Modeling Optimal Age-Specific Vaccination Strategies Against Pandemic Influenza

Optimality of Maximal-Effort Vaccination

It is widely acknowledged that vaccinating at maximal effort in the face of an ongoing epidemic is the best strategy to minimise infections and deaths from the disease. Despite this, no one has proved that this is guaranteed to be true if the disease follows multi-group SIR (Susceptible-Infected-Recovered) dynamics. This paper provides a novel proof of this principle for the existing SIR framework, showing that the total number of deaths or infections from an epidemic is decreasing in vaccination effort. Furthermore, it presents a novel model for vaccination which assumes that vaccines assigned to a subgroup are distributed randomly to the unvaccinated population of that subgroup. It suggests, using COVID-19 data, that this more accurately captures vaccination dynamics than the model commonly found in the literature. However, as the novel model provides a strictly larger set of possible vaccination policies, the results presented in this paper hold for both models.

Understanding the impact of disease and vaccine mechanisms on the importance of optimal vaccine allocation

Vaccination is an important epidemic intervention strategy. However, it is generally unclear how the outcomes of different vaccine strategies change depending on population characteristics, vaccine mechanisms and allocation objective. In this paper we develop a conceptual mathematical model to simulate strategies for pre-epidemic vaccination. We extend the SEIR model to incorporate a range of vaccine mechanisms and disease characteristics. We then compare the outcomes of optimal and suboptimal vaccination strategies for three public health objectives (total infections, total symptomatic infections and total deaths) using numerical optimisation. Our comparison shows that the difference in outcomes between vaccinating optimally and suboptimally depends on vaccine mechanisms, disease characteristics, and objective considered. Our modelling finds vaccines that impact transmission produce better outcomes as transmission is reduced for all strategies. For vaccines that impact the likelihood of symptomatic disease or dying due to infection, the improvement in outcome as we decrease these variables is dependent on the strategy implemented. Through a principled model-based process, this work highlights the importance of designing effective vaccine allocation strategies. We conclude that efficient allocation of resources can be just as crucial to the success of a vaccination strategy as the vaccine effectiveness and/or amount of vaccines available.

Asymptotic Analysis of Optimal Vaccination Policies

Targeted vaccination policies can have a significant impact on the number of infections and deaths in an epidemic. However, optimising such policies is complicated, and the resultant solution may be difficult to explain to policy-makers and to the public. The key novelty of this paper is a derivation of the leading-order optimal vaccination policy under multi-group susceptible-infected-recovered dynamics in two different cases. Firstly, it considers the case of a small vulnerable subgroup in a population and shows that (in the asymptotic limit) it is optimal to vaccinate this group first, regardless of the properties of the other groups. Then, it considers the case of a small vaccine supply and transforms the optimal vaccination problem into a simple knapsack problem by linearising the final size equations. Both of these cases are then explored further through numerical examples, which show that these solutions are also directly useful for realistic parameter values. Moreover, the findings of this paper give some general principles for optimal vaccination policies which will help policy-makers and the public to understand the reasoning behind optimal vaccination programs in more generic cases.

Pharmacological, Non-Pharmacological Policies and Mutation: An Artificial Intelligence Based Multi-Dimensional Policy Making Algorithm for Controlling the Casualties of the Pandemic Diseases

Fighting against the pandemic diseases with unique characters requires new sophisticated approaches like the artificial intelligence. This paper develops an artificial intelligence algorithm to produce multi-dimensional policies for controlling and minimizing the pandemic casualties under the limited pharmacological resources. In this respect, a comprehensive parametric model with a priority and age-specific vaccination policy and a variety of non-pharmacological policies are introduced. This parametric model is utilized for constructing an artificial intelligence algorithm by following the exact analogy of the model-based solution. Also, this parametric model is manipulated by the artificial intelligence algorithm to seek for the best multi-dimensional non-pharmacological policies that minimize the future pandemic casualties as desired. The role of the pharmacological and non-pharmacological policies on the uncertain future casualties are extensively addressed on the real data. It is shown that the developed artificial intelligence algorithm is able to produce efficient policies which satisfy the particular optimization targets such as focusing on minimization of the death casualties more than the infected casualties or considering the curfews on the people age over 65 rather than the other non-pharmacological policies. The paper finally analyses a variety of the mutant virus cases and the corresponding non-pharmacological policies aiming to reduce the morbidity and mortality rates.

