A Mathematical Model of the Eyam Plague using R Programming Language (original) (raw)

The Eyam plague revisited: did the village isolation change transmission from fleas to pulmonary

Back in the 17th century the Derbyshire village of Eyam fell victim to the Black Death, which is thought to have arrived from London in some old clothes brought by a travelling tailor. The village population was 350 at the commencement of plague, of which only 83 survived. Led by the church leaders, the village community realized that the whole surrounding region was at risk from the epidemic, and therefore decided to seal themselves off from the other surrounding villages. In the first 275 days of the outbreak, transmission was predominantly from infected fleas to susceptible humans. From then onward, mortality sharply increased, which indicates a changing in transmission pattern. We hypothesize that the confinement facilitated the spread of the infection by increasing the contact rate through direct transmission among humans. This would be more consistent with pulmonary plague, a deadlier form of the disease. In order to test the above hypothesis we designed a mathematical model for plague dynamics, incorporating both the indirect (fleas-rats-humans) and direct (human-to-human) transmissions of the infection. Our results show remarkable agreement between data and the model, lending support to our hypotheses. The Eyam plague episode is celebrated as a remarkable act of collective self-sacrifice. However, to the best of our knowledge, there were no evidence before that the confinement actually increased the burden payed by the commoners.

Modelling the dynamics of bubonic plague with Yersinia pestis in the environment

Communications in mathematical biology and neuroscience, 2016

Bubonic plague is an infectious disease that is caused by the bacteria Yersinia pestis when it affect a part of circulatory system namely lymphatic system. It is mainly transferred between populations through flea bites. In this paper we develop a deterministic model that includes four compartments namely Human, Rodent, Flea and pathogens in the environment to study the dynamics and spread of bubonic plague. The model is analyzed to determine the role and magnitude of the involvement of the four sub-populations in the transmission and spread of the disease. We use the next generation method to find the disease threshold R 0 which is then used to examine the local stability of the equilibrium points. A sensitivity analysis is carried out to determine the most, medium and least sensitive model parameters that affect negatively or positively the basic reproduction number. The result call for attention to most sensitive parameter which is the progression rate of flea from susceptible state to infected state (β). Other significant parameters are adequate contact rates (Γ f h), (Γ h f), (Γ f r), (Γ f r) and (Γ r f); progression rates (α 1), (α 2), (γ 1) and (γ 2); and the pathogens in the environment under the condition that the their survival is favored by the environment. The numerical simulation also support the analytical solution which then cement the argument that the fruitful strategy to combat bubonic plague disease will be the one that will consider the factors that have shown to have a significant contribution to the increase of the basic reproduction number.

Bubonic plague: a metapopulation model of a zoonosis

Proceedings of the Royal Society B: Biological Sciences, 2000

Bubonic plague (Yersinia pestis) is generally thought of as a historical disease; however, it is still responsible for around 1000^3000 deaths each year worldwide. This paper expands the analysis of a model for bubonic plague that encompasses the disease dynamics in rat, £ea and human populations. Some key variables of the deterministic model, including the force of infection to humans, are shown to be robust to changes in the basic parameters, although variation in the £ea searching e¤ciency, and the movement rates of rats and £eas will be considered throughout the paper. The stochastic behaviour of the corresponding metapopulation model is discussed, with attention focused on the dynamics of rats and the force of infection at the local spatial scale. Short-lived local epidemics in rats govern the invasion of the disease and produce an irregular pattern of human cases similar to those observed. However, the endemic behaviour in a few rat subpopulations allows the disease to persist for many years. This spatial stochastic model is also used to identify the criteria for the spread to human populations in terms of the rat density. Finally, the full stochastic model is reduced to the form of a probabilistic cellular automaton, which allows the analysis of a large number of replicated epidemics in large populations. This simpli¢ed model enables us to analyse the spatial properties of rat epidemics and the e¡ects of movement rates, and also to test whether the emergent metapopulation behaviour is a property of the local dynamics rather than the precise details of the model.

Modelling outbreak control for pneumonic plague

Epidemiology and Infection, 2007

SUMMARYAlthough pneumonic plague is listed by the Centers of Disease Control in the leading ‘critical biological agents’, very few studies exist on this subject. In this study, a mathematical compartment model was used to describe the geographical and temporal spread of an epidemic of pneumonic plague following its use as a biological weapon. Univariate and multivariate analyses were performed in order to assess the key parameters for the control of an outbreak in France. If interventions were taken 10 days after an attack, a reference scenario of 1000 index cases in Paris would lead to 2500 deaths. The results of the study indicate that the rapidity of onset of interventions has the largest effect on the final size of the epidemic, followed by wearing masks, treating contacts preventively and quarantine. Limiting inter-regional mixing does little to reduce casualties, although it does confine them to a single region.

