The failure of R0

Approximating reproduction numbers: a general numerical method for age-structured models

In this paper, we introduce a general numerical method to approximate the reproduction numbers of a large class of multi-group, age-structured, population models with a finite age span. To provide complete flexibility in the definition of the birth and transition processes, we propose an equivalent formulation for the age-integrated state within the extended space framework. Then, we discretize the birth and transition operators via pseudospectral collocation. We discuss applications to epidemic models with continuous and piecewise continuous rates, with different interpretations of the age variable (e.g., demographic age, infection age and disease age) and the transmission terms (e.g., horizontal and vertical transmission). The tests illustrate that the method can compute different reproduction numbers, including the basic and type reproduction numbers as special cases.

Herd immunity, vaccination and moral obligation

The public health benefits of herd immunity are often used as the justification for coercive vaccine policies. Yet, 'herd immunity' as a term has multiple referents, which can result in ambiguity, including regarding its role in ethical arguments. The term 'herd immunity' can refer to (1) the herd immunity threshold, at which models predict the decline of an epidemic; (2) the percentage of a population with immunity, whether it exceeds a given threshold or not; and/or (3) the indirect benefit afforded by collective immunity to those who are less immune. Moreover, the accumulation of immune individuals in a population can lead to two different outcomes: elimination (for measles, smallpox, etc) or endemic equilibrium (for COVID-19, influenza, etc). We argue that the strength of a moral obligation for individuals to contribute to herd immunity through vaccination, and by extension the acceptability of coercion, will depend on how 'herd immunity' is interpreted as well as facts about a given disease or vaccine. Among other things, not all uses of 'herd immunity' are equally valid for all pathogens. The optimal conditions for herd immunity threshold effects, as illustrated by measles, notably do not apply to the many pathogens for which reinfections are ubiquitous (due to waning immunity and/or antigenic variation). For such pathogens, including SARS-CoV-2, mass vaccination can only be expected to delay rather than prevent new infections, in which case the obligation to contribute to herd immunity is much weaker, and coercive policies less justifiable.

Spatiotemporal reproduction number with Bayesian model selection for evaluation of emerging infectious disease transmissibility: an application to COVID-19 national surveillance data

BACKGROUND

To control emerging diseases, governments often have to make decisions based on limited evidence. The effective or temporal reproductive number is used to estimate the expected number of new cases caused by an infectious person in a partially susceptible population. While the temporal dynamic is captured in the temporal reproduction number, the dominant approach is currently based on modeling that implicitly treats people within a population as geographically well mixed.

METHODS

In this study we aimed to develop a generic and robust methodology for estimating spatiotemporal dynamic measures that can be instantaneously computed for each location and time within a Bayesian model selection and averaging framework. A simulation study was conducted to demonstrate robustness of the method. A case study was provided of a real-world application to COVID-19 national surveillance data in Thailand.

RESULTS

Overall, the proposed method allowed for estimation of different scenarios of reproduction numbers in the simulation study. The model selection chose the true serial interval when included in our study whereas model averaging yielded the weighted outcome which could be less accurate than model selection. In the case study of COVID-19 in Thailand, the best model based on model selection and averaging criteria had a similar trend to real data and was consistent with previously published findings in the country.

CONCLUSIONS

The method yielded robust estimation in several simulated scenarios of force of transmission with computing flexibility and practical benefits. Thus, this development can be suitable and practically useful for surveillance applications especially for newly emerging diseases. As new outbreak waves continue to develop and the risk changes on both local and global scales, our work can facilitate policymaking for timely disease control.

A discrete model for the evaluation of public policies: The case of Colombia during the COVID-19 pandemic

In mathematical epidemiology, it is usual to implement compartmental models to study the transmission of diseases, allowing comprehension of the outbreak dynamics. Thus, it is necessary to identify the natural history of the disease and to establish promissory relations between the structure of a mathematical model, as well as its parameters, with control-related strategies (real interventions) and relevant socio-cultural behaviors. However, we identified gaps between the model creation and its implementation for the use of decision-makers for policy design. We aim to cover these gaps by proposing a discrete mathematical model with parameters having intuitive meaning to be implemented to help decision-makers in control policy design. The model considers novel contagion probabilities, quarantine, and diffusion processes to represent the recovery and mortality dynamics. We applied mathematical model for COVID-19 to Colombia and some of its localities; moreover, the model structure could be adapted for other diseases. Subsequently, we implemented it on a web platform (MathCOVID) for the usage of decision-makers to simulate the effect of policies such as lock-downs, social distancing, identification in the contagion network, and connectivity among populations. Furthermore, it was possible to assess the effects of migration and vaccination strategies as time-dependent inputs. Finally, the platform was capable of simulating the effects of applying one or more policies simultaneously.

Spatio-temporal modeling of infectious diseases by integrating compartment and point process models

Infectious disease modeling plays an important role in understanding disease spreading dynamics and can be used for prevention and control. The well-known SIR (Susceptible, Infected, and Recovered) compartment model and spatial and spatio-temporal statistical models are common choices for studying problems of this kind. This paper proposes a spatio-temporal modeling framework to characterize infectious disease dynamics by integrating the SIR compartment and log-Gaussian Cox process (LGCP) models. The method's performance is assessed via simulation using a combination of real and synthetic data for a region in São Paulo, Brazil. We also apply our modeling approach to analyze COVID-19 dynamics in Cali, Colombia. The results show that our modified LGCP model, which takes advantage of information obtained from the previous SIR modeling step, leads to a better forecasting performance than equivalent models that do not do that. Finally, the proposed method also allows the incorporation of age-stratified contact information, which provides valuable decision-making insights.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s00477-022-02354-4.

Ground reality versus model-based computation of basic reproductive numbers in epidemics

Computation of basic reproductive numbers is one of the primary goals of epidemic modelers. There are several challenges in such computations, especially when the data from the virus transmission networks are not so easy to collect; this makes model validation almost impossible. We provide a technical comment on the precautions to be taken while computing model-based basic reproductive numbers so that the ground realities of such computation are maintained. Basic reproductive numbers need to be adjusted retrospectively to compensate for reporting errors within the epidemic spread networks. Such an adjustment would lead to revised pandemic preparedness and mitigation plans.

