Estimating the probability of an extinction or major outbreak for an environmentally transmitted infectious disease

Semi-Quantitative Biosecurity Assessment Framework Targeting Prevention of the Introduction and Establishment of Salmonella Dublin in Dairy Cattle Herds

An increasing average herd size and complexity in farm structures call for a higher level of biosecurity. It can reduce the risk of introducing and establishing pathogens with multiple-pathway and indirect spread mechanisms, such as Salmonella Dublin, a pathogen with an increasing occurrence in dairy cattle farms across different countries and continents. Therefore, this study aimed to use existing knowledge to develop a framework with a supporting tool allowing for a time-efficient, yet comprehensive, assessment of biosecurity measures that can help prevent the introduction and establishment of S. Dublin in dairy herds. Based on the literature review, a seven-step biosecurity assessment framework was developed and evaluated in collaboration with biosecurity experts. The resulting framework includes a weighted semi-quantitative assessment method with a scoring guide in an electronic supporting tool for 12 biosecurity sections assessed through on-farm observations and farmer interviews. The framework and tool provide a novel approach to comprehensively assess the overall (mainly external) on-farm biosecurity level by a trained biosecurity assessor. They can be used for systematic data collection in epidemiological studies on risk factors for the introduction and establishment of S. Dublin in dairy farms. Preliminary interrater reliability testing indicated moderate reliability between assessors with varying biosecurity skills.

Modeling cryptosporidiosis in humans and cattle: Deterministic and stochastic approaches

Cryptosporidiosis is a zoonotic disease caused by Cryptosporidium. The disease poses a public and veterinary health problem worldwide. A deterministic model and its corresponding continuous time Markov chain (CTMC) stochastic model are developed and analyzed to investigate cryptosporidiosis transmission dynamics in humans and cattle. The basic reproduction number R 0 for the deterministic model and stochastic threshold for the CTMC stochastic model are computed by the next generation matrix method and multitype branching process, respectively. The normalized forward sensitivity index method is used to determine the sensitivity index for each parameter in R 0 . Per capita birth rate of cattle, the rate of cattle to acquire cryptosporidiosis infection from the environment and the rate at which infected cattle shed Cryptosporidium oocysts in the environment play an important role in the persistence of the disease whereas Cryptosporidium oocysts natural death rate, cattle recovery rate and cattle natural death rate are most negative sensitive parameters in the dynamics of cryptosporidiosis. Numerical results for CTMC stochastic model show that the likelihood of cryptosporidiosis extinction is high when it arises from an infected human. However, there is a major outbreak if cryptosporidiosis emerges either from infected cattle or from Cryptosporidium oocysts in the environment or when it emerges from all three infectious compartments. Therefore to control the disease, control measures should focus on maintaining personal and cattle farm hygiene and decontaminating the environment to destroy Cryptosporidium oocysts.

A practical guide to mathematical methods for estimating infectious disease outbreak risks

Mathematical models are increasingly used throughout infectious disease outbreaks to guide control measures. In this review article, we focus on the initial stages of an outbreak, when a pathogen has just been observed in a new location (e.g., a town, region or country). We provide a beginner's guide to two methods for estimating the risk that introduced cases lead to sustained local transmission (i.e., the probability of a major outbreak), as opposed to the outbreak fading out with only a small number of cases. We discuss how these simple methods can be extended for epidemiological models with any level of complexity, facilitating their wider use, and describe how estimates of the probability of a major outbreak can be used to guide pathogen surveillance and control strategies. We also give an overview of previous applications of these approaches. This guide is intended to help quantitative researchers develop their own epidemiological models and use them to estimate the risks associated with pathogens arriving in new host populations. The development of these models is crucial for future outbreak preparedness. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Multi-season transmission model of Eastern Equine Encephalitis

