Optimal control of epidemics with limited resources

We extend the existing work on the time-optimal control of the basic SIR epidemic model with mass action contact rate. Previous results have focused on minimizing an objective function that is a linear combination of the cost associated with using control and either the outbreak size or the infectious burden. We instead, provide analytic solutions for the control that minimizes the outbreak size (or infectious burden) under the assumption that there are limited control resources. We provide optimal control policies for an isolation only model, a vaccination only model and a combined isolation–vaccination model (or mixed model). The optimal policies described here contain many interesting features especially when compared to previous analyses. For example, under certain circumstances the optimal isolation only policy is not unique. Furthermore the optimal mixed policy is not simply a combination of the optimal isolation only policy and the optimal vaccination only policy. The results presented here also highlight a number of areas that warrant further study and emphasize that time-optimal control of the basic SIR model is still not fully understood.     

Optimal intervention for epidemic models with general infection and removal rate functions

 In this paper, known results on optimal intervention policies for the general stochastic epidemic model are extended to epidemic models with more general infection and removal rate functions. We consider first policies allowing for the isolation of any number of infectives from the susceptible population at any time, secondly policies allowing for the immunisation of the entire susceptible population at any time, and finally policies allowing for either of these interventions. In each case the costs of infection, isolation and immunisation are assumed to have a particular, rather simple, form. Sufficient conditions are given on the infection and removal rate functions of the model for the optimal policies to take the same simple form as in the case of the general stochastic epidemic model. More general costs are briefly discussed, and some numerical examples given. Finally, we discuss possible directions for further work. Received: 16 February 1999     

Optimal control for pandemic influenza: The role of limited antiviral treatment and isolation

The implementation of optimal control strategies involving antiviral treatment and/or isolation measures can reduce significantly the number of clinical cases of influenza. Pandemic-level control measures must be carefully assessed specially in resource-limited situations. A model for the transmission dynamics of influenza is used to evaluate the impact of isolation and/or antiviral drug delivery measures during an influenza pandemic. Five pre-selected control strategies involving antiviral treatment and isolation are tested under the “unlimited” resource assumption followed by an exploration of the impact of these “optimal” policies when resources are limited in the context of a 1918-type influenza pandemic scenario. The implementation of antiviral treatment at the start of a pandemic tends to reduce the magnitude of epidemic peaks, spreading the maximal impact of an outbreak over an extended window in time. Hence, the controls’ timing and intensity can reduce the pressures placed on the health care infrastructure by a pandemic reducing the stress put on the system during epidemic peaks. The role of isolation strategies is highlighted in this study particularly when access to antiviral resources is limited.     

Control of emerging infectious diseases using responsive imperfect vaccination and isolation

This paper is concerned with a stochastic model for the spread of an SEIR (susceptible --> exposed (= latent) --> infective --> removed) epidemic among a population partitioned into households, featuring different rates of infection for within and between households. The model incorporates responsive vaccination and isolation policies, based upon the appearance of diagnosed cases in households. Different models for imperfect vaccine response are considered. A threshold parameter R*, which determines whether or not a major epidemic can occur, and the probability of a major epidemic are obtained for different infectious and latent period distributions. Simpler expressions for these quantities are obtained in the limiting case of infinite within-household infection rate. Numerical studies suggest that the choice of infectious period distribution and whether or not latent individuals are vaccine-sensitive have a material influence on the spread of the epidemic, while, for given vaccine efficacy, the choice of vaccine action model is less influential. They also suggest that an effective isolation policy has a more significant impact than vaccination. The results show that R* alone is not sufficient to summarise the potential for an epidemic.     

A comparison of a deterministic and stochastic model for Hepatitis C with an isolation stage

We formulate a deterministic epidemic model for the spread of Hepatitis C containing an acute, chronic and isolation class and analyse the effects of the isolation class on the transmission dynamics of the disease. We calculate the basic reproduction number R₀ and show that for R₀≤1, the disease-free equilibrium is globally asymptotically stable. In addition, it is shown that for a special case when R₀>1, the endemic equilibrium is locally asymptotically stable. Furthermore, an analogous stochastic epidemic model for Hepatitis C is formulated using a continuous time Markov chain. Numerical simulations are used to estimate the mean, variance and probability distributions of the discrete random variables and these are compared to the steady-state solutions of the deterministic model. Finally, the expected time to disease extinction is estimated for the stochastic model and the impact of isolation on the time to extinction is explored.     

