Reed-Frost Model Research Articles

Analysing the COVID-19 epidemic in Italy through the Reed-Frost model: A methodology to delimit epidemic waves over time.

to set out a method based on the Reed Frost model to delimit over time COVID-19 epidemic waves in Italy. the available national epidemic reports published by the Protezione Civile (Italian civil defence) from 24.02.2020 to 16.022022 were used to collect data on COVID-19 epidemic in Italy. Then, the Reed-Frost model was applied to develop a methodology based on the calculation of the effective contact probability, i.e., the probability of contact. in Italy, a daily report related to the epidemic was immediately available, including main epidemiological data (point and periodic infection prevalence, mortality, etc), which made it possible for researchers from different institutions to perform analyses about the epidemic. an iterative methodology was developed resulting in the identification of the start-of-wave, end-of-wave, and inter-wave periods and of the starting and ending days of the COVID-19 epidemic waves in Italy (first wave: from 26±2 February 2020 to 28±2 June 2020). this study led to the development of an accessible and reproducible method to determine the start-of-wave and end-of-wave dates of an epidemic, starting only from the number of cases and susceptible people. The main implications of the method mainly consist in allowing benchmarking and forecasting analyses of the epidemic trend to be carried out to support policy and decision-making processes.

О ПРИМЕНИМОСТИ ОДНОЙ МАРКОВСКОЙ МОДЕЛИ РАСПРОСТРАНЕНИЯ КОМПЬЮТЕРНЫХ ВИРУСОВ К E-MAIL-ГРАФАМ

Mathematical modeling of computer virus epidemics is the most important area of theoretical research in the field of information security. The objective of this paper is to analyze the applicability of the modified Reed-Frost model for epidemic analysis on real-life graphs. We compare the results calculated using the model with the results obtained during the simulation experiment with different epidemic parameters. As a result, the epidemic parameters, in which both methods are in good agreement with each other, have been identified.

Transmission of SARS-CoV-2 in a primary school setting with and without public health measures using real-world contact data: A modelling study.

BackgroundStringent public health measures have been shown to influence the transmission of SARS-CoV-2 within school environments. We investigated the potential transmission of SARS-CoV-2 in a primary school setting with and without public health measures, using fine-grained physical positioning traces captured before the COVID-19 pandemic.MethodsApproximately 172.63 million position data from 98 students and six teachers from an open-plan primary school were used to predict a potential transmission of SARS-CoV-2 in primary school settings. We first estimated the daily average number of contacts of students and teachers with an infected individual during the incubation period. We then used the Reed-Frost model to estimate the probability of transmission per contact for the SARS-CoV-2 Alpha (B.1.1.7), Delta (B.1.617.2), and Omicron variant (B.1.1.529). Finally, we built a binomial distribution model to estimate the probability of onward transmission in schools with and without public health measures, including face masks and physical distancing.ResultsAn infectious student would have 49.1 (95% confidence interval (CI) = 46.1-52.1) contacts with their peers and 2.00 (95% CI = 1.82-2.18) contacts with teachers per day. An infectious teacher would have 47.6 (95% CI = 45.1-50.0) contacts with students and 1.70 (95% CI = 1.48-1.92) contacts with their colleague teachers per day. While the probability of onward SARS-CoV-2 transmission was relatively low for the Alpha and Delta variants, the risk increased for the Omicron variant, especially in the absence of public health measures. Onward teacher-to-student transmission (88.9%, 95% CI = 88.6%-89.1%) and teacher-to-teacher SARS-CoV-2 transmission (98.4%, 95% CI = 98.5%-98.6%) were significantly higher for the Omicron variant without public health measures in place.ConclusionsOur findings illustrate that, despite a lower frequency of close contacts, teacher-to-teacher close contacts demonstrated a higher risk of transmission per contact of SARS-CoV-2 compared to student-to-student close contacts. This was especially significant with the Omicron variant, with onward transmission more likely occurring from teacher index cases than student index cases. Public health measures (eg, face masks and physical distance) seem essential in reducing the risk of onward transmission within school environments.