Eastern Equine Encephalitis (EEE) is an arbovirus that, while it has been known to exist since the 1930's, recently had a spike in cases. This increased prevalence is particularly concerning due to the severity of the disease with 1 in 3 symptomatic patients dying. The cause of this peak is currently unknown but could be due to changes in climate, the virus itself, or host behavior. In this paper we propose a novel multi-season deterministic model of EEE spread and its stochastic counterpart. Models were parameterized using a dataset from the Florida Department of Health with sixteen years of sentinel chicken seroconversion rates. The different roles of the enzootic and bridge mosquito vectors were explored. As expected, enzootic mosquitoes like Culiseta melanura were more important for EEE persistence, while bridge vectors were implicated in the disease burden in humans. These models were used to explore hypothetical viral mutations and host behavior changes, including increased infectivity, vertical transmission, and host feeding preferences. Results showed that changes in the enzootic vector transmission increased cases among birds more drastically than equivalent changes in the bridge vector. Additionally, a 5% difference in the bridge vector's bird feeding preference can increase cumulative dead-end host infections more than 20-fold. Taken together, this suggests changes in many parts of the transmission cycle can augment cases in birds, but the bridge vectors feeding preference acts as a valve limiting the enzootic circulation from its impact on dead-end hosts, such as humans. Our what-if scenario analysis reveals and measures possible threats regarding EEE and relevant environmental changes and hypothetically suggests how to prevent potential damage to public health and the equine economy.

Impact of demographic variability on the disease dynamics for honeybee model

For the last few years, annual honeybee colony losses have been center of key interest for many researchers throughout the world. The spread of the parasitic mite and its interaction with specific honeybee viruses carried by Varroa mites has been linked to the decline of honeybee colonies. In this investigation, we consider honeybee-virus and honeybee-infected mite-virus models. We perform sensitivity analysis locally and globally to see the effect of the parameters on the basic reproduction number for both models and to understand the disease dynamics in detail. We use the continuous-time Markov chain model to develop and analyze stochastic epidemic models corresponding to both deterministic models. By using the disease extinction process, we compare both deterministic and stochastic models. We have observed that the numerically approximated probability of disease extinction based on 30 000 sample paths agrees well with the calculated probability using multitype branching process approximation. In particular, it is observed that the disease extinction probability is higher when infected honeybees spread the disease instead of infected mites. We conduct a sensitivity analysis for the stochastic model also to examine how the system parameters affect the probability of disease extinction. We have also derived the equation for the expected time required to reach disease-free equilibrium for stochastic models. Finally, the effect of the parameters on the expected time is represented graphically.

A Comparison of Deterministic and Stochastic Plant-Vector-Virus Models Based on Probability of Disease Extinction and Outbreak

In this investigation, we formulate and analyse a stochastic epidemic model using the continuous-time Markov chain model for the propagation of a vector-borne cassava mosaic disease in a single population. The stochastic model is based upon a pre-existing deterministic plant-vector-virus model. To see how demographic stochasticity affects the vector-borne cassava mosaic disease dynamics, we compare the disease dynamics of both deterministic and stochastic models through disease extinction process. The probability of disease extinction and therefore the major outbreak are estimated analytically using the multitype Galton-Watson branching process (GWbp) approximation. Also, we have found the approximate probabilities of disease extinction numerically based on 30000 sample paths, and it is shown to be good estimate with the calculated probabilities from GWbp approximation. In particular, it is observed that there is a very high probability of disease extinction when the disease is introduced via the infected vectors rather than through infected plants.

Determining the effects of wind-aided midge movement on the outbreak and coexistence of multiple bluetongue virus serotypes in patchy environments

Bluetongue virus (BTV) has 27 serotypes with some of them coexisting in different environments which make its control difficult. Wind-aided midge movement is a known mechanism in the spread of BTV. However, its effects on the dynamics of multiple BTV serotypes are not clear. Ordinary differential equation (ODE) and continuous-time Markov chain (CTMC) models for two BTV serotypes in an environment divided into two patches depending on the risk of infection are formulated and analysed. By approximating the CTMC model with a multitype branching process, an estimate for the probability of a major outbreak of two BTV serotypes is obtained. It is shown that without movement a major outbreak occurs in the high-risk patch, but with cattle or midge movement it occurs in both patches. When a major outbreak occurs, numerical simulations of the ODE model illustrate possible coexistence in both patches if the patches are connected by midge or cattle movement. Sensitivity analysis, based on the Latin hypercube sampling method, identified midge mortality and biting rates as being the most important in determining the magnitude of the probability of a major outbreak. These results indicate the significance of wind-aided midge movement on the outbreak and coexistence of multiple BTV serotypes in patchy environments.