Spatial effects in parasite-induced marine diseases of immobile hosts

Emerging marine infectious diseases pose a substantial threat to marine ecosystems and the conservation of their biodiversity. Compartmental models of epidemic transmission in marine sessile organisms, available only recently, are based on non-spatial descriptions in which space is homogenized and parasite mobility is not explicitly accounted for. However, in realistic scenarios epidemic transmission is conditioned by the spatial distribution of hosts and the parasites' mobility patterns, calling for an explicit description of space. In this work, we develop a spatially explicit individual-based model to study disease transmission by waterborne parasites in sessile marine populations. We investigate the impact of spatial disease transmission through extensive numerical simulations and theoretical analysis. Specifically, the effects of parasite mobility into the epidemic threshold and the temporal progression of the epidemic are assessed. We show that larger values of pathogen mobility imply more severe epidemics, as the number of infections increases, and shorter timescales to extinction. An analytical expression for the basic reproduction number of the spatial model, R ~ 0 , is derived as a function of the non-spatial counterpart, R ₀, which characterizes a transition between a disease-free and a propagation phase, in which the disease propagates over a large fraction of the system.

Computing R0 of dynamic models by a definition-based method

Objectives

Computing the basic reproduction number (R₀) in deterministic dynamical models is a hot topic and is frequently demanded by researchers in public health. The next-generation methods (NGM) are widely used for such computation, however, the results of NGM are usually not to be the true R₀ but only a threshold quantity with little interpretation. In this paper, a definition-based method (DBM) is proposed to solve such a problem.

Methods

Start with the definition of R₀, consider different states that one infected individual may develop into, and take expectations. A comparison with NGM has proceeded. Numerical verification is performed using parameters fitted by data of COVID-19 in Hunan Province.

Results

DBM and NGM give identical expressions for single-host models with single-group and interactive R_ij of single-host models with multi-groups, while difference arises for models partitioned into subgroups. Numerical verification showed the consistencies and differences between DBM and NGM, which supports the conclusion that R₀ derived by DBM with true epidemiological interpretations are better.

Conclusions

DBM is more suitable for single-host models, especially for models partitioned into subgroups. However, for multi-host dynamic models where the true R₀ is failed to define, we may turn to the NGM for the threshold R₀.

A review on COVID-19 transmission, epidemiological features, prevention and vaccination

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has caused hundreds of millions of infections and millions of deaths over past two years. Currently, many countries have still not been able to take the pandemic under control. In this review, we systematically summarized what we have done to mitigate the COVID-19 pandemic, from the perspectives of virus transmission, public health control measures, to the development and vaccination of COVID-19 vaccines. As a virus most likely coming from bats, the SARS-CoV-2 may transmit among people via airborne, faecal-oral, vertical or foodborne routes. Our meta-analysis suggested that the R₀ of COVID-19 was 2.9 (95% CI: 2.7–3.1), and the estimates in Africa and Europe could be higher. The median R_t could decrease by 23–96% following the nonpharmacological interventions, including lockdown, isolation, social distance, and face mask, etc. Comprehensive intervention and lockdown were the most effective measures to control the pandemic. According to the pooled R₀ in our meta-analysis, there should be at least 93.3% (95% CI: 89.9–96.2%) people being vaccinated around the world. Limited amount of vaccines and the inequity issues in vaccine allocation call for more international cooperation to achieve the anti-epidemic goals and vaccination fairness.

Modeling the evolution of SARS-CoV-2 under non-pharmaceutical interventions and testing

Background and objectives

Social and behavioral non-pharmaceutical interventions (NPIs), such as mask-wearing, social distancing and travel restrictions, as well as diagnostic tests, have been broadly implemented in response to the COVID-19 pandemic. Epidemiological models and data analysis affirm that wide adoption of NPIs helps to control the pandemic. However, SARS-CoV-2 has extensively demonstrated its ability to evolve. Therefore, it is crucial to examine how NPIs may affect the evolution of the virus. Such evolution could have important effects on the spread and impact of the pandemic.

Methodology

We used evo-epidemiological models to examine the effect of NPIs and testing on two evolutionary trajectories for SARS-CoV-2: attenuation and test evasion.

Results

Our results show that when stronger measures are taken, selection may act to reduce disease severity. Additionally, the timely application of NPIs could significantly affect the competition between viral strains, favoring the milder strain. Furthermore, a higher testing rate can select for a test-evasive viral strain, even if that strain is less infectious than the detectable competing strain. Importantly, if a less detectable strain evolves, epidemiological metrics such as confirmed daily cases may distort our assessment of the pandemic.

Conclusions and implications

Our results highlight the important implications NPIs can have on the evolution of SARS-CoV-2.

Lay Summary

We used evo-epidemiological models to examine the effect of non-pharmaceutical interventions and testing on two evolutionary trajectories for SARS-CoV-2: attenuation and test evasion. Our results show that when stronger measures are taken, selection may act to reduce disease severity.

On the reproduction number in epidemics

This note provides an elementary derivation of the basic reproduction number and the effective reproduction number from the discrete Kermack-McKendrick epidemic model. The derived formulae match those derived from the continuous version of the model; however, the derivation from discrete model is a bit more intuitive. The MATLAB functions for its calculation are given. A real case example is considered and the results are compared with those obtained by the R0 and the EpiEstim software packages.

Modeling the Spread and Control of COVID-19

Data-centric models of COVID-19 have been attempted, but have certain limitations. In this work, we propose an agent-based model of the epidemic in a confined space of agents representing humans. An extension to the SEIR model allows us to consider the difference between the appearance (black-box view) of the spread of disease and the real situation (glass-box view). Our model allows for simulations of lockdowns, social distancing, personal hygiene, quarantine, and hospitalization, with further considerations of different parameters, such as the extent to which hygiene and social distancing are observed in a population. Our results provide qualitative indications of the effects of various policies and parameters, for instance, that lockdowns by themselves are extremely unlikely to bring an end to an epidemic and may indeed make things worse, that social distancing is more important than personal hygiene, and that the growth of infection is significantly reduced for moderately high levels of social distancing and hygiene, even in the absence of herd immunity.