Приближённая оценка среднего числа заражённых узлов в марковской модели распространения компьютерных вирусов

In this work, we consider a model of computer virus spreading, which is amodification of the well-known Reed-Frost model. The main result of the paper is anapproximate reccurence formula for determining the mean number of infected nodesat any moment of time. Using this formula, we describe a method for calculating thelimit value of the mean number of infected nodes, and then we derive an approximatecriterion for the existence of the so-called extincion regime.

Simulating the mechanics behind sub-optimal mobility and the associated economic losses in dairy production

Hoof disorders and sub-optimal mobility (SOM) are economically important health issues in dairy farming. Although the dynamics of hoof disorders have an important effect on cow mobility, they have not been considered in previous simulation models that estimate the economic loss of SOM. Furthermore, these models do not consider the varying severities of SOM. The objective of this study was to develop a novel bio-economic simulation model to simulate the dynamics of 8 hoof disorders: digital dermatitis (DD), interdigital hyperplasia (HYP), interdigital dermatitis/heel-horn erosion (IDHE), interdigital phlegmon (IP), overgrown hoof (OH), sole haemorrhage (SH), sole ulcer (SU) and white-line disease (WLD), their role in SOM, and estimate the economic loss of SOM in a herd of 125 dairy cows. A Reed-Frost model was used for DD and a Greenwood model for the other 7 hoof disorders. Economic analysis was conducted per mobility score according to a 5-point mobility scoring method (1 = perfect mobility; 5 = severely impaired mobility) by comparing a scenario with SOM and one without SOM. Parameters used in the model were based on literature and expert opinion and deemed credible during model validation rounds. Results showed that the mean cumulative incidence for maximum mobility scores 2–5 SOM episodes were respectively 34, 16, 7 and <1 episodes per 100 cows per pasture period and 39, 19, 8, <1 episodes per 100 cows per housing period. The mean total annual economic loss due to SOM resulting from the hoof disorders under study was €15,342: €122 per cow per year. The economic analysis uncovered direct economic losses that could be directly linked to SOM episodes and indirect economic losses that could not be directly linked to SOM episodes but arose due to the presence of SOM. The mean total annual direct economic loss for maximum mobility score 2–5 SOM episodes was €1129, €3098, €4354 and €480, respectively. The mean total annual indirect economic loss varied considerably between the 5th and 95th percentiles: €−6174 and €19,499, and had a mean of €6281. This loss was composed of additional indirect culling due to SOM (∼65%) and changes in the overall herd milk production (∼35%) because of additional younger replacement heifers entering the herd due to increased culling rates. The bio-economic model presented novel results with respect to indirect economic losses arising due to SOM. The results can be used to stimulate farmer awareness and promote better SOM management.

SARS-CoV-2 infection risk during delivery of childhood vaccination campaigns: a modelling study

BackgroundThe COVID-19 pandemic has disrupted the delivery of immunisation services globally. Many countries have postponed vaccination campaigns out of concern about infection risks to the staff delivering vaccination, the children being vaccinated, and their families. The World Health Organization recommends considering both the benefit of preventive campaigns and the risk of SARS-CoV-2 transmission when making decisions about campaigns during COVID-19 outbreaks, but there has been little quantification of the risks.MethodsWe modelled excess SARS-CoV-2 infection risk to vaccinators, vaccinees, and their caregivers resulting from vaccination campaigns delivered during a COVID-19 epidemic. Our model used population age structure and contact patterns from three exemplar countries (Burkina Faso, Ethiopia, and Brazil). It combined an existing compartmental transmission model of an underlying COVID-19 epidemic with a Reed-Frost model of SARS-CoV-2 infection risk to vaccinators and vaccinees. We explored how excess risk depends on key parameters governing SARS-CoV-2 transmissibility, and aspects of campaign delivery such as campaign duration, number of vaccinations, and effectiveness of personal protective equipment (PPE) and symptomatic screening.ResultsInfection risks differ considerably depending on the circumstances in which vaccination campaigns are conducted. A campaign conducted at the peak of a SARS-CoV-2 epidemic with high prevalence and without special infection mitigation measures could increase absolute infection risk by 32 to 45% for vaccinators and 0.3 to 0.5% for vaccinees and caregivers. However, these risks could be reduced to 3.6 to 5.3% and 0.1 to 0.2% respectively by use of PPE that reduces transmission by 90% (as might be achieved with N95 respirators or high-quality surgical masks) and symptomatic screening.ConclusionsSARS-CoV-2 infection risks to vaccinators, vaccinees, and caregivers during vaccination campaigns can be greatly reduced by adequate PPE, symptomatic screening, and appropriate campaign timing. Our results support the use of adequate risk mitigation measures for vaccination campaigns held during SARS-CoV-2 epidemics, rather than cancelling them entirely.