Epidemiology and Transmission Dynamics of Infectious Diseases and Control Measures

The epidemiology and transmission dynamics of infectious diseases must be understood at the individual and community levels to improve public health decision-making for real-time and integrated community-based control strategies. Herein, we explore the epidemiological characteristics for assessing the impact of public health interventions in the community setting and their applications. Computational statistical methods could advance research on infectious disease epidemiology and accumulate scientific evidence of the potential impacts of pharmaceutical/nonpharmaceutical measures to mitigate or control infectious diseases in the community. Novel public health threats from emerging zoonotic infectious diseases are urgent issues. Given these direct and indirect mitigating impacts at various levels to different infectious diseases and their burdens, we must consider an integrated assessment approach, 'One Health', to understand the dynamics and control of infectious diseases.

COVID-19 vaccine distribution planning using a congested queuing system-A real case from Australia

Crisis-induced vaccine supply chain management has recently drawn attention to the importance of immediate responses to a crisis (e.g., the COVID-19 pandemic). This study develops a queuing model for a crisis-induced vaccine supply chain to ensure efficient coordination and distribution of different COVID-19 vaccine types to people with various levels of vulnerability. We define a utility function for queues to study the changes in arrival rates related to the inventory level of vaccines, the efficiency of vaccines, and a risk aversion coefficient for vaccinees. A multi-period queuing model considering congestion in the vaccination process is proposed to minimise two contradictory objectives: (i) the expected average wait time of vaccinees and (ii) the total investment in the holding and ordering of vaccines. To develop the bi-objective non-linear programming model, the goal attainment algorithm and the non-dominated sorting genetic algorithm (NSGA-II) are employed for small- to large-scale problems. Several solution repairs are also implemented in the classic NSGA-II algorithm to improve its efficiency. Four standard performance metrics are used to investigate the algorithm. The non-parametric Friedman and Wilcoxon signed-rank tests are applied on several numerical examples to ensure the privilege of the improved algorithm. The NSGA-II algorithm surveys an authentic case study in Australia, and several scenarios are created to provide insights for an efficient vaccination program.

A data-driven robust optimization model by cutting hyperplanes on vaccine access uncertainty in COVID-19 vaccine supply chain

The worldwide COVID-19 pandemic sparked such a wave of concern that made access to vaccines more necessary than before. As the vaccine inaccessibility in developing countries has made pandemic eradication more difficult, this study has presented a mathematical model of a sustainable SC for the COVID-19 vaccine that covers the economic, environmental and social aspects and provides vaccine both domestically and internationally. It has also proposed a robust data-driven model based on a polyhedral uncertainty set to address the unjust worldwide vaccine distribution as an uncertain parameter. It is acceptably robust and is also less conservative than its classical counterparts. For validation, the model has been implemented in a real case in Iran, and the results have shown that it is 21% less conservative than its classical rivals (Box and Polyhedral convex uncertainty sets) in facing the uncertain parameter. As a result, the model proposes the construction of two domestic vaccine production centers, including Pasteur Institute and Razi Institute, and five foreign distributors in Tehran, Isfahan, Ahvaz, Kermanshah, and Bandar Abbas strategically.

Home quarantine: A numerical evaluation of SARS-CoV-2 spread in a single-family house

The worldwide pandemic of SARS-CoV-2 which causes coronavirus disease 2019 (COVID-19) has resulted in first-time responses in recent history as most of affected countries are confining residents to their homes. When sick people are detected, home quarantine is usually recommended because of the lack of available hospital rooms for first-stage symptoms that do not require constant medical monitoring. In this situation, one can thus wonder about the transmission of the disease to other household members. In this paper, we numerically investigate the transport of the aerosol generated by an infected person quarantined to his bedroom to the other rooms in a typical French detached house by performing TRNSYS-CONTAM simulations. The intent here is to assess the exposure concentration to the virus of the other household members when simple strategies are employed to reduce the risk of airborne transmission. Due to the uncertainty regarding the evaluation of different parameters such as the emission of contaminated droplets and the infectivity of the occupants, critical cases have been considered in this study. The main results show that, even if remediation actions lower the exposure of other occupants, the risk of contamination remains high even if the contagious person is quarantined to its bedroom.