Probability of Disease Extinction or Outbreak in a Stochastic Epidemic Model for West Nile Virus Dynamics in Birds

Thresholds for disease extinction provide essential information for the prevention and control of diseases. In this paper, a stochastic epidemic model, a continuous-time Markov chain, for the transmission dynamics of West Nile virus in birds is developed based on the assumptions of its analogous deterministic model. The branching process is applied to derive the extinction threshold for the stochastic model and conditions for disease extinction or persistence. The probability of disease extinction computed from the branching process is shown to be in good agreement with the probability approximated from numerical simulations. The disease dynamics of both models are compared to ascertain the effect of demographic stochasticity on West Nile virus dynamics. Analytical and numerical results show differences in model predictions and asymptotic dynamics between stochastic and deterministic models that are crucial for the prevention of disease outbreaks. It is found that there is a high probability of disease extinction if the disease emerges from exposed mosquitoes unlike if it emerges from infectious mosquitoes and birds. Finite-time to disease extinction is estimated using sample paths and it is shown that the epidemic duration is shortest if the disease is introduced by exposed mosquitoes.

Stochastic models of infectious diseases in a periodic environment with application to cholera epidemics

Seasonal variation affects the dynamics of many infectious diseases including influenza, cholera and malaria. The time when infectious individuals are first introduced into a population is crucial in predicting whether a major disease outbreak occurs. In this investigation, we apply a time-nonhomogeneous stochastic process for a cholera epidemic with seasonal periodicity and a multitype branching process approximation to obtain an analytical estimate for the probability of an outbreak. In particular, an analytic estimate of the probability of disease extinction is shown to satisfy a system of ordinary differential equations which follows from the backward Kolmogorov differential equation. An explicit expression for the mean (resp. variance) of the first extinction time given an extinction occurs is derived based on the analytic estimate for the extinction probability. Our results indicate that the probability of a disease outbreak, and mean and standard derivation of the first time to disease extinction are periodic in time and depend on the time when the infectious individuals or free-living pathogens are introduced. Numerical simulations are then carried out to validate the analytical predictions using two examples of the general cholera model. At the end, the developed theoretical results are extended to more general models of infectious diseases.

Disease Emergence in Multi-Patch Stochastic Epidemic Models with Demographic and Seasonal Variability

Factors such as seasonality and spatial connectivity affect the spread of an infectious disease. Accounting for these factors in infectious disease models provides useful information on the times and locations of greatest risk for disease outbreaks. In this investigation, stochastic multi-patch epidemic models are formulated with seasonal and demographic variability. The stochastic models are used to investigate the probability of a disease outbreak when infected individuals are introduced into one or more of the patches. Seasonal variation is included through periodic transmission and dispersal rates. Multi-type branching process approximation and application of the backward Kolmogorov differential equation lead to an estimate for the probability of a disease outbreak. This estimate is also periodic and depends on the time, the location, and the number of initial infected individuals introduced into the patch system as well as the magnitude of the transmission and dispersal rates and the connectivity between patches. Examples are given for seasonal transmission and dispersal in two and three patches.

Modelling the Transmission Dynamics of Tuberculosis in the Ashanti Region of Ghana

Mathematical models can aid in elucidating the spread of infectious disease dynamics within a given population over time. In an attempt to model tuberculosis (TB) dynamics among high-burden districts in the Ashanti Region of Ghana, the SEIR epidemic model with demography was employed within both deterministic and stochastic settings for comparison purposes. The deterministic model showed success in modelling TB infection in the region to the transmission dynamics of the stochastic SEIR model over time. It predicted tuberculosis dying out in ten of twelve high-burden districts in the Ashanti Region, but an outbreak in Obuasi municipal and Amansie West district. The effect of introducing treatment at the incubation stage of TB transmission was also investigated, and it was discovered that treatment introduced at the exposed stage decreased the spread of TB. Branching process approximation was used to derive explicit forms of relevant epidemiological quantities of the deterministic SEIR model for stability analysis of equilibrium points. Numerical simulations were performed to validate the overall infection rate, basic reproductive number, herd immunity threshold, and Malthusian parameter based on bootstrapping, jackknife, and Latin Hypercube sampling schemes. It was recommended that the Ghana Health Service should find a good mechanism to detect TB in the early stages of infection in the region. Public health attention must also be given to districts with a potentially higher risk of experiencing endemic TB even though the estimates of the overall epidemic thresholds from our SEIR model suggested that the Ashanti Region as a whole had herd immunity against TB infection.