Modeling Effectiveness of Testing Strategies to Prevent Coronavirus Disease 2019 (COVID-19) in Nursing Homes-United States, 2020.

BackgroundSARS-CoV-2 outbreaks in nursing homes can be large with high case fatality. Identifying asymptomatic individuals early through serial testing is recommended to control COVID-19 in nursing homes, both in response to an outbreak (“outbreak testing” of residents and healthcare personnel) and in facilities without outbreaks (“non-outbreak testing” of healthcare personnel). The effectiveness of outbreak testing and isolation with or without non-outbreak testing was evaluated.MethodsUsing published SARS-CoV-2 transmission parameters, the fraction of SARS-CoV-2 transmissions prevented through serial testing (weekly, every three days, or daily) and isolation of asymptomatic persons compared to symptom-based testing and isolation was evaluated through mathematical modeling using a Reed-Frost model to estimate the percentage of cases prevented (i.e., “effectiveness”) through either outbreak testing alone or outbreak plus non-outbreak testing. The potential effect of simultaneous decreases (by 10%) in the effectiveness of isolating infected individuals when instituting testing strategies was also evaluated.ResultsModeling suggests that outbreak testing could prevent 54% (weekly testing with 48-hour test turnaround) to 92% (daily testing with immediate results and 50% relative sensitivity) of SARS-CoV-2 infections. Adding non-outbreak testing could prevent up to an additional 8% of SARS-CoV-2 infections (depending on test frequency and turnaround time). However, added benefits of non-outbreak testing were mostly negated if accompanied by decreases in infection control practice.ConclusionsWhen combined with high-quality infection control practices, outbreak testing could be an effective approach to preventing COVID-19 in nursing homes, particularly if optimized through increased test frequency and use of tests with rapid turnaround.

Fractional seroprevalence rates in common prey species can cause more than half of feral cats to be exposed to Toxoplasma gondii annually

Rodents comprise a major component of cat (Felis catus) diets in many ecosystems, and life cycle diagrams of Toxoplasma gondii typically depict small rodents as quintessential intermediate hosts. Counter-intuitively, small rodents often experience a lower T. gondii seroprevalence than do larger sympatric mammals. This observation has repeatedly caused confusion about the relative importance of small rodents to the ecology of T. gondii. To address this confusion, we modified the Reed-Frost epidemic model to develop a simple binomial equation to model T. gondii transmission from prey to feline predators. This equation takes into account variations in prey seroprevalence and the frequency with which they are consumed by felids. Even when T. gondii seroprevalence in prey is < 1%, computation reveals that the risk of feline exposure to T. gondii can easily exceed 50 % annually. For example, if cats eat an average of 1 mouse per day, a seroprevalence of 0.2 % (1/500) in mice will cause 51.9 % of cats to be exposed to T. gondii annually. Our simple equation demonstrates that both prey seroprevalence and the rate at which prey are consumed are of approximately equal importance to the ecology of T. gondii. When inferring the importance of various prey species to the ecology of T. gondii, researchers must consider the predation and dietary habits of felids from within their study system. Our simple binomial equation could also be used to predict T. gondii exposure rates of humans or other carnivorous animals from various dietary sources or be applied to other predator-prey parasite life cycles.