A comprehensive estimation of country-level basic reproduction numbers R0 for COVID-19: Regime regression can automatically estimate the end of the exponential phase in epidemic data

In a compartmental epidemic model, the initial exponential phase reflects a fixed interaction between an infectious agent and a susceptible population in steady state, so it determines the basic reproduction number R₀ on its own. After the exponential phase, dynamic complexities like societal responses muddy the practical interpretation of many estimated parameters. The computer program ARRP, already available from sequence alignment applications, automatically estimated the end of the exponential phase in COVID-19 and extracted the exponential growth rate r for 160 countries. By positing a gamma-distributed generation time, the exponential growth method then yielded R₀ estimates for COVID-19 in 160 countries. The use of ARRP ensured that the R₀ estimates were largely freed from any dependency outside the exponential phase. The Prem matrices quantify rates of effective contact for infectious disease. Without using any age-stratified COVID-19 data, but under strong assumptions about the homogeneity of susceptibility, infectiousness, etc., across different age-groups, the Prem contact matrices also yielded theoretical R₀ estimates for COVID-19 in 152 countries, generally in quantitative conflict with the R₀ estimates derived from the exponential growth method. An exploratory analysis manipulating only the Prem contact matrices reduced the conflict, suggesting that age-groups under 20 years did not promote the initial exponential growth of COVID-19 as much as other age-groups. The analysis therefore supports tentatively and tardily, but independently of age-stratified COVID-19 data, the low priority given to vaccinating younger age groups. It also supports the judicious reopening of schools. The exploratory analysis also supports the possibility of suspecting differences in epidemic spread among different age-groups, even before substantial amounts of age-stratified data become available.

A simple but complex enough θ -SIR type model to be used with COVID-19 real data. Application to the case of Italy

Since the start of the COVID-19 pandemic in China many models have appeared in the literature, trying to simulate its dynamics. Focusing on modeling the biological and sociological mechanisms which influence the disease spread, the basic reference example is the SIR model. However, it is too simple to be able to model those mechanisms (including the three main types of control measures: social distancing, contact tracing and health system measures) to fit real data and to simulate possible future scenarios. A question, then, arises: how much and how do we need to complexify a SIR model? We develop a θ -SEIHQRD model, which may be the simplest one satisfying the mentioned requirements for arbitrary territories and can be simplified in particular cases. We show its very good performance in the Italian case and study different future scenarios.

A statistical analysis of the novel coronavirus (COVID-19) in Italy and Spain

The novel coronavirus (COVID-19) that was first reported at the end of 2019 has impacted almost every aspect of life as we know it. This paper focuses on the incidence of the disease in Italy and Spain-two of the first and most affected European countries. Using two simple mathematical epidemiological models-the Susceptible-Infectious-Recovered model and the log-linear regression model, we model the daily and cumulative incidence of COVID-19 in the two countries during the early stage of the outbreak, and compute estimates for basic measures of the infectiousness of the disease including the basic reproduction number, growth rate, and doubling time. Estimates of the basic reproduction number were found to be larger than 1 in both countries, with values being between 2 and 3 for Italy, and 2.5 and 4 for Spain. Estimates were also computed for the more dynamic effective reproduction number, which showed that since the first cases were confirmed in the respective countries the severity has generally been decreasing. The predictive ability of the log-linear regression model was found to give a better fit and simple estimates of the daily incidence for both countries were computed.

Modeling COVID-19 Nonpharmaceutical Interventions: Exploring periodic NPI strategies

We developed a COVID-19 transmission model used as part of RAND's web-based COVID-19 decision support tool that compares the effects of nonpharmaceutical public health interventions (NPIs) on health and economic outcomes. An interdisciplinary approach informed the selection and use of multiple NPIs, combining quantitative modeling of the health/economic impacts of interventions with qualitative assessments of other important considerations (e.g., cost, ease of implementation, equity). This paper provides further details of our model, describes extensions, presents sensitivity analyses, and analyzes strategies that periodically switch between a base NPI level and a higher NPI level. We find that a periodic strategy, if implemented with perfect compliance, could have produced similar health outcomes as static strategies but might have produced better outcomes when considering other measures of social welfare. Our findings suggest that there are opportunities to shape the tradeoffs between economic and health outcomes by carefully evaluating a more comprehensive range of reopening policies.

Monitoring and Tracking the Evolution of a Viral Epidemic Through Nonlinear Kalman Filtering: Application to the Covid-19 Case

This work presents a novel methodology for systematically processing the time series that report the number of positive, recovered and deceased cases from a viral epidemic, such as Covid-19. The main objective is to unveil the evolution of the number of real infected people, and consequently to predict the peak of the epidemic and subsequent evolution. For this purpose, an original nonlinear model relating the raw data with the time-varying geometric ratio of infected people is elaborated, and a Kalman Filter is used to estimate the involved state variables. A hypothetical simulated case is used to show the adequacy and limitations of the proposed method. Then, several countries, including China, South Korea, Italy, Spain, U.K. and the USA, are tested to illustrate its behavior when real-life data are processed. The results obtained clearly show the beneficial effect of the severe lockdowns imposed by many countries worldwide, but also that the softer social distancing measures adopted afterwards have been almost always insufficient to prevent the subsequent virus waves.

What the reproductive number R0 can and cannot tell us about COVID-19 dynamics

The reproductive number R (or R0, the initial reproductive number in an immune-naïve population) has long been successfully used to predict the likelihood of pathogen invasion, to gauge the potential severity of an epidemic, and to set policy around interventions. However, often ignored complexities have generated confusion around use of the metric. This is particularly apparent with the emergent pandemic virus SARS-CoV-2, the causative agent of COVID-19. We address some misconceptions about the predictive ability of the reproductive number, focusing on how it changes over time, varies over space, and relates to epidemic size by referencing the mathematical definition of R and examples from the current pandemic. We hope that a better appreciation of the uses, nuances, and limitations of R and R0 facilitates a better understanding of epidemic spread, epidemic severity, and the effects of interventions in the context of SARS-CoV-2.

Understanding biological control with entomopathogenic fungi-Insights from a stochastic pest-pathogen model

In this study, an individual-based model is proposed to investigate the effect of demographic stochasticity on biological control using entomopathogenic fungi. The model is formulated as a continuous time Markov process, which is then decomposed into a deterministic dynamics using stochastic corrections and system size expansion. The stability and bifurcation analysis shows that the system dynamic is strongly affected by the contagion rate and the basic reproduction number. However, sensitivity analysis of the extinction probability shows that the persistence of a biological control agent depends to the proportion of spores collected from insect cadavers as well as their ability to be reactivated and infect insects. When considering the migration of each species within a set of patches, the dispersion relation shows a Hopf-damped Turing mode for a threshold contagion rate. A large size population led to a spatial and temporal resonant stochasticity and also induces an amplification effect on power spectrum density.