Risk assessment and evaluation of China’s policy to prevent COVID-19 cases imported by plane

As of October 5, 2020, China has reported 2,921 cases imported from overseas. Assessing the effectiveness of China''s current policies on imported cases abroad is very important for China and other countries that are facing or will face overseas imported cases. In April, we used a susceptible-exposed-infectious-recovered metapopulation model to simulate the epidemic in seven foreign countries, China and the three Chinese key cities. Based on the model outside China, we estimated the proportion of people in incubation period and calculated the risk indexes for Chinese cities through analyzing aviation traffic data from these countries. Based on the model in China and the three key cities, we collected information on control measures and quantified the effectiveness of implementing the current policies at different times and intensities. Our model results showed that Shanghai, Beijing, Qingdao, Guangzhou, and Tianjin have the top five risk indexes. As of April 20, 2020, under current measures, the number of confirmed cases could be reduced by 99% compared with no air traffic restrictions and isolation measures; the reduction could be 93% with isolation of passengers only from key countries. If the current policy were postponed for 7, 10, or 20 days, the increase in the number of confirmed cases would be 1,329, 5,524, and 779,245 respectively, which is 2^.1, 5^.7, and 662^.9 times the number of confirmed cases under current measures. Our research indicates that the importation control measures currently taken by China were implemented at an appropriate time to prevent the epidemic spreading and have achieved relatively good control results. However, it is necessary to remain vigilant; otherwise, another outbreak peak could occur.     

Epidemiologic and economic effectiveness of school closure during influenza epidemics and pandemics

Epidemiologic and economic effectiveness of school closure during influenza epidemics and pandemics is discussed. Optimal effect of school closure is observed when this measure is taken at the start of the epidemic or pandemic and for a sufficiently long time. School closure during high morbidity among schoolchildren, in the middle (at the peak) and by the end of epidemic or pandemic does not influence significantly the spread of influenza or morbidity. Significant economic losses and other negative consequences of school closure are noted. School closure may be the most appropriate during the emergence of influenza pandemic when the pandemic vaccine is not yet available, however timely mass immunization of schoolchildren against influenza may be a more appropriate measure than school closure for the reduction of influenza morbidity and spread during seasonal influenza epidemics.     

综述与专论: 针对严重急性呼吸综合征冠状病毒2变异株Omicron的疫苗研究进展

2021年底,严重急性呼吸综合征冠状病毒2 Omicron变异株迅速取代Delta突变株在世界范围内广泛流行,其S蛋白具有36个位点突变,导致致病力和传播力发生明显变化,并且具备了免疫逃逸的能力。疫苗接种是目前疫情防控最普适的手段,研究发现,现有疫苗针对Omicron突变株的保护效果明显下降。新的免疫策略或特异性疫苗/多价疫苗针对Omicron有效性的评估均需要动物模型的支撑。在实验室条件下,利用动物模型进行活病毒攻击实验,是在体内验证保护性中和抗体、疫苗有效性的关键技术手段,本文将从动物模型方向综述国内外针对Omicron变异株的疫苗研究进展。     

Measles epidemic from failure to immunize.

During 1988 through 1990, California experienced its worst measles epidemic in more than a decade, with 16,400 reported cases, 3,390 hospital admissions, and 75 deaths. More than half of the patients were younger than 5 years; the highest incidence was among infants younger than 12 months. The epidemic centered in low-income Hispanic communities in southern and central California. The major cause of the epidemic was low immunization levels among preschool-aged children and young adults. Rates of complications, admission to hospital, and death were surprisingly high. Outbreak control efforts met with indeterminate success. Problems with these efforts included insufficient funding early in the epidemic and disappointing public response to community-based immunization campaigns. The cost of medical care and outbreak control for the epidemic is conservatively estimated at $30.9 million. Unless the level of immunization in preschool-aged children is increased, this type of epidemic will probably recur.     