Modeling of the Plague Epizootic Process

2018

Commonly-used mathematical models that rely on differential equations require input from many data sources. While, at the same time, the availability of plague surveillance source data is rather insufficient.  Temporal models, (based on differential equations or other methods), are being used for modeling without consideration of spatial distribution. In these models, it is assumed that all events are happening at an abstract point rather than considering plague focus, landscape region, or primary square.  If plague simulates nature, it analyzes only a static structure of plague focus or its parts. The probabilistic, cellular automaton simulation allows the user to combine all of these rationales. The epizootic process of plague in Great gerbil settlements was a good subject for this experiment. The gerbil's burrow system represents a time-spatial, discrete unit of an epizootic process (Sedin, 1985). The data on burrow systems distribution within a plague focus was determined by GIS and remote sensing. Satellite images of the burrow systems provide a coherent picture of the colony structure (Burdelov et al., 2007; Addink et al., 2010) at many plague foci (Fig 1.).

Empirical assessment of a threshold model for sylvatic plague

Journal of The Royal Society Interface, 2007

Plague surveillance programmes established in Kazakhstan, Central Asia, during the previous century, have generated large plague archives that have been used to parameterize an abundance threshold model for sylvatic plague in great gerbil (Rhombomys opimus) populations. Here, we assess the model using additional data from the same archives. Throughout the focus, population levels above the threshold were a necessary condition for an epizootic to occur. However, there were large numbers of occasions when an epizootic was not observed even though great gerbils were, and had been, abundant. We examine six hypotheses that could explain the resulting false positive predictions, namely (i) including end-of-outbreak data erroneously lowers the estimated threshold, (ii) too few gerbils were tested, (iii) plague becomes locally extinct, (iv) the abundance of fleas was too low, (v) the climate was unfavourable, and (vi) a high proportion of gerbils were resistant. Of these, separate thresholds, fleas and climate received some support but accounted for few false positives and can be disregarded as serious omissions from the model. Small sample size and local extinction received strong support and can account for most of the false positives. Host resistance received no support here but should be subject to more direct experimental testing.

MODELING PLAGUE PERSISTENCE IN HOST-VECTOR COMMUNITIES IN CALIFORNIA

2007

Plague is an enzootic disease in the western United States, even though long-term persistent infections do not seem to occur. Enzootic persistence may occur as a function of dynamic interactions between flea vectors and transiently infected hosts, but the specific levels of vector competence, host competence, and transmission and recovery rates that would promote persistence and emergence among wild hosts and vectors are not known. We developed a mathematical model of enzootic plague in the western United States and implemented the model with the following objectives: 1) to use matrix manipulation within a classic susceptibleRinfecti-veRresistantRsusceptible (SIRS) model framework to describe transmission of the plague bacterium Yersinia pestis among rodents and fleas in California, 2) to perform sensitivity analysis with model parameters and variables to indicate which values tended to dominate model output, and 3) to determine whether enzootic maintenance would be predicted with realistic parameter values obtained from the literature for Y. pestis in California rodents and fleas. The model PlagueSIRS was implemented in discrete time as a computer simulation incorporating environmental stochasticity and seasonality, by using matrix functions in the computer language R, allowing any number of rodent and flea species to interact through parasitism and disease transmission. Sensitivity analysis indicated that the model was sensitive to flea attack rate, host recovery rate, and rodent host carrying capacity but relatively insensitive to changes in the duration of latent infection in the flea, host and vector competence, flea recovery from infection, and host mortality attributable to plague. Realistic parameters and variable values did allow for the model to predict enzootic plague in some combinations, specifically when rodent species that were susceptible to infection but resistant to morbidity were parasitized by multiple poorly competent flea species, including some that were present year-round. This model could be extended to similar vectorborne disease systems and could be used iteratively with data collection in sylvatic plague studies to better understand plague persistence and emergence in nature.

Plague Epidemic, Response and tips of Prevention

2017

World memory still tired with Ebola, the last outbreak accrued in west Africa and now we are facing another outbreak started slowly and again in Africa but in the east part. Black Death or plague it`s a one of the historic disease known even before 6th century CE. History told much about the epidemic and pandemic over centuries. Plague is caused by the bacteria Yersinia pestis, a zoonotic bacteria usually found in small mammals and their fleas, this is could be found everywhere since there is no a particular home land for this causative agent. Infected persons started developing sign and symptoms after 7 days and the main sings are respiratory and lymphnodes changes. Plague is associated with high fatality ratio and it could reach 30% to 60% for the bubonic type, and is always fatal for the pneumonic kind when left untreated. World health organization mentioned in their report that from 2010 to 2015 there were 3248 cases reported worldwide, including 584 deaths (1). Plague is associ...