Assessing the impact of vaccination in a COVID-19 compartmental model

BACKGROUND

The aim of this research is to understand the role played by vaccination in the dynamics of a given COVID-19 compartmental model. Most of all, how vaccination modifies the stability, sensitivity, and the reproduction number of a susceptible population.

METHODS

The proposed COVID-19 compartmental model (SVEIRD) has seven compartments. Namely, susceptible (S), vaccinated (V), exposed (E, infected but not yet infectious), symptomatic infectious (I _s ), asymptomatic infectious (I _a ), recovered (R), and dead by Covid-19 disease (D).We have developed a computational code to mimic the first wave of the coronavirus pandemic in a state like New York (NYS).

FINDINGS

First a stability analysis was carried out. Next, a sensitivity analysis showed that the more relevant parameters are birth rate, transmission coefficient, and vaccine failure. We found an alternative procedure to easily calculate the vaccinated reproductive number of the proposed SVEIRD model. Our graphical results allow to make a comparison between unvaccinated (SEIRD) and vaccinated (SVEIRD) populations. In the peak of the first wave, we estimated 21% (2.5%) and 6% (0.8%) of the unvaccinated (vaccinated) susceptible population was symptomatic and asymptomatic, respectively. At 180 days of the NYS pandemic, the model forecast about 25786 deaths by coronavirus. A vaccine with 95% efficacy could reduce the number of deaths from 25786 to 3784.

CONCLUSION

The proposed compartmental model can be used to mimic different possible scenarios of the pandemic not only in NYS, but in any country or region. Further, for an unvaccinated reproductive number R > 1, we showed that the vaccine's efficacy must be greater than the herd immunity to stop the spread of the COVID-19 disease.

Deriving risk maps from epidemiological models of vector borne diseases: State-of-the-art and suggestions for best practice

Epidemiological models (EMs) are widely used to predict the temporal outbreak risk of vector-borne diseases (VBDs). EMs typically use the basic reproduction number (R₀), a threshold quantity, to indicate risk. To provide an overall view of the risk, these model outputs can be transformed into spatial risk maps, using various aggregation methods (e.g. average R₀ over time, cumulative number of days with R₀ > 1). However, there is no standardized methodology available for this. Depending on the specific aggregation methods used, the yielded spatial risk maps may have considerably different interpretations. Additionally, the method used to visualize the aggregated data also affects the perceived spatial patterns. In this review, we compare commonly used aggregation and visualization methods and discuss the respective interpretation of risk maps. Research publications using epidemiological modelling methods were drawn from Web of Science. Only publications containing maps of R₀ transformed from EMs were considered for the analysis. An example EM was applied to illustrate how aggregation and visualization methods affect the final presentations of risk maps. Risk maps can be generated to show duration, intensity and spatio-temporal dynamics of potential outbreak risk of VBDs. We show that 1) different temporal aggregation methods lead to different interpretations; 2) similar spatial patterns do not necessarily bear the same meaning; 3) visualization methods considerably affect how results are perceived, and thus should be applied with caution. We recommend mapping both intensity and duration of the VBD outbreak risk, using small time-steps to show spatio-temporal dynamics when possible.

Identifying Spanish Areas at More Risk of Monthly BTV Transmission with a Basic Reproduction Number Approach

Bluetongue virus (BTV) causes a disease that is endemic in Spain and its two major biological vector species, C. imicola and the Obsoletus complex species, differ greatly in their ecology and distribution. Understanding the seasonality of BTV transmission in risk areas is key to improving surveillance and control programs, as well as to better understand the pathogen transmission networks between wildlife and livestock. Here, monthly risk transmission maps were generated using risk categories based on well-known BTV R0 equations and predicted abundances of the two most relevant vectors in Spain. Previously, Culicoides spp. predicted abundances in mainland Spain and the Balearic Islands were obtained using remote sensing data and random forest machine learning algorithm. Risk transmission maps were externally assessed with the estimated date of infection of BTV-1 and BTV-4 historical outbreaks. Our results highlight the differences in risk transmission during April-October, June-August being the period with higher R0 values. Likewise, a natural barrier has been identified between northern and central-southern areas at risk that may hamper BTV spread between them. Our results can be relevant to implement risk-based interventions for the prevention, control and surveillance of BTV and other diseases shared between livestock and wildlife host populations.

Preliminary estimation of temporal and spatiotemporal dynamic measures of COVID-19 transmission in Thailand

BACKGROUND

As a new emerging infectious disease pandemic, there is an urgent need to understand the dynamics of COVID-19 in each country to inform planning of emergency measures to contain its spread. It is essential that appropriate disease control activities are planned and implemented in a timely manner. Thailand was one of the first countries outside China to be affected with subsequent importation and domestic spread in most provinces in the country.

METHOD

A key ingredient to guide planning and implementation of public health measures is a metric of transmissibility which represents the infectiousness of a disease. Ongoing policies can utilize this information to plan appropriately with updated estimates of disease transmissibility. Therefore we present descriptive analyses and preliminary statistical estimation of reproduction numbers over time and space to facilitate disease control activities in Thailand.

RESULTS

The estimated basic reproduction number for COVID-19 during the study ranged from 2.23-5.90, with a mean of 3.75. We also tracked disease dynamics over time using temporal and spatiotemporal reproduction numbers. The results suggest that the outbreak was under control since the middle of April. After the boxing stadium and entertainment venues, the numbers of new cases had increased and spread across the country.

DISCUSSION

Although various scenarios about assumptions were explored in this study, the real situation was difficult to determine given the limited data. More thorough mathematical modelling would be helpful to improve the estimation of transmissibility metrics for emergency preparedness as more epidemiological and clinical information about this new infection becomes available. However, the results can be used to guide interventions directly and to help parameterize models to predict the impact of these interventions.

The Role of Vector Trait Variation in Vector-Borne Disease Dynamics

Many important endemic and emerging diseases are transmitted by vectors that are biting arthropods. The functional traits of vectors can affect pathogen transmission rates directly and also through their effect on vector population dynamics. Increasing empirical evidence shows that vector traits vary significantly across individuals, populations, and environmental conditions, and at time scales relevant to disease transmission dynamics. Here, we review empirical evidence for variation in vector traits and how this trait variation is currently incorporated into mathematical models of vector-borne disease transmission. We argue that mechanistically incorporating trait variation into these models, by explicitly capturing its effects on vector fitness and abundance, can improve the reliability of their predictions in a changing world. We provide a conceptual framework for incorporating trait variation into vector-borne disease transmission models, and highlight key empirical and theoretical challenges. This framework provides a means to conceptualize how traits can be incorporated in vector borne disease systems, and identifies key areas in which trait variation can be explored. Determining when and to what extent it is important to incorporate trait variation into vector borne disease models remains an important, outstanding question.