A simple vaccination model with multiple endemic states

A simple two-dimensional SIS model with vaccination exhibits a backward bifurcation for some parameter values. A two-population version of the model leads to the consideration of vaccination policies in paired border towns. The results of our mathematical analysis indicate that a vaccination campaign φ meant to reduce a disease's reproduction number R(φ) below one may fail to control the disease. If the aim is to prevent an epidemic outbreak, a large initial number of infective persons can cause a high endemicity level to arise rather suddenly even if the vaccine-reduced reproduction number is below threshold. If the aim is to eradicate an already established disease, bringing the vaccine-reduced reproduction number below one may not be sufficient to do so. The complete bifurcation analysis of the model in terms of the vaccine-reduced reproduction number is given, and some extensions are considered.     

A general model for stochastic SIR epidemics with two levels of mixing

This paper is concerned with a general stochastic model for susceptible→infective→removed epidemics, among a closed finite population, in which during its infectious period a typical infective makes both local and global contacts. Each local contact of a given infective is with an individual chosen independently according to a contact distribution ‘centred’ on that infective, and each global contact is with an individual chosen independently and uniformly from the whole population. The asymptotic situation in which the local contact distribution remains fixed as the population becomes large is considered. The concepts of local infectious clump and local susceptibility set are used to develop a unified approach to the threshold behaviour of this class of epidemic models. In particular, a threshold parameter R_* governing whether or not global epidemics can occur, the probability that a global epidemic occurs and the mean proportion of initial susceptibles ultimately infected by a global epidemic are all determined. The theory is specialised to (i) the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective’s household; (ii) the overlapping groups model, in which the population is partitioned in several ways, with local uniform mixing within the elements of the partitions; and (iii) the great circle model, in which individuals are equally spaced on a circle and local contacts are nearest-neighbour.     

Epidemiological Models with Non-Exponentially Distributed Disease Stages and Applications to Disease Control

SEIR epidemiological models with the inclusion of quarantine and isolation are used to study the control and intervention of infectious diseases. A simple ordinary differential equation (ODE) model that assumes exponential distribution for the latent and infectious stages is shown to be inadequate for assessing disease control strategies. By assuming arbitrarily distributed disease stages, a general integral equation model is developed, of which the simple ODE model is a special case. Analysis of the general model shows that the qualitative disease dynamics are determined by the reproductive number , which is a function of control measures. The integral equation model is shown to reduce to an ODE model when the disease stages are assumed to have a gamma distribution, which is more realistic than the exponential distribution. Outcomes of these models are compared regarding the effectiveness of various intervention policies. Numerical simulations suggest that models that assume exponential and non-exponential stage distribution assumptions can produce inconsistent predictions.     

A Statistical Framework for the Adaptive Management of Epidemiological Interventions

Background

Epidemiological interventions aim to control the spread of infectious disease through various mechanisms, each carrying a different associated cost.

Methodology

We describe a flexible statistical framework for generating optimal epidemiological interventions that are designed to minimize the total expected cost of an emerging epidemic while simultaneously propagating uncertainty regarding the underlying disease model parameters through to the decision process. The strategies produced through this framework are adaptive: vaccination schedules are iteratively adjusted to reflect the anticipated trajectory of the epidemic given the current population state and updated parameter estimates.

Conclusions

Using simulation studies based on a classic influenza outbreak, we demonstrate the advantages of adaptive interventions over non-adaptive ones, in terms of cost and resource efficiency, and robustness to model misspecification.     

