Branching Process Approximation Research Articles

Seasonal variability and stochastic branching process in malaria outbreak probability

Background: Malaria is the world’s most fatal and challenging parasitic disease, caused by the Plasmodium parasite, which is transmitted to humans by the bites of infected female mosquitoes. Bangladesh is the most vulnerable region to spread malaria because of its geographic position. In this paper, we have considered the dynamics of vector-host models and observed the stochastic behavior. This study elaborates on the seasonal variability and calculates the probability of disease outbreaks. Methods: We present a model for malaria disease transmission and develop its corresponding continuous-time Markov chain (CTMC) representation. The proposed vector-host models illustrate the malaria transmission model along with sensitivity analysis. The deterministic model with CTMC curves is depicted to show the randomness in real scenarios. Sequentially, we expand these studies to a time-varying stochastic vector-host model that incorporates seasonal variability. Phase plane analysis is conducted to explore the characteristics of the disease, examine interactions among various compartments, and evaluate the impact of key parameters. The branching process approximation is developed for the corresponding vector-host model to calculate the probability outbreak. Numerous numerical results are accomplished to observe the analytical investigation. Results: Seasonality and contact patterns affect the dynamics of disease outbreaks. The numerical illustration provides that the probability of a disease outbreak depends on the infected host or vector. Additionally, periodic transmission rates have a great influence on the probability outbreak. The basic reproduction number (R0) is derived, which is the main justification for studying the dynamical behavior of epidemic models. Conclusions: Seasonal variability significantly impacts malaria transmission, and the probability of disease outbreaks is influenced by time and the initial number of infected individuals. Moreover, the branching process approximation is applicable when the population size is large enough and the basic reproduction number is less than 1. In the future, such analysis can help decision-makers understand the impact of various parameters and their stochastic behavior in the vector-host model to prevent such types of disease outbreaks.

The effect of demographic stochasticity on Zika virus transmission dynamics: Probability of disease extinction, sensitivity analysis, and mean first passage time.

Viral infections spread by mosquitoes are a growing threat to human health and welfare. Zika virus (ZIKV) is one of them and has become a global worry, particularly for women who are pregnant. To study ZIKV dynamics in the presence of demographic stochasticity, we consider an established ZIKV transmission model that takes into consideration the disease transmission from human to mosquito, mosquito to human, and human to human. In this study, we look at the local stability of the disease-free and endemic equilibriums. By conducting the sensitivity analysis both locally and globally, we assess the effect of the model parameters on the model outcomes. In this work, we use the continuous-time Markov chain (CTMC) process to develop and analyze a stochastic model. The main distinction between deterministic and stochastic models is that, in the absence of any preventive measures such as avoiding travel to infected areas, being careful from mosquito bites, taking precautions to reduce the risk of sexual transmission, and seeking medical care for any acute illness with a rash or fever, the stochastic model shows the possibility of disease extinction in a finite amount of time, unlike the deterministic model shows disease persistence. We found that the numerically estimated disease extinction probability agrees well with the analytical probability obtained from the Galton-Watson branching process approximation. We have discovered that the disease extinction probability is high if the disease emerges from infected mosquitoes rather than infected humans. In the context of the stochastic model, we derive the implicit equation of the mean first passage time, which computes the average amount of time needed for a system to undergo its first state transition.

INVASION AND SUPERINFECTION IN DETERMINISTIC AND STOCHASTIC TWO-STRAIN DENGUE MODELS WITH DEMOGRAPHIC AND SEASONAL VARIATION

The effects of seasonality on disease invasion and superinfection are investigated in dengue epidemic models with two viral strains. A two-strain host–vector nonautonomous ODE is formulated with seasonal periodicity in either vector recruitment or transmission. The basic reproduction number is derived and the dynamics of the single-strain periodic subsystem are summarized. Conditions for uniform persistence and existence of a single-strain periodic solution are verified and the invasion reproduction number is derived. A time-nonhomogeneous, continuous-time Markov chain (CTMC) is formulated from the ODE framework. The basic and invasion reproduction numbers apply to the ODE and CTMC. In the case of invasion, after one strain is established and the invasion reproduction number exceeds one, the ODE predicts a successful invasion, but in the CTMC, there is a periodic probability of invasion that depends on time and size of the invasion. A multitype branching process approximation of the CTMC provides an analytical method to predict the periodic probability of invasion. The numerical results show that the peak time of invasion is related to the current endemic strain and the dominant seasonal driver but it often precedes the peak time of the seasonal driver.