Early network properties of the COVID-19 pandemic - The Chinese scenario

OBJECTIVES

To control epidemics, sites more affected by mortality should be identified.

METHODS

Defining epidemic nodes as areas that included both most fatalities per time unit and connections, such as highways, geo-temporal Chinese data on the COVID-19 epidemic were investigated with linear, logarithmic, power, growth, exponential, and logistic regression models. A z-test compared the slopes observed.

RESULTS

Twenty provinces suspected to act as epidemic nodes were empirically investigated. Five provinces displayed synchronicity, long-distance connections, directionality and assortativity - network properties that helped discriminate epidemic nodes. The rank I node included most fatalities and was activated first. Fewer deaths were reported, later, by rank II and III nodes, while the data from rank I-III nodes exhibited slopes, the data from the remaining provinces did not. The power curve was the best fitting model for all slopes. Because all pairs (rank I vs. rank II, rank I vs. rank III, and rank II vs. rank III) of epidemic nodes differed statistically, rank I-III epidemic nodes were geo-temporally and statistically distinguishable.

CONCLUSIONS

The geo-temporal progression of epidemics seems to be highly structured. Epidemic network properties can distinguish regions that differ in mortality. This real-time geo-referenced analysis can inform both decision-makers and clinicians.

A Mathematical Framework for Predicting Lifestyles of Viral Pathogens

Despite being similar in structure, functioning, and size, viral pathogens enjoy very different, usually well-defined ways of life. They occupy their hosts for a few days (influenza), for a few weeks (measles), or even lifelong (HCV), which manifests in acute or chronic infections. The various transmission routes (airborne, via direct physical contact, etc.), degrees of infectiousness (referring to the viral load required for transmission), antigenic variation/immune escape and virulence define further aspects of pathogenic lifestyles. To survive, pathogens must infect new hosts; the success determines their fitness. Infection happens with a certain likelihood during contact of hosts, where contact can also be mediated by vectors. Besides structural aspects of the host-contact network, three parameters appear to be key: the contact rate and the infectiousness during contact, which encode the mode of transmission, and third the immunity of susceptible hosts. On these grounds, what can be said about the reproductive success of viral pathogens? This is the biological question addressed in this paper. The answer extends earlier results of the author and makes explicit connection to another basic work on the evolution of pathogens. A mathematical framework is presented that models intra- and inter-host dynamics in a minimalistic but unified fashion covering a broad spectrum of viral pathogens, including those that cause flu-like infections, childhood diseases, and sexually transmitted infections. These pathogens turn out as local maxima of numerically simulated fitness landscapes. The models involve differential and integral equations, agent-based simulation, networks, and probability.

The challenging management of Rift Valley Fever in humans: literature review of the clinical disease and algorithm proposal

Rift Valley Fever (RVF) is an emerging zoonotic arbovirus with a complex cycle of transmission that makes difficult the prediction of its expansion. Recent outbreaks outside Africa have led to rediscover the human disease but it remains poorly known. The wide spectrum of acute and delayed manifestations with potential unfavorable outcome much complicate the management of suspected cases and prediction of morbidity and mortality during an outbreak. We reviewed literature data on bio-clinical characteristics and treatments of RVF human illness. We identified gaps in the field and provided a practical algorithm to assist clinicians in the cases assessment, determination of setting of care and prolonged follow-up.

Estimation of the Hidden Population with COVID-19 Disease

The population with emerging diseases such as COVID-19, which is used to calculate the basic reproduction number of epidemic outbreak (R₀ ) cannot be simply observed. In this article, we have proposed a method for estimating the hidden population of people with COVID-19 disease. Knowing the number of people with COVID-19 disease is very important for health policy. The provision of medical equipment (e.g., masks, alcohol, ventilators, medication, etc.), the reopening of schools and universities, the start of tourism and public gatherings, the provision of medical staff and preventive planning depend on the number of patients with the disease. Therefore, it is very important to estimate the number of patients.

Spatiotemporal multi-disease transmission dynamic measure for emerging diseases: an application to dengue and zika integrated surveillance in Thailand

Background

New emerging diseases are public health concerns in which policy makers have to make decisions in the presence of enormous uncertainty. This is an important challenge in terms of emergency preparation requiring the operation of effective surveillance systems. A key concept to investigate the dynamic of infectious diseases is the basic reproduction number. However it is difficult to be applicable in real situations due to the underlying theoretical assumptions.

Methods

In this paper we propose a robust and flexible methodology for estimating disease strength varying in space and time using an alternative measure of disease transmission within the hierarchical modeling framework. The proposed measure is also extended to allow for incorporating knowledge from related diseases to enhance performance of surveillance system.

Results

A simulation was conducted to examine robustness of the proposed methodology and the simulation results demonstrate that the proposed method allows robust estimation of the disease strength across simulation scenarios. A real data example is provided of an integrative application of Dengue and Zika surveillance in Thailand. The real data example also shows that combining both diseases in an integrated analysis essentially decreases variability of model fitting.

Conclusions

The proposed methodology is robust in several simulated scenarios of spatiotemporal transmission force with computing flexibility and practical benefits. This development has potential for broad applicability as an alternative tool for integrated surveillance of emerging diseases such as Zika.

Combining serological and contact data to derive target immunity levels for achieving and maintaining measles elimination

BACKGROUND

Vaccination has reduced the global incidence of measles to the lowest rates in history. However, local interruption of measles virus transmission requires sustained high levels of population immunity that can be challenging to achieve and maintain. The herd immunity threshold for measles is typically stipulated at 90-95%. This figure does not easily translate into age-specific immunity levels required to interrupt transmission. Previous estimates of such levels were based on speculative contact patterns based on historical data from high-income countries. The aim of this study was to determine age-specific immunity levels that would ensure elimination of measles when taking into account empirically observed contact patterns.

METHODS

We combined estimated immunity levels from serological data in 17 countries with studies of age-specific mixing patterns to derive contact-adjusted immunity levels. We then compared these to case data from the 10 years following the seroprevalence studies to establish a contact-adjusted immunity threshold for elimination. We lastly combined a range of hypothetical immunity profiles with contact data from a wide range of socioeconomic and demographic settings to determine whether they would be sufficient for elimination.