О ВЕРОЯТНОСТИ ЗАРАЖЕНИЯ ВОСПРИИМЧИВОГО УЗЛА В МОДЕЛИ РИДА-ФРОСТА

We investigate a model of computer virus propagation based on the Reed-Frost model and the random graph model. A simplified formula for the probability of infection of a susceptible node is obtained, which significantly simplifies the software implementation of the model under consideration. Then a simulation experiment was carried out, which showed that the resulting formula gives a good estimate for the results obtained experimentally.

Multiple exposures, reinfection and risk of progression to active tuberculosis.

A recent study reported on a tuberculosis (TB) outbreak in a largely Inuit village. Among newly infected individuals, exposure to additional active cases was associated with an increasing probability of developing active disease within a year. Using binomial risk models, we evaluated two potential mechanisms by which multiple infections during the first year following initial infection could account for increasing disease risk with increasing exposures. In the reinfection model, each infectious contact confers an independent risk of an infection, and infections contribute independently to active disease. In the threshold model, disease risk follows a sigmoidal function with small numbers of infectious contacts conferring a low risk of active disease and large numbers of contacts conferring a high risk. To determine the dynamic impact of reinfection during the early phase of infection, we performed simulations from a modified Reed–Frost model of TB dynamics following spread from an initial number of cases. We parametrized this model with the maximum-likelihood estimates from the reinfection and threshold models in addition to the observed distribution of exposures among new infections. We find that both models can plausibly account for the observed increase in disease risk with increasing infectious contacts, but the threshold model confers a better fit than a nested model without a threshold (p = 0.04). Our simulations indicate that multiple exposures to infectious individuals during this critical time period can lead to dramatic increases in outbreak size. In order to decrease TB burden in high-prevalence settings, it may be necessary to implement measures aimed at preventing repeated exposures, in addition to preventing primary infection.

Who is the infector? Epidemic models with symptomatic and asymptomatic cases

What role do asymptomatically infected individuals play in the transmission dynamics? There are many diseases, such as norovirus and influenza, where some infected hosts show symptoms of the disease while others are asymptomatically infected, i.e. do not show any symptoms. The current paper considers a class of epidemic models following an SEIR (Susceptible → Exposed → Infectious → Recovered) structure that allows for both symptomatic and asymptomatic cases. The following question is addressed: what fraction ρ of those individuals getting infected are infected by symptomatic (asymptomatic) cases? This is a more complicated question than the related question for the beginning of the epidemic: what fraction of the expected number of secondary cases of a typical newly infected individual, i.e. what fraction of the basic reproduction number R0, is caused by symptomatic individuals? The latter fraction only depends on the type-specific reproduction numbers, while the former fraction ρ also depends on timing and hence on the probabilistic distributions of latent and infectious periods of the two types (not only their means). Bounds on ρ are derived for the situation where these distributions (and even their means) are unknown. Special attention is given to the class of Markov models and the class of continuous-time Reed–Frost models as two classes of distribution functions for latent and infectious periods. We show how these two classes of models can exhibit very different behaviour.

Counting the dead to determine the source and transmission of the marine herpesvirus OsHV-1 in Crassostrea gigas

Marine herpesviruses are responsible for epizootics in economically, ecologically and culturally significant taxa. The recent emergence of microvariants of Ostreid herpesvirus 1 (OsHV-1) in Pacific oysters Crassostrea gigas has resulted in socioeconomic losses in Europe, New Zealand and Australia however, there is no information on their origin or mode of transmission. These factors need to be understood because they influence the way the disease may be prevented and controlled. Mortality data obtained from experimental populations of C. gigas during natural epizootics of OsHV-1 disease in Australia were analysed qualitatively. In addition we compared actual mortality data with those from a Reed–Frost model of direct transmission and analysed incubation periods using Sartwell’s method to test for the type of epizootic, point source or propagating. We concluded that outbreaks were initiated from an unknown environmental source which is unlikely to be farmed oysters in the same estuary. While direct oyster-to-oyster transmission may occur in larger oysters if they are in close proximity (< 40 cm), it did not explain the observed epizootics, point source exposure and indirect transmission being more common and important. A conceptual model is proposed for OsHV-1 index case source and transmission, leading to endemicity with recurrent seasonal outbreaks. The findings suggest that prevention and control of OsHV-1 in C. gigas will require multiple interventions. OsHV-1 in C. gigas, which is a sedentary animal once beyond the larval stage, is an informative model when considering marine host-herpesvirus relationships.