Multi-period vaccine allocation model in a pandemic: A case study of COVID-19 in Australia

While the swift development and production of a COVID-19 vaccine has been a remarkable success, it is equally crucial to ensure that the vaccine is allocated and distributed in a timely and efficient manner. Prior research on pandemic supply chain has not fully incorporated the underlying factors and constraints in designing a vaccine allocation model. This study proposes an innovative vaccine allocation model to contain the spread of infectious diseases incorporating key contributing factors to the risk of uninoculated people including susceptibility rate and exposure risk. Analyses of the data collected from the state of Victoria in Australia show that a vaccine allocation model can deliver a superior performance in minimizing the risk of unvaccinated people when a multi-period approach is employed and augmenting operational mechanisms including transshipment between medical centers, capacity sharing, and mobile units being integrated into the vaccine allocation model.

Optimal control of a discrete-time plant-herbivore/pest model with bistability in fluctuating environments

Motivated by regulating/eliminating the population of herbivorous pests, we investigate a discrete-time plant-herbivore model with two different constant control strategies (removal versus reduction), and formulate the corresponding optimal control problems when its dynamics exhibits varied types of bi-stability and fluctuating environments. We provide basic analysis and identify the critical factors to characterize the optimal controls and the corresponding plant-herbivore dynamics such as the control upper bound (the effectiveness level of the implementation of control measures) and the initial conditions of the plant and herbivore. Our results show that optimal control could be easier when the model has simple dynamics such as stable equilibrium dynamics under constant environment or the model exhibits chaotic dynamics under fluctuating environments. Due to bistability, initial conditions are important for optimal controls. Regardless of with or without fluctuating environments, initial conditions taken from the near the boundary makes optimal control easier. In general, the pest is hard to be eliminated when the control upper bound is not large enough. However, as the control upper bound is increased or the initial conditions are chosen from near the boundary of the basin of attractions, the pest can be manageable regardless of the fluctuating environments.

Optimal control and cost-effective analysis of an age-structured emerging infectious disease model

Emerging infectious diseases are one of the global public health problems which may lead to widespread epidemics and potentially life-threatening infection. Integrated vaccination and physical distancing interventions are two elementary methods for preventing infectious diseases transmission. In this paper, we construct a continuous age-structured model for investigating the transmission dynamics of an emerging infection disease during a short period. We derive the basic regeneration number R0, the spectral radius of the next generation operator K, which determines the disease outbreak or not. Furthermore, we propose an optimal control problem to take account for the cost-effectiveness of social distancing intervention and vaccination. We rigorously obtain sufficient conditions for a L¹ control problem. Numerical simulations show that coupling integrated vaccination and physical distancing intervention could effectively eliminate the infection, and such control strategy is more sensitive for people aged 10–39 and over 60.

Where to locate COVID-19 mass vaccination facilities?

The outbreak of COVID-19 led to a record-breaking race to develop a vaccine. However, the limited vaccine capacity creates another massive challenge: how to distribute vaccines to mitigate the near-end impact of the pandemic? In the United States in particular, the new Biden administration is launching mass vaccination sites across the country, raising the obvious question of where to locate these clinics to maximize the effectiveness of the vaccination campaign. This paper tackles this question with a novel data-driven approach to optimize COVID-19 vaccine distribution. We first augment a state-of-the-art epidemiological model, called DELPHI, to capture the effects of vaccinations and the variability in mortality rates across age groups. We then integrate this predictive model into a prescriptive model to optimize the location of vaccination sites and subsequent vaccine allocation. The model is formulated as a bilinear, nonconvex optimization model. To solve it, we propose a coordinate descent algorithm that iterates between optimizing vaccine distribution and simulating the dynamics of the pandemic. As compared to benchmarks based on demographic and epidemiological information, the proposed optimization approach increases the effectiveness of the vaccination campaign by an estimated 20%, saving an extra 4000 extra lives in the United States over a 3-month period. The proposed solution achieves critical fairness objectives-by reducing the death toll of the pandemic in several states without hurting others-and is highly robust to uncertainties and forecast errors-by achieving similar benefits under a vast range of perturbations.