RESULTS

We found that contact-adjusted immunity levels were able to predict whether countries would experience outbreaks in the decade following the serological studies in about 70% of countries. The corresponding threshold level of contact-adjusted immunity was found to be 93%, corresponding to an average basic reproduction number of approximately 14. Testing different scenarios of immunity with this threshold level using contact studies from around the world, we found that 95% immunity would have to be achieved by the age of five and maintained across older age groups to guarantee elimination. This reflects a greater level of immunity required in 5-9-year-olds than established previously.

CONCLUSIONS

The immunity levels we found necessary for measles elimination are higher than previous guidance. The importance of achieving high immunity levels in 5-9-year-olds presents both a challenge and an opportunity. While such high levels can be difficult to achieve, school entry provides an opportunity to ensure sufficient vaccination coverage. Combined with observations of contact patterns, further national and sub-national serological studies could serve to highlight key gaps in immunity that need to be filled in order to achieve national and regional measles elimination.

Seasonal host life-history processes fuel disease dynamics at different spatial scales

Understanding the drivers underlying disease dynamics is still a major challenge in disease ecology, especially in the case of long-term disease persistence. Even though there is a strong consensus that density-dependent factors play an important role for the spread of diseases, the main drivers are still discussed and, more importantly, might differ between invasion and persistence periods. Here, we analysed long-term outbreak data of classical swine fever, an important disease in both wild boar and livestock, prevalent in the wild boar population from 1993 to 2000 in Mecklenburg-Vorpommern, Germany. We report outbreak characteristics and results from generalized linear mixed models to reveal what factors affected infection risk on both the landscape and the individual level. Spatiotemporal outbreak dynamics showed an initial wave-like spread with high incidence during the invasion period followed by a drop of incidence and an increase in seroprevalence during the persistence period. Velocity of spread increased with time during the first year of outbreak and decreased linearly afterwards, being on average 7.6 km per quarter. Landscape- and individual-level analyses of infection risk indicate contrasting seasonal patterns. During the persistence period, infection risk on the landscape level was highest during autumn and winter seasons, probably related to spatial behaviour such as increased long-distance movements and contacts induced by rutting and escaping movements. In contrast, individual-level infection risk peaked in spring, probably related to the concurrent birth season leading to higher densities, and was significantly higher in piglets than in reproductive animals. Our findings highlight that it is important to investigate both individual- and landscape-level patterns of infection risk to understand long-term persistence of wildlife diseases and to guide respective management actions. Furthermore, we highlight that exploring different temporal aggregation of the data helps to reveal important seasonal patterns, which might be masked otherwise.

Complexity of the Basic Reproduction Number (R0)

The basic reproduction number (R₀), also called the basic reproduction ratio or rate or the basic reproductive rate, is an epidemiologic metric used to describe the contagiousness or transmissibility of infectious agents. R₀ is affected by numerous biological, sociobehavioral, and environmental factors that govern pathogen transmission and, therefore, is usually estimated with various types of complex mathematical models, which make R₀ easily misrepresented, misinterpreted, and misapplied. R₀ is not a biological constant for a pathogen, a rate over time, or a measure of disease severity, and R₀ cannot be modified through vaccination campaigns. R₀ is rarely measured directly, and modeled R₀ values are dependent on model structures and assumptions. Some R₀ values reported in the scientific literature are likely obsolete. R₀ must be estimated, reported, and applied with great caution because this basic metric is far from simple.

Analysis of Malaria Control Measures' Effectiveness Using Multistage Vector Model

We analyze an epidemiological model to evaluate the effectiveness of multiple means of control in malaria-endemic areas. The mathematical model consists of a system of several ordinary differential equations and is based on a multi-compartment representation of the system. The model takes into account the multiple resting-questing stages undergone by adult female mosquitoes during the period in which they function as disease vectors. We compute the basic reproduction number [Formula: see text] and show that if [Formula: see text], the disease-free equilibrium is globally asymptotically stable (GAS) on the nonnegative orthant. If [Formula: see text], the system admits a unique endemic equilibrium (EE) that is GAS. We perform a sensitivity analysis of the dependence of [Formula: see text] and the EE on parameters related to control measures, such as killing effectiveness and bite prevention. Finally, we discuss the implications for a comprehensive, cost-effective strategy for malaria control.

Trait-Based Modeling of Multihost Pathogen Transmission: Plant-Pollinator Networks

Epidemiological models for multihost pathogen systems often classify individuals taxonomically and use species-specific parameter values, but in species-rich communities that approach may require intractably many parameters. Trait-based epidemiological models offer a potential solution but have not accounted for within-species trait variation or between-species trait overlap. Here we propose and study trait-based models with host and vector communities represented as trait distributions without regard to species identity. To illustrate this approach, we develop susceptible-infectious-susceptible models for disease spread in plant-pollinator networks with continuous trait distributions. We model trait-dependent contact rates in two common scenarios: nested networks and specialized plant-pollinator interactions based on trait matching. We find that disease spread in plant-pollinator networks is impacted the most by selective pollinators, universally attractive flowers, and cospecialized plant-pollinator pairs. When extreme pollinator traits are rare, pollinators with common traits are most important for disease spread, whereas when extreme flower traits are rare, flowers with uncommon traits impact disease spread the most. Greater nestedness and specialization both typically promote disease persistence. Given recent pollinator declines caused in part by pathogens, we discuss how trait-based models could inform conservation strategies for wild and managed pollinators. Furthermore, while we have applied our model to pollinators and pathogens, its framework is general and can be transferred to any kind of species interactions in any community.

Flooding in Townsville, North Queensland, Australia, in February 2019 and Its Effects on Mosquito-Borne Diseases

In February 2019, a major flooding event occurred in Townsville, North Queensland, Australia. Here we present a prediction of the occurrence of mosquito-borne diseases (MBDs) after the flooding. We used a mathematical modelling approach based on mosquito population abundance, survival, and size as well as current infectiousness to predict the changes in the occurrences of MBDs due to flooding in the study area. Based on 2019 year-to-date number of notifiable MBDs, we predicted an increase in number of cases, with a peak at 104 by one-half month after the flood receded. The findings in this study indicate that Townsville may see an upsurge in the cases of MBDs in the coming days. However, the burden of diseases will go down again if the mosquito control program being implemented by the City Council continues. As our predictions focus on the near future, longer term effects of flooding on the occurrence of mosquito-borne diseases need to be studied further.