The size of a Markovian SIR epidemic given only removal data

AbstractDuring an epidemic outbreak, typically only partial information about the outbreak is known. A common scenario is that the infection times of individuals are unknown, but individuals, on displaying symptoms, are identified as infectious and removed from the population. We study the distribution of the number of infectives given only the times of removals in a Markovian susceptible–infectious–removed (SIR) epidemic. Primary interest is in the initial stages of the epidemic process, where a branching (birth–death) process approximation is applicable. We show that the number of individuals alive in a time-inhomogeneous birth–death process at time $t \geq 0$ , given only death times up to and including time t, is a mixture of negative binomial distributions, with the number of mixing components depending on the total number of deaths, and the mixing weights depending upon the inter-arrival times of the deaths. We further consider the extension to the case where some deaths are unobserved. We also discuss the application of the results to control measures and statistical inference.

The fate of recessive deleterious or overdominant mutations near mating-type loci under partial selfing

Large regions of suppressed recombination having extended over time occur in many organisms around genes involved in mating compatibility (sex-determining or mating-type genes). The sheltering of deleterious alleles has been proposed to be involved in such expansions. However, the dynamics of deleterious mutations partially linked to genes involved in mating compatibility are not well understood, especially in finite populations. In particular, under what conditions deleterious mutations are likely to be maintained for long enough near mating-compatibility genes remains to be evaluated, especially under selfing, which generally increases the purging rate of deleterious mutations. Using a branching process approximation, we studied the fate of a new deleterious or overdominant mutation in a diploid population, considering a locus carrying two permanently heterozygous mating-type alleles, and a partially linked locus at which the mutation appears. We obtained analytical and numerical results on the probability and purging time of the new mutation. We investigated the impact of recombination between the two loci and of the mating system (outcrossing, intra and inter-tetrad selfing) on the maintenance of the mutation. We found that the presence of a fungal-like mating-type locus (i.e. not preventing diploid selfing) always sheltered the mutation under selfing, i.e. it decreased the purging probability and increased the purging time of the mutations. The sheltering effect was higher in case of automixis (intra-tetrad selfing). This may contribute to explain why evolutionary strata of recombination suppression near the mating-type locus are found mostly in automictic (pseudo-homothallic) fungi. We also showed that rare events of deleterious mutation maintenance during strikingly long evolutionary times could occur, suggesting that deleterious mutations can indeed accumulate near the mating-type locus over evolutionary time scales. In conclusion, our results show that, although selfing purges deleterious mutations, these mutations can be maintained for very long times near a mating-type locus, which may contribute to promote the evolution of recombination suppression in sex-related chromosomes.

An epidemic model with short-lived mixing groups

Almost all epidemic models make the assumption that infection is driven by the interaction between pairs of individuals, one of whom is infectious and the other of whom is susceptible. However, in society individuals mix in groups of varying sizes, at varying times, allowing one or more infectives to be in close contact with one or more susceptible individuals at a given point in time. In this paper we study the effect of mixing groups beyond pairs on the transmission of an infectious disease in an SIR (susceptible rightarrow infective rightarrow recovered) model, both through a branching process approximation for the initial stages of an epidemic with few initial infectives and a functional central limit theorem for the trajectories of the numbers of infectives and susceptibles over time for epidemics with many initial infectives. We also derive central limit theorems for the final size of (i) an epidemic with many initial infectives and (ii) a major outbreak triggered by few initial infectives. We show that, for a given basic reproduction number R_0, the distribution of the size of mixing groups has a significant impact on the probability and final size of a major epidemic outbreak. Moreover, the standard pair-based homogeneously mixing epidemic model is shown to represent the worst case scenario, with both the highest probability and the largest final size of a major epidemic.

Exploring the dynamics of African swine fever transmission cycles at a wildlife-livestock interface

African swine fever (ASF), a contagious disease of pigs, can be economically devastating. At a wildlife–livestock interface, depending on the presence and interactions of domestic pigs, wild pigs, and soft ticks, three transmission cycles have been proposed for the persistence of ASF: a wild pig-domestic pig-tick cycle, a domestic pig-tick cycle, and a domestic pig cycle, but to date, their relative importance on the outbreak and persistence of the disease is not clear. A stochastic model that considers these cycles is formulated and analysed to assess their impacts on the outbreak and persistence of the disease. Using a multitype branching process approximation, it is shown that all cycles can result in very large probabilities of disease outbreaks. Numerical results show that the disease can be endemic if the ticks are not controlled. If ticks are controlled, disease extinction is certain, regardless of the cycle, but the time to extinction can be long. These findings have important implications for the prevention and control of ASF outbreaks at a wildlife–livestock interface.