Production of highly concentrated, heat-stable hepatitis B surface antigen in maize

Plant‐based oral vaccines are a promising emergent technology that could help alleviate disease burden worldwide by providing a low‐cost, heat‐stable, oral alternative to parenterally administered commercial vaccines. Here, we describe high‐level accumulation of the hepatitis B surface antigen (HBsAg) at a mean concentration of 0.51%TSP in maize T1 seeds using an improved version of the globulin1 promoter. This concentration is more than fourfold higher than any previously reported lines. HBsAg expressed in maize seeds was extremely heat stable, tolerating temperatures up to 55 °C for 1 month without degradation. Optimal heat stability was achieved after oil extraction of ground maize material, either by supercritical fluid extraction or hexane treatment. The contributions of this material towards the development of a practical oral vaccine delivery system are discussed.     

Conditions for the persistence of partial zygotic reproductive isolation in an island-continent model

A theoretical study has been made of the contact between two populations (island-continent model) which are partially isolated insofar as they are monomorphic for different alleles at one or more strictly underdominant loci with multiplicative heterozygote fitnesses. This genetic model of an isolating mechanism is the simplest model of zygotic isolation and is also a very good representation of underdominant chromosomal rearrangements. The particular problem tackled is how the threshold value of the migration rate (m_c), below which differentiation at the underdominant loci can persist, varies as a function of the number of underdominant loci involved (n), for a given level of selection against hybrids (α). It has been found that the stability of genetic differentiation causing a given level α of reproductive isolation decreases as the number of loci involved increases. This effect is particularly noticeable for low levels of isolation. These results have some relevance with regard to the question of whether the establishment of zygotic reproductive isolation between populations occurs more probably as a result of many mutations with small effects on fitness or of only a few mutations with significant effects.     

The impact of local heterogeneity on alternative control strategies for foot-and-mouth disease

The 2001 epidemic of foot-and-mouth disease (FMD) in the UK resulted in the death of nearly 10 million livestock at a cost that was estimated to be up to 8 billion pounds. Owing to the controversy surrounding the epidemic, the question of whether or not alternative policies would have resulted in significantly better control of the epidemic remains of great interest. A hexagonal lattice simulation of FMD in Cumbria is used to address the central question of whether or not better use could have been made of expert knowledge of FMD transmission to target pre-emptive culling, by assuming that the premises at greatest risk of becoming infected can be targeted for culling. The 2000 UK census and the epidemiological database collected during the epidemic are used to describe key characteristics of disease transmission, and the model is fit to the epidemic time-series. Under the assumptions of the model, the parameters that best fit the epidemic in Cumbria indicate that a policy based on expert knowledge would have exacerbated the epidemic compared with the policy as implemented. However, targeting more distant, high-risk farms could be more valuable under different epidemic conditions, notably, if risk factors of sufficient magnitude could be identified to aid in prioritizing vaccination or culling of farms at high risk of becoming infected.     

Optimal harvesting and optimal vaccination

Two optimization problems are considered: Harvesting from a structured population with maximal gain subject to the condition of non-extinction, and vaccinating a population with prescribed reduction of the reproduction number of the disease at minimal costs. It is shown that these problems have a similar structure and can be treated by the same mathematical approach. The optimal solutions have a 'two-window' structure: Optimal harvesting and vaccination strategies or policies are concentrated on one or two preferred age classes. The results are first shown for a linear age structure problem and for an epidemic situation at the uninfected state (minimize costs for a given reduction of the reproduction number) and then extended to populations structured by size, to harvesting at Gurtin-MacCamy equilibria and to vaccination at infected equilibria.     

An integral equation model for the control of a smallpox outbreak

An integral equation model of a smallpox epidemic is proposed. The model structures the incidence of infection among the household, the workplace, the wider community and a health-care facility; and incorporates a finite incubation period and plausible infectivity functions. Linearisation of the model is appropriate for small epidemics, and enables analytic expressions to be derived for the basic reproduction number and the size of the epidemic. The effects of control interventions (vaccination, isolation, quarantine and public education) are explored for a smallpox epidemic following an imported case. It is found that the rapid identification and isolation of cases, the quarantine of affected households and a public education campaign to reduce contact would be capable of bringing an epidemic under control. This could be used in conjunction with the vaccination of healthcare workers and contacts. Our results suggest that prior mass vaccination would be an inefficient method of containing an outbreak.