Modeling R₀ for Pathogens with Environmental Transmission: Animal Movements, Pathogen Populations, and Local Infectious Zones

How a disease is transmitted affects our ability to determine R₀, the average number of new cases caused by an infectious host at the onset of an epidemic. R₀ becomes progressively more difficult to compute as transmission varies from directly transmitted diseases to diseases that are vector-borne to environmentally transmitted diseases. Pathogens responsible for diseases with environmental transmission are typically maintained in environmental reservoirs that exhibit a complex spatial distribution of local infectious zones (LIZs). Understanding host encounters with LIZs and pathogen persistence within LIZs is required for an accurate R₀ and modeling these contacts requires an integrated geospatial and dynamical systems approach. Here we review how interactions between host and pathogen populations and environmental reservoirs are driven by landscape-level variables, and synthesize the quantitative framework needed to formulate outbreak response and disease control.

Modification of the Conventional Influenza Epidemic Models Using Environmental Parameters in Iran

Objectives

The association between the spread of infectious diseases and climate parameters has been widely studied in recent decades. In this paper, we formulate, exploit, and compare three variations of the susceptible-infected-recovered (SIR) model incorporating climate data. The SIR model is a well-studied model to investigate the dynamics of influenza viruses; however, the improved versions of the classic model have been developed by introducing external factors into the model.

Methods

The modification models are derived by multiplying a linear combination of three complementary factors, namely, temperature (T), precipitation (P), and humidity (H) by the transmission rate. The performance of these proposed models is evaluated against the standard model for two outbreak seasons.

Results

The values of the root-mean-square error (RMSE) and the Akaike information criterion (AIC) improved as they declined from 8.76 to 7.05 and from 98.12 to 93.01 for season 2013/14, respectively. Similarly, for season 2014/15, the RMSE and AIC decreased from 8.10 to 6.45 and from 117.73 to 107.91, respectively. The estimated values of R(t) in the framework of the standard and modified SIR models are also compared.

Conclusions

Through simulations, we determined that among the studied environmental factors, precipitation showed the strongest correlation with the transmission dynamics of influenza. Moreover, the SIR+P+T model is the most efficient for simulating the behavioral dynamics of influenza in the area of interest.

Complexity of the Basic Reproduction Number (R0)

The basic reproduction number (R₀), also called the basic reproduction ratio or rate or the basic reproductive rate, is an epidemiologic metric used to describe the contagiousness or transmissibility of infectious agents. R₀ is affected by numerous biological, sociobehavioral, and environmental factors that govern pathogen transmission and, therefore, is usually estimated with various types of complex mathematical models, which make R₀ easily misrepresented, misinterpreted, and misapplied. R₀ is not a biological constant for a pathogen, a rate over time, or a measure of disease severity, and R₀ cannot be modified through vaccination campaigns. R₀ is rarely measured directly, and modeled R₀ values are dependent on model structures and assumptions. Some R₀ values reported in the scientific literature are likely obsolete. R₀ must be estimated, reported, and applied with great caution because this basic metric is far from simple.

A comparison of Zika and dengue outbreaks using national surveillance data in the Dominican Republic

BACKGROUND

Aedes-borne arboviruses continue to precipitate epidemics worldwide. In Dominican Republic, the appearance of Zika virus cases that closely followed a large dengue epidemic provided an opportunity to study the different transmission drivers behind these two flaviviruses. Retrospective datasets were used to collect information on the populations at risk and descriptive statistics were used to describe the outbreaks on a national scale.

METHODOLOGY/ PRINCIPAL FINDINGS

Expectedly, box plots showed that 75% of dengue was reported in those aged <20 years while Zika infections were more widely dispersed among the population. Dengue attack rates were marginally higher among males at 25.9 per 10,000 population vs. 21.5 per 10,000 population for females. Zika infections appeared to be highly clustered among females (73.8% (95% CI 72.6%, 75.0%; p<0.05)); age-adjusted Zika attack rates among females were 7.64 per 10,000 population compared with 2.72 per 10,000 population among males. R0 calculations stratified by sex also showed a significantly higher metric among females: 1.84 (1.82, 1.87; p<0.05) when compared to males at 1.72 (1.69, 1.75; p<0.05). However, GBS attack rates stratified by sex revealed slightly higher risk in males vs. females, at 0.62 and 0.57 per 10,000 population respectively.

CONCLUSIONS/ SIGNIFICANCE

Evidence suggests little impact of existing dengue immunity on reported attack rates of Zika at the population level. Confounding of R0 and incident risk calculations by sex-specific over-reporting can alter the reliability of epidemiological metrics, which could be addressed using associated proxy syndromes or conditions to explore seemingly sex-skewed incidence. The findings indicate that community awareness campaigns, through influencing short-term health seeking behaviour, remain the most plausible mechanism behind increased reporting among women of reproductive age, although biological susceptibility cannot yet be ruled out. Media campaigns and screening are therefore recommended for women of reproductive age during Zika outbreaks. Future research should focus on clinical Zika outcomes among dengue seropositive individuals.

Effects of complexity and seasonality on backward bifurcation in vector-host models

We study implications of complexity and seasonality in vector-host epidemiological models exhibiting backward bifurcation. Vector-host diseases represent complex infection systems that can vary in the transmission processes and population stages involved in disease progression. Seasonal fluctuations in external forcing factors can also interact in a complex way with internal host factors to govern the transmission dynamics. In backward bifurcation, the insufficiency of R0 < 1 for predicting the stability of the disease-free equilibrium (DFE) state arises due to existence of bistability (coexisting DFE and endemic equilibria) for a range of R0 values below one. Here we report that this region of bistability decreases with increasing complexity of vector-borne disease transmission as well as with increasing seasonality strength. The decreases in the bistability region are accompanied by a reduced force of infection acting on primary hosts. As a consequence, we show counterintuitively that a more complex vector-borne disease may be easier to control in settings of high seasonality.