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.

Bistability in deterministic and stochastic SLIAR-type models with imperfect and waning vaccine protection

Various vaccines have been approved for use to combat COVID-19 that offer imperfect immunity and could furthermore wane over time. We analyze the effect of vaccination in an SLIARS model with demography by adding a compartment for vaccinated individuals and considering disease-induced death, imperfect and waning vaccination protection as well as waning infections-acquired immunity. When analyzed as systems of ordinary differential equations, the model is proven to admit a backward bifurcation. A continuous time Markov chain (CTMC) version of the model is simulated numerically and compared to the results of branching process approximations. While the CTMC model detects the presence of the backward bifurcation, the branching process approximation does not. The special case of an SVIRS model is shown to have the same properties.

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.

Estimation of the probability of epidemic fade-out from multiple outbreak data

Deterministic epidemic models that allow for replenishment of susceptibles typically display damped oscillatory behaviour. If the population is initially fully susceptible, once an epidemic takes off a distinct trough will exist between the first and second waves of infection. Epidemic dynamics are, however, influenced by stochastic effects, particularly when the prevalence is low. At the beginning of an epidemic, stochastic die-out is possible and well characterised through use of a branching process approximation. Conditional on an epidemic taking off, stochastic extinction is highly unlikely during the first epidemic wave, but the probability of extinction increases again as the wave declines. Extinction during this period, prior to a potential second wave of infection, is defined as ‘epidemic fade-out’. We consider a set of observed epidemics, each distinct and having evolved independently, in which some display fade-out and some do not. While fade-out is necessarily a stochastic phenomenon, the probability of fade-out will depend on the model parameters associated with each epidemic. Accordingly, we ask whether time-series data for the epidemics contain sufficient information to identify the key driver(s) of different outcomes—fade-out or otherwise—across the sub-populations supporting each epidemic. We apply a Bayesian hierarchical modelling framework to synthetic data from an SIRS model of epidemic dynamics and demonstrate that we can (1) identify when the sub-population specific model parameters supporting each epidemic have significant variability and (2) estimate the probability of epidemic fade-out for each sub-population. We demonstrate that a hierarchical analysis can provide precise estimates of the probability of fade-out than is possible if considering each epidemic in isolation. Our methods may be applied to both epidemiological and other biological data to identify where differences in outcome—fade-out or recurrent infection/waves are purely due to chance or driven by underlying changes in the parameters driving the dynamics.

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.

Modified Branching Process and Saddlepoint Approximations

This article presents a brief on using a saddlepoint approximations technique to solve some branching process problems; this method of approximation provides a good understanding of branching process behaviors such as probability density, mass function, and cumulative distribution function based on a moment generating function for complicated models such as nuclear chain reaction, survival of family name problems, and extinction probability problem.

Probability of a zoonotic spillover with seasonal variation

Zoonotic infectious diseases are spread from animals to humans. It is estimated that over 60% of human infectious diseases are zoonotic and 75% of them are emerging zoonoses. The majority of emerging zoonotic infectious diseases are caused by viruses including avian influenza, rabies, Ebola, coronaviruses and hantaviruses. Spillover of infection from animals to humans depends on a complex transmission pathway, which is influenced by epidemiological and environmental processes. In this investigation, the focus is on direct transmission between animals and humans and the effects of seasonal variations on the transmission and recovery rates. Fluctuations in transmission and recovery, besides being influenced by physiological processes and behaviors of pathogen and host, are driven by seasonal variations in temperature, humidity or rainfall. A new time-nonhomogeneous stochastic process is formulated for infectious disease spread from animals to humans when transmission and recovery rates are time-periodic. A branching process approximation is applied near the disease-free state to predict the probability of the first spillover event from animals to humans. This probability is a periodic function of the time when infection is introduced into the animal population. It is shown that the highest risk of a spillover depends on a combination of animal to human transmission, animal to animal transmission and animal recovery. The results are applied to a stochastic model for avian influenza with spillover from domestic poultry to humans.

Stochastic investigation of HIV infection and the emergence of drug resistance.