A behavioural modelling approach to assess the impact of COVID-19 vaccine hesitancy

We introduce a compartmental epidemic model to describe the spread of COVID-19 within a population, assuming that a vaccine is available, but vaccination is not mandatory. The model takes into account vaccine hesitancy and the refusal of vaccination by individuals, which take their decision on vaccination based on both the present and past information about the spread of the disease. Theoretical analysis and simulations show that voluntary vaccination can certainly reduce the impact of the disease but is unable to eliminate it. We also demonstrate how the information-related parameters affect the dynamics of the disease. In particular, vaccine hesitancy and refusal are better contained in case of widespread information coverage and short-term memory. Finally, the possible impact of seasonality on the spread of the disease is investigated.

Priority and age specific vaccination algorithm for the pandemic diseases: a comprehensive parametric prediction model

Background

There have been several destructive pandemic diseases in the human history. Since these pandemic diseases spread through human-to-human infection, a number of non-pharmacological policies has been enforced until an effective vaccine has been developed. In addition, even though a vaccine has been developed, due to the challenges in the production and distribution of the vaccine, the authorities have to optimize the vaccination policies based on the priorities. Considering all these facts, a comprehensive but simple parametric model enriched with the pharmacological and non-pharmacological policies has been proposed in this study to analyse and predict the future pandemic casualties.

Method

This paper develops a priority and age specific vaccination policy and modifies the non-pharmacological policies including the curfews, lockdowns, and restrictions. These policies are incorporated with the susceptible, suspicious, infected, hospitalized, intensive care, intubated, recovered, and death sub-models. The resulting model is parameterizable by the available data where a recursive least squares algorithm with the inequality constraints optimizes the unknown parameters. The inequality constraints ensure that the structural requirements are satisfied and the parameter weights are distributed proportionally.

Results

The results exhibit a distinctive third peak in the casualties occurring in 40 days and confirm that the intensive care, intubated, and death casualties converge to zero faster than the susceptible, suspicious, and infected casualties with the priority and age specific vaccination policy. The model also estimates that removing the curfews on the weekends and holidays cause more casualties than lifting the restrictions on the people with the chronic diseases and age over 65.

Conclusion

Sophisticated parametric models equipped with the pharmacological and non-pharmacological policies can predict the future pandemic casualties for various cases.

Age Structured Mathematical Modeling Studies on COVID-19 with respect to Combined Vaccination and Medical Treatment Strategies

In this study, we develop a mathematical model incorporating age-specific transmission dynamics of COVID-19 to evaluate the role of vaccination and treatment strategies in reducing the size of COVID-19 burden. Initially, we establish the positivity and boundedness of the solutions of the non controlled model and calculate the basic reproduction number and do the stability analysis. We then formulate an optimal control problem with vaccination and treatment as control variables and study the same. Pontryagin’s Minimum Principle is used to obtain the optimal vaccination and treatment rates. Optimal vaccination and treatment policies are analysed for different values of the weight constant associated with the cost of vaccination and different efficacy levels of vaccine. Findings from these suggested that the combined strategies (vaccination and treatment) worked best in minimizing the infection and disease induced mortality. In order to reduce COVID-19 infection and COVID-19 induced deaths to maximum, it was observed that optimal control strategy should be prioritized to the population with age greater than 40 years. Varying the cost of vaccination it was found that sufficient implementation of vaccines (more than 77 %) reduces the size of COVID-19 infections and number of deaths. The infection curves varying the efficacies of the vaccines against infection were also analysed and it was found that higher efficacy of the vaccine resulted in lesser number of infections and COVID induced deaths. The findings would help policymakers to plan effective strategies to contain the size of the COVID-19 pandemic.