Detecting the impact of temperature on transmission of Zika, dengue, and chikungunya using mechanistic models

Recent epidemics of Zika, dengue, and chikungunya have heightened the need to understand the seasonal and geographic range of transmission by Aedes aegypti and Ae. albopictus mosquitoes. We use mechanistic transmission models to derive predictions for how the probability and magnitude of transmission for Zika, chikungunya, and dengue change with mean temperature, and we show that these predictions are well matched by human case data. Across all three viruses, models and human case data both show that transmission occurs between 18-34°C with maximal transmission occurring in a range from 26-29°C. Controlling for population size and two socioeconomic factors, temperature-dependent transmission based on our mechanistic model is an important predictor of human transmission occurrence and incidence. Risk maps indicate that tropical and subtropical regions are suitable for extended seasonal or year-round transmission, but transmission in temperate areas is limited to at most three months per year even if vectors are present. Such brief transmission windows limit the likelihood of major epidemics following disease introduction in temperate zones.

The Role of Hyalomma Truncatum on the Dynamics of Rift Valley Fever: Insights from a Mathematical Epidemic Model

To date, our knowledge of Rift Valley fever (RVF) disease spread and maintenance is still limited, as flooding, humid weather and presence of biting insects such as mosquitoes, have not completely explained RVF outbreaks. We propose a model that includes livestock, mosquitoes and ticks compartments structured according to their questing and feeding behaviour in order to study the possible role of ticks on the dynamics of RVF. To quantify disease transmission at the initial stage of the epidemic, we derive an explicit formula of the basic reproductive number, [Formula: see text]. Using the concept of Metzler matrix, we state necessary conditions for global asymptotic stability of the disease-free equilibrium. Results suggest that although host-ticks interactions may serve as disease reservoirs or disease amplifiers, the Aedes reproductive number should be kept under unity if disease post-epizootics activities are to be controlled. Results of both local and global sensitivity analysis of selected model parameters indicate that [Formula: see text] is more sensitive to the ticks attachment and detachment rates, probability of transmission from ticks to host and from host to ticks, length of infection in livestock and ticks death rate. Furthermore, when comparing the mean value of [Formula: see text] with that from previous studies which did not include ticks we found that our [Formula: see text] is very much larger resulting in an increase in the exponential phase of an outbreak. These findings suggest that if ticks are capable of transmitting the virus, they may be contributing to disease outbreaks and endemicity.

Toward Standardizing a Lexicon of Infectious Disease Modeling Terms

Disease modeling is increasingly being used to evaluate the effect of health intervention strategies, particularly for infectious diseases. However, the utility and application of such models are hampered by the inconsistent use of infectious disease modeling terms between and within disciplines. We sought to standardize the lexicon of infectious disease modeling terms and develop a glossary of terms commonly used in describing models’ assumptions, parameters, variables, and outcomes. We combined a comprehensive literature review of relevant terms with an online forum discussion in a virtual community of practice, mod4PH (Modeling for Public Health). Using a convergent discussion process and consensus amongst the members of mod4PH, a glossary of terms was developed as an online resource. We anticipate that the glossary will improve inter- and intradisciplinary communication and will result in a greater uptake and understanding of disease modeling outcomes in heath policy decision-making. We highlight the role of the mod4PH community of practice and the methodologies used in this endeavor to link theory, policy, and practice in the public health domain.

Rift Valley fever trasmission dynamics described by compartmental models

Rift Valley fever (RVF) is one of the most important zoonotic Transboundary Animal Diseases able to cross international borders and cause devastating effect on animal health and food security. Climate changes and the presence of competent vectors in the most of the current RVF-free temperate countries strongly support the inclusion of RVF virus (RVFV) among the most significant emerging viral threats for public and animal health. The transmission of RVFV is driven by complex eco-climatic factors making the epidemiology of RVF infection difficult to study and to understand. Mathematical, statistical and spatial models are often used to explain the mechanisms underlying these biological processes, providing new and effective tools to plan measures for public health protection. In this paper we performed a systematic literature review on RVF published papers with the aim of identifying and describing the most recent papers developing compartmental models for the study of RVFV transmission dynamics.

A comparative analysis of three different methods for the estimation of the basic reproduction number of dengue

The basic reproduction number, R₀, is defined as the expected number of secondary cases of a disease produced by a single infection in a completely susceptible population, and can be estimated in several ways. For example, from the stability analysis of a compartmental model; through the use of the matrix of next generation, or from the final size of an epidemic, etc. In this paper we applied the method for estimating R₀ of dengue fever from the initial growth phase of an outbreak, without assuming exponential growth of cases, a common assumption in many studies. We used three different methods of calculating R₀ to compare the techniques' details and to evaluate how these techniques estimate the value of R₀ of dengue using data from the city of Ribeirão Preto (SE of Brazil) in two outbreaks. The results of the three methods are numerically different but, when we compare them using a system of differential equations developed for modeling only the first generation time, we can observe that the methods differ little in the initial growth phase. We conclude that the methods predict that dengue will spread in the city studied and the analysis of the data shows that the estimated values of R₀ have an equal pattern overtime.

The many guises of R0 (a didactic note)

The basic reproduction number R0 is, by definition, the expected life time number of offspring of a newborn individual. An operationalization entails a specification of what events are considered as "reproduction" and what events are considered as "transitions from one individual-state to another". Thus, an element of choice can creep into the concretization of the definition. The aim of this note is to clearly expose this possibility by way of examples from both population dynamics and infectious disease epidemiology.

Modeling the Epidemiology of Cholera to Prevent Disease Transmission in Developing Countries

Cholera remains an important global cause of morbidity and mortality, which is capable of causing periodic epidemic disease. A number of mathematical models have been developed to help in understanding the dynamics of cholera outbreaks and for use as a tool in planning interventions, including vaccination campaigns. We have explored the utility of models in assessing the spread of cholera in the recent epidemics in Zimbabwe and Haiti. In both instances, a mathematical model was formulated and fitted to cumulative cholera cases to estimate the basic reproductive number ℛ0, and the partial reproductive numbers reflecting potential differences in environmental-to-human versus human-to-human transmission were quantified. In Zimbabwe, estimated ℛ0 for the epidemic using aggregated data at the national level was 1.15; in Haiti, it was 1.55. However, when calculated at a provincial/departmental level, estimated basic reproductive numbers were highly heterogeneous, with a range of 1.11 to 2.72 in Zimbabwe and 1.06 to 2.63 in Haiti. Our models suggest that the underlying patterns of cholera transmission varied widely from region to region, with a corresponding variation in the amenability of outbreaks to control measures such as immunization. These data underscore the heterogeneity of transmission dynamics, potentially linked to differences in environment, socio-economic conditions, and cultural practices. They also highlight the potential utility of these types of models in guiding development of public health intervention strategies.