Reed - Frost 모형을 이용한 전염병 감염 확률 추정

질병의 확산 과정을 설명하기 위한 모형으로 가장 대표적인 방법은 Kermack과 McKendrick (1927)에 의해 제안된 SIR (susceptible - infectious - recovered) 모형이다. SIR 모형을 구축하기 위해서는 질병의 감염률 (transition rate)과 회복률 (recovery rate)이 주어져 있거나 질병의 전체 확산 과정이 데이터로 주어진 경우 추정을 통하여 구할 수 있다. 하지만 데이터가 제한적으로 관찰된 경우 직접적인 감염률와 회복률의 계산이 불가능 하다. 본 연구에서는 관찰된 자료가 가지는 한계점을 고려하여 질병의 초기 확산과정에서 질병 감염 확률을 추정하기 위하여 리드-프로스트 (Reed-Frost) 모형 (Andersson과 Britton, 2000)을 적용하였다. 리드-프로스트 모형은 질병의 최초 감염자 수, 최종 감염자 수, 그리고 최초 감염대상자의 수가 주어졌을 때 이를 통하여 감염 확률을 추정하기 위한 모형이다. 본 연구에서는 서아프리카의 카메룬 공화국에서 조사된 역학 조사 자료를 이용하여 콜레라의 초기 감염 확률을 추정하였다. 그리고 추정된 결과를 이용하여 다시 SIR 모형에 적용하여 질병의 확산 경로에 대한 예측을 수행하였다. 예측 결과 조사 지역의 주민 가운데 50% 이상이 감염될 것으로 예측되었으며 질병의 전파는 약 한달 정도 지속될 것으로 예측 되었다. SIR model (Kermack and McKendrik, 1927) is one of the most popular method to explain the spread of disease, In order to construct SIR model, we need to estimate transition rate parameter and recovery rate parameter. If we don't have any information of the two rate parameters, we should estimate using observed whole trajectory of pandemic of disease. Thus, with restricted observed data, we can't estimate rate parameters. In this research, we introduced Reed-Frost model (Andersson and Britton, 2000) to calculate the probability of infection in the early stage of pandemic with the restriction of data. When we have an initial number of susceptible and infected, and a final number of infected, we can apply Reed - Frost model and we can get the probability of infection. We applied the Reed - Frost model to the Vibrio cholerae pandemic data from Republic of the Cameroon and calculated the probability of infection at the early stage. We also construct SIR model using the result of Reed - Frost model.

Analysis of household data on influenza epidemic with Bayesian hierarchical model

Data used for modelling the household transmission of infectious diseases, such as influenza, have inherent multilevel structures and correlated property, which make the widely used conventional infectious disease transmission models (including the Greenwood model and the Reed–Frost model) not directly applicable within the context of a household (due to the crowded domestic condition or socioeconomic status of the household). Thus, at the household level, the effects resulting from individual-level factors, such as vaccination, may be confounded or modified in some way. We proposed the Bayesian hierarchical random-effects (random intercepts and random slopes) model under the context of generalised linear model to capture heterogeneity and variation on the individual, generation, and household levels. It was applied to empirical surveillance data on the influenza epidemic in Taiwan. The parameters of interest were estimated by using the Markov chain Monte Carlo method in conjunction with the Bayesian directed acyclic graphical models. Comparisons between models were made using the deviance information criterion. Based on the result of the random-slope Bayesian hierarchical method under the context of the Reed–Frost transmission model, the regression coefficient regarding the protective effect of vaccination varied statistically significantly from household to household. The result of such a heterogeneity was robust to the use of different prior distributions (including non-informative, sceptical, and enthusiastic ones). By integrating out the uncertainty of the parameters of the posterior distribution, the predictive distribution was computed to forecast the number of influenza cases allowing for random-household effect.