Modelling lockdown measures in epidemic outbreaks using selective socio-economic containment with uncertainty

After the introduction of drastic containment measures aimed at stopping the epidemic contagion from SARS-CoV2, many governments have adopted a strategy based on a periodic relaxation of such measures in the face of a severe economic crisis caused by lockdowns. Assessing the impact of such openings in relation to the risk of a resumption of the spread of the disease is an extremely difficult problem due to the many unknowns concerning the actual number of people infected, the actual reproduction number and infection fatality rate of the disease. In this work, starting from a SEIRD compartmental model with a social structure based on the age of individuals and stochastic inputs that account for data uncertainty, the effects of containment measures are introduced via an optimal control problem dependent on specific social activities, such as home, work, school, etc. Through a short time horizon approximation, we derive models with multiple feedback controls depending on social activities that allow us to assess the impact of selective relaxation of containment measures in the presence of uncertain data. After analyzing the effects of the various controls, results from different scenarios concerning the first wave of the epidemic in some major countries, including Germany, France, Italy, Spain, the United Kingdom and the United States, are presented and discussed. Specific contact patterns in the home, work, school and other locations have been considered for each country. Numerical simulations show that a careful strategy of progressive relaxation of containment measures, such as that adopted by some governments, may be able to keep the epidemic under control by restarting various productive activities.

Control with uncertain data of socially structured compartmental epidemic models

The adoption of containment measures to reduce the amplitude of the epidemic peak is a key aspect in tackling the rapid spread of an epidemic. Classical compartmental models must be modified and studied to correctly describe the effects of forced external actions to reduce the impact of the disease. The importance of social structure, such as the age dependence that proved essential in the recent COVID-19 pandemic, must be considered, and in addition, the available data are often incomplete and heterogeneous, so a high degree of uncertainty must be incorporated into the model from the beginning. In this work we address these aspects, through an optimal control formulation of a socially structured epidemic model in presence of uncertain data. After the introduction of the optimal control problem, we formulate an instantaneous approximation of the control that allows us to derive new feedback controlled compartmental models capable of describing the epidemic peak reduction. The need for long-term interventions shows that alternative actions based on the social structure of the system can be as effective as the more expensive global strategy. The timing and intensity of interventions, however, is particularly relevant in the case of uncertain parameters on the actual number of infected people. Simulations related to data from the first wave of the recent COVID-19 outbreak in Italy are presented and discussed.

Optimal Management of Public Perceptions During A Flu Outbreak: A Game-Theoretic Perspective

Public perceptions and sentiments play a crucial role in the success of vaccine uptake in the community. While vaccines have proven to be the best preventive method to combat the flu, the attitude and knowledge about vaccines are a major hindrance to higher uptake in most of the countries. The yearly coverage, especially in the vulnerable groups in the population, often remains below the herd immunity level despite the Flu Awareness Campaign organized by WHO every year worldwide. This brings immense challenges to the nation's public health protection agency for strategic decision-making in controlling the flu outbreak every year. To understand the impact of public perceptions and vaccination decisions while designing optimal immunization policy, we model the individual decision-making as a two-strategy pairwise contest game, where pay-off is considered as a function of public health effort for the campaign. We use Pontryagin's maximum principle to identify the best possible strategy for public health to implement vaccination and reduce infection at a minimum cost. Our optimal analysis shows that the cost of public health initiatives is qualitatively and quantitatively different under different public perceptions and attitudes towards vaccinations. When individual risk perception evolves with vaccine uptake or disease induced death, our model demonstrates a feed-forward mechanism in the dynamics of vaccination and exhibits an increase in vaccine uptake. Using numerical simulation, we also observe that the optimal cost can be minimized by putting the effort in the beginning and later part of the outbreak rather than during the peak. It confers that public health efforts towards disseminating disease severity or actual vaccination risk might accelerate the vaccination coverage and mitigate the infection faster.

Optimal vaccine allocation during the mumps outbreak in two SIR centres

The aim of this