.0041 for every unit increase (day) in the extrinsic incubation period (EIP), which is itself directly dependent on temperature. Moreover, within the range of values at which BTV-1 transmission occurred in Andalusia (20.6°C to 29.5°C) there was a positive correlation between temperature and R_t values, although the relationship was not linear, probably as a result of the complex relationship between temperature and the different parameters affecting BTV transmission. R_t values for BTV-1 in Andalusia fell below the threshold of 1 when temperatures dropped below 21°C, a much higher threshold than that reported in other BTV outbreaks, such as the BTV-8 epidemic in Northern Europe. This divergence may be explained by differences in the adaptation to temperature of the main vectors of the BTV-1 epidemic in Andalusia (Culicoides imicola) compared those of the BTV-8 epidemic in Northern Europe (Culicoides obsoletus). Importantly, we found that BTV transmission (R_t value) increased significantly in areas with higher densities of sheep. Our analysis also established that control of BTV-1 in Andalusia was complicated by the simultaneous establishment of several distant foci at the start of the epidemic, which may have been caused by several independent introductions of infected vectors from the North of Africa. We discuss the implications of these findings for BTV surveillance and control in this region of Europe.     

Impact of group mixing on disease dynamics

A general mathematical model is proposed to study the impact of group mixing in a heterogeneous host population on the spread of a disease that confers temporary immunity upon recovery. The model contains general distribution functions that account for the probabilities that individuals remain in the recovered class after recovery. For this model, the basic reproduction number R₀ is identified. It is shown that if R₀<1, then the disease dies out in the sense that the disease free equilibrium is globally asymptotically stable; whereas if R₀>1, this equilibrium becomes unstable. In this latter case, depending on the distribution functions and the group mixing strengths, the disease either persists at a constant endemic level or exhibits sustained oscillatory behavior.     

The effects of human movement on the persistence of vector-borne diseases

With the recent resurgence of vector-borne diseases due to urbanization and development there is an urgent need to understand the dynamics of vector-borne diseases in rapidly changing urban environments. For example, many empirical studies have produced the disturbing finding that diseases continue to persist in modern city centers with zero or low rates of transmission. We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian models that mimic human commuting behavior and (ii) Eulerian models that mimic human migration. We determine the basic reproduction number R₀ for both modeling approaches. We show that for both approaches that if the disease-free equilibrium is stable (R₀<1) then it is globally stable and if the disease-free equilibrium is unstable (R₀>1) then there exists a unique positive (endemic) equilibrium that is globally stable among positive solutions. Finally, we prove in general that Lagrangian and Eulerian modeling approaches are not equivalent. The modeling approaches presented provide a framework to explore spatial vector-borne disease dynamics and control in heterogeneous environments. As an example, we consider two patches in which the disease dies out in both patches when there is no movement between them. Numerical simulations demonstrate that the disease becomes endemic in both patches when humans move between the two patches.     

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%).     

On the Exact Measure of Disease Spread in Stochastic Epidemic Models

The basic reproduction number, R ₀, is probably the most important quantity in epidemiology. It is used to measure the transmission potential during the initial phase of an epidemic. In this paper, we are specifically concerned with the quantification of the spread of a disease modeled by a Markov chain. Due to the occurrence of repeated contacts taking place between a typical infective individual and other individuals already infected before, R ₀ overestimates the average number of secondary infections. We present two alternative measures, namely, the exact reproduction number, R _e0, and the population transmission number, R _p , that overcome this difficulty and provide valuable insight. The applicability of R _e0 and R _p to control of disease spread is also examined.     

Multiple blood feeding in mosquitoes shortens the Plasmodium falciparum incubation period and increases malaria transmission potential

Many mosquito species, including the major malaria vector Anopheles gambiae, naturally undergo multiple reproductive cycles of blood feeding, egg development and egg laying in their lifespan. Such complex mosquito behavior is regularly overlooked when mosquitoes are experimentally infected with malaria parasites, limiting our ability to accurately describe potential effects on transmission. Here, we examine how Plasmodium falciparum development and transmission potential is impacted when infected mosquitoes feed an additional time. We measured P. falciparum oocyst size and performed sporozoite time course analyses to determine the parasite’s extrinsic incubation period (EIP), i.e. the time required by parasites to reach infectious sporozoite stages, in An. gambiae females blood fed either once or twice. An additional blood feed at 3 days post infection drastically accelerates oocyst growth rates, causing earlier sporozoite accumulation in the salivary glands, thereby shortening the EIP (reduction of 2.3 ± 0.4 days). Moreover, parasite growth is further accelerated in transgenic mosquitoes with reduced reproductive capacity, which mimic genetic modifications currently proposed in population suppression gene drives. We incorporate our shortened EIP values into a measure of transmission potential, the basic reproduction number R₀, and find the average R₀ is higher (range: 10.1%–12.1% increase) across sub-Saharan Africa than when using traditional EIP measurements. These data suggest that malaria elimination may be substantially more challenging and that younger mosquitoes or those with reduced reproductive ability may provide a larger contribution to infection than currently believed. Our findings have profound implications for current and future mosquito control interventions.