Comparison of intervention methods for reducing human exposure to Mycobacterium bovis through milk in pastoralist households of Tanzania

Bovine tuberculosis (bTB), caused by Mycobacterium bovis, is a disease of zoonotic concern, especially in countries with no control programs in livestock and where routine pasteurization of milk is not practiced. In Tanzania, bTB is widespread in livestock and has been diagnosed in humans; however, herd bTB testing is primarily carried out for bTB-free certification in commercial dairy herds at the expense of the dairy cattle owner. For rural livestock holders, such an expense is prohibitive, and consequently there is no control of bTB in most areas. Although effective long-term solutions to control bTB in livestock are desirable, there is a need to assess the effect of preventive measures on reducing human exposure to bTB in such settings. We utilized locally relevant cattle herd characteristics and management data from the Health for Animals and Livelihood Improvement (HALI) project in south-central Tanzania to build a Reed-Frost model that compared the efficacy of alternative methods aimed at reducing the exposure of humans to infectious milk from a typical pastoralist cattle herd. During a 10-year simulation period, the model showed that boiling milk 80% of the time is necessary to obtain a reduction in liters of infectious milk approximately equivalent to what would be obtained with a standard 2-year testing and removal regimen, and that boiling milk was more effective than animal test and removal early in the time period.In addition, even with testing and removing infected cattle, a residual risk of exposure to infectious milk remained due to imperfect sensitivity of the skin test and a continuous risk of introduction of infectious animals from other herds. The model was sensitive to changes in initial bTB prevalence but not to changes in herd size. In conclusion, continuous complimentary treatment of milk may be an effective strategy to reduce human exposure to M. bovis-infected milk in settings where bTB is endemic and a comprehensive bTB control program is yet to be implemented.

Prevalence and incidence of intramammary infections in lactating dairy goats

AIM: To determine the prevalence of intramammary infection (IMI) of lactating dairy goats between 0 and 4 days postpartum, the prevalence and incidence rate of new IMI at Weeks 2, 14 and 27 of lactation, and the relationship between herd-level prevalence of IMI and bulk tank somatic cell count (BTSCC).METHODS: Milk samples were collected from 8% of a herd (total 624 does) from 18 dairy goat herds in the Waikato region of New Zealand, for bacteriology and somatic cell count (SCC) determination, from both glands within 4 days of kidding (Week 0) and again at Weeks 2, 14 and 27. Prevalence of IMI was determined at each time point and incidence rate calculated for Weeks 0–2, 2–14, and 14–27. Greenwood and Reed-Frost models were compared for estimation of the transmission parameter for all pathogens, and for Staphylococcus aureus, coagulase negative staphylococci (CNS) and Corynebacterium spp. separately.RESULTS: Bacteria were isolated from 1,122/4,814 (23.3%) glands, with CNS (13.4%) and Corynebacterium spp. (7.3%) being the most common isolates. Prevalence of any IMI increased with stage of lactation, varied among herds, and increased with age (all p<0.05). Incidence rate was 80, 24 and 7 new IMI/10,000 gland days for Weeks 0–2, 2–14 and 14–27, respectively. Incidence rate for any IMI increased with age and with the presence of an IMI in the contralateral gland, and varied among herds (p<0.001). The transmission of each pathogen was better modelled assuming contagiousness (Reed-Frost models), than not (Greenwood models).At gland level, IMI increased SCC at all stages of lactation (p<0.001). The gland prevalence of IMI within herds was positively associated with ln BTSCC at Week 2 (p=0.02), but not Weeks 14 or 27 (p>0.05).CONCLUSIONS: Prevalence of IMI increased with stage of lactation, but the highest incidence rate of new IMI occurred in early lactation. Models accounting for the contagious nature of infection fitted better than those not accounting for contagiousness. BTSCC was only associated with prevalence of IMI in early lactation.CLINICAL RELEVANCE: Reduction of BTSCC in dairy goats may be best achieved by minimising the prevalence and incidence of new IMI in early lactation. Further studies are required to define management factors associated with the between herd, and stage of lactation, effects on prevalence and incidence rate in order to reduce BTSCC throughout lactation.

A Branching Process Approximation of the Final Size of a Multitype Collective Reed-Frost Model

We consider the asymptotic behavior of the final size of a multitype collective Reed-Frost process. This type of models was introduced by (9) and include most known epidemic models of the type SIR (Susceptible, Infected, Removed) as special cases. Under certain conditions, we show that, when the initial number of susceptible is very large and the initial number of infected individuals is finite, the infection process behaves as a multitype Galton-Watson process. This fact is proved using a simple argument based on Bernstein polynomials. We use this result to study the final size of the epidemic.

Approximating the epidemic curve

Many models of epidemic spread have a common qualitative structure. The numbers of infected individuals during the initial stages of an epidemic can be well approximated by a branching process, after which the proportion of individuals that are susceptible follows a more or less deterministic course. In this paper, we show that both of these features are consequences of assuming a locally branching structure in the models, and that the deterministic course can itself be determined from the distribution of the limiting random variable associated with the backward, susceptibility branching process. Examples considered includea stochastic version of the Kermack & McKendrick model, the Reed-Frost model, and the Volz configuration model.

VARIABILITY FOR CARRIER-BORNE EPIDEMICS AND REED–FROST MODELS INCORPORATING UNCERTAINTIES AND DEPENDENCIES FROM SUSCEPTIBLES AND INFECTIVES

This article provides analytical results on which are the implications of the statistical dependencies among certain random parameters on the variability of the number of susceptibles of the carrier-borne epidemic model with heterogeneous populations and of the number of infectives under the Reed–Frost model with random infection rates. We consider dependencies among the random infection rates, among the random infectious times, and among random initial susceptibles of several carrier-borne epidemic models. We obtain conditions for the variability ordering between the number of susceptibles for carrier-borne epidemics under two different random environments, at any time-scale value. These results are extended to multivariate comparisons of the random vectors of populations in the strata. We also obtain conditions for the increasing concave order between the number of infectives in the Reed–Frost model under two different random environments, for any generation. Variability bounds are obtained for different epidemic models from modeling dependencies for a range of special cases that are useful for risk assessment of disease propagation.

Selfish Drug Allocation for Containing an International Influenza Pandemic at the Onset

Recent epidemiologic studies have suggested that the prophylactic use of antiviral drugs could slow down the spread of an influenza epidemic. Because drug stockpiles are presently scattered in different countries, the outbreak of an epidemic gives rise to a game in which each country must make decisions about how best to allocate its own stockpile in order to protect its population. We develop a two-period multivariate Reed-Frost model to represent the spread of the epidemic within and across countries at its onset. We consider the first two periods only to mimic the exponential growth of an epidemic in its early stage, while keeping the model tractable. Preliminary numerical studies suggest that insights from the two-period model hold in general when considering the entire time horizon. Our model captures three critical sources of uncertainty: the number of initial infections, the spread of the disease, and drug efficacy. We show that for small probabilities of between-country infections, the underlying game is supermodular, Nash equilibrium exists, and there is a unique one that is Pareto optimal among all existing equilibria. Further, we identify sufficient conditions under which the optimal solution of a central planner (such as the World Health Organization) constitutes a Pareto improvement over the decentralized equilibrium, suggesting that countries should agree on an allocation scheme that would benefit everyone. By contrast, when the central planner's solution does not constitute a Pareto improvement, minimizing the total number of infected persons globally requires some countries to sacrifice part of their own population, which raises intriguing ethical issues.