Within-host dynamics and random duration of pathogen infection: Implications for between-host transmission

Taking an ecological perspective, this paper reports theoretical and empirical results concerning fatal bacterial infections of adult insects. Two models, each combining deterministic and stochastic elements, characterize how the pathogen's dynamics might govern an infected host's mortality rate. We analyze the models in detail for exponential pathogen growth, and apply them to observed insect mortality when the pathogen's growth is unregulated. We then allow bacteriophage to generate fluctuations in the within-host pathogen density; we demonstrate that only one of our models matches host mortality rates when pathogen growth is regulated by phage. We generalize our results on mortality hazard of individual hosts to analyze how random duration of the infectious period can combine with probabilistic transmission events to affect between-host transmission.

A Stochastic Tick-Borne Disease Model: Exploring the Probability of Pathogen Persistence

We formulate and analyse a stochastic epidemic model for the transmission dynamics of a tick-borne disease in a single population using a continuous-time Markov chain approach. The stochastic model is based on an existing deterministic metapopulation tick-borne disease model. We compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in tick-borne disease dynamics. The probability of disease extinction and that of a major outbreak are computed and approximated using the multitype Galton-Watson branching process and numerical simulations, respectively. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that a disease outbreak is more likely if the disease is introduced by infected deer as opposed to infected ticks. These insights demonstrate the importance of host movement in the expansion of tick-borne diseases into new geographic areas.

Modeling the intrinsic dynamics of foot-and-mouth disease

We propose a new mathematical modeling framework to investigate the transmission and spread of foot-and-mouth disease. Our models incorporate relevant biological and ecological factors, vaccination effects, and seasonal impacts during the complex interaction among susceptible, vaccinated, exposed, infected, carrier, and recovered animals. We conduct both epidemic and endemic analysis, with a focus on the threshold dynamics characterized by the basic reproduction numbers. In addition, numerical simulation results are presented to demonstrate the analytical findings.

Environmentally transmitted parasites: Host-jumping in a heterogeneous environment

Groups of chronically infected reservoir-hosts contaminate resource patches by shedding a parasite׳s free-living stage. Novel-host groups visit the same patches, where they are exposed to infection. We treat arrival at patches, levels of parasite deposition, and infection of the novel host as stochastic processes, and derive the expected time elapsing until a host-jump (initial infection of a novel host) occurs. At stationarity, mean parasite densities are independent of reservoir-host group size. But within-patch parasite-density variances increase with reservoir group size. The probability of infecting a novel host declines with parasite-density variance; consequently larger reservoir groups extend the mean waiting time for host-jumping. Larger novel-host groups increase the probability of a host-jump during any single patch visit, but also reduce the total number of visits per unit time. Interaction of these effects implies that the waiting time for the first infection increases with the novel-host group size. If the reservoir-host uses resource patches in any non-uniform manner, reduced spatial overlap between host species increases the waiting time for host-jumping.

Modeling the Effects of Multiple Intervention Strategies on Controlling Foot-and-Mouth Disease

Foot-and-mouth disease (FMD) is a threat to economic security and infrastructure as well as animal health, in both developed and developing countries. We propose and analyze an optimal control problem where the control system is a mathematical model for FMD that incorporates vaccination and culling of infectious animals. The control functions represent the fraction of animals that are vaccinated during an outbreak, infectious symptomatic animals that are detected and culled, and infectious nonsymptomatic animals that are detected and culled. Our aim was to study how these control measures should be implemented for a certain time period, in order to reduce or eliminate FMD in the community, while minimizing the interventions implementation costs. A cost-effectiveness analysis is carried out, to compare the application of each one of the control measures, separately or in combination.