Drug-resistant HIV-1 has caused a growing concern in clinic and public health. Although combination antiretroviral therapy can contribute massively to the suppression of viral loads in patients with HIV-1, it cannot lead to viral eradication. Continuing viral replication during sub-optimal therapy (due to poor adherence or other reasons) may lead to the accumulation of drug resistance mutations, resulting in an increased risk of disease progression. Many studies also suggest that events occurring during the early stage of HIV-1 infection (i.e., the first few hours to days following HIV exposure) may determine whether the infection can be successfully established. However, the numbers of infected cells and viruses during the early stage are extremely low and stochasticity may play a critical role in dictating the fate of infection. In this paper, we use stochastic models to investigate viral infection and the emergence of drug resistance of HIV-1. The stochastic model is formulated by a continuous-time Markov chain (CTMC), which is derived based on an ordinary differential equation model proposed by Kitayimbwa et al. that includes both forward and backward mutations. An analytic estimate of the probability of the clearance of HIV infection of the CTMC model near the infection-free equilibrium is obtained by a multitype branching process approximation. The analytical predictions are validated by numerical simulations. Unlike the deterministic dynamics where the basic reproduction number R0 serves as a sharp threshold parameter (i.e., the disease dies out if R0<1 and persists if R0>1), the stochastic models indicate that there is always a positive probability for HIV infection to be eradicated in patients. In the presence of antiretroviral therapy, our results show that the chance of clearance of the infection tends to increase although drug resistance is likely to emerge.

The effect of demographic and environmental variability on disease outbreak for a dengue model with a seasonally varying vector population

Seasonal changes in temperature, humidity, and rainfall affect vector survival and emergence of mosquitoes and thus impact the dynamics of vector-borne disease outbreaks. Recent studies of deterministic and stochastic epidemic models with periodic environments have shown that the average basic reproduction number is not sufficient to predict an outbreak. We extend these studies to time-nonhomogeneous stochastic dengue models with demographic variability wherein the adult vectors emerge from the larval stage vary periodically. The combined effects of variability and periodicity provide a better understanding of the risk of dengue outbreaks. A multitype branching process approximation of the stochastic dengue model near the disease-free periodic solution is used to calculate the probability of a disease outbreak. The approximation follows from the solution of a system of differential equations derived from the backward Kolmogorov differential equation. This approximation shows that the risk of a disease outbreak is also periodic and depends on the particular time and the number of the initial infected individuals. Numerical examples are explored to demonstrate that the estimates of the probability of an outbreak from that of branching process approximations agree well with that of the continuous-time Markov chain. In addition, we propose a simple stochastic model to account for the effects of environmental variability on the emergence of adult vectors from the larval stage.

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.

Stochastic modeling approach of infectious disease with sir epidemiological compartment model

The aim of this paper is to present a review on the stochastic version of the deterministic SIR (Susceptible – Infectious - Recovery) epidemiological compartment model through the branching process approximation. The stochastic process (branching process) approximation was developed using the Continuous Time Markov Chains where the time variable is continuous and the state variable is discrete. The state random variables are the compartments: S(t), I(t) and R(t). In this review two ways of estimating the state transition probability has been provided and some stochastic thresholds of the branching process (basic reproduction number, Malthusian parameter and the average number of infections produced by an infectious individual in a single generation) have also been deduced. Finally, the probability of major and minor outbreak of the branching process (epidemic process) has been presented. The theoretical methods have also been validated with some examples of numerical simulations.

On the probability of strain invasion in endemic settings: Accounting for individual heterogeneity and control in multi-strain dynamics

Pathogen evolution is an imminent threat to global health that has warranted, and duly received, considerable attention within the medical, microbiological and modelling communities. Outbreaks of new pathogens are often ignited by the emergence and transmission of mutant variants descended from wild-type strains circulating in the community. In this work we investigate the stochastic dynamics of the emergence of a novel disease strain, introduced into a population in which it must compete with an existing endemic strain. In analogy with past work on single-strain epidemic outbreaks, we apply a branching process approximation to calculate the probability that the new strain becomes established. As expected, a critical determinant of the survival prospects of any invading strain is the magnitude of its reproduction number relative to that of the background endemic strain. Whilst in most circumstances this ratio must exceed unity in order for invasion to be viable, we show that differential control scenarios can lead to less-fit novel strains invading populations hosting a fitter endemic one. This analysis and the accompanying findings will inform our understanding of the mechanisms that have led to past instances of successful strain invasion, and provide valuable lessons for thwarting future drug-resistant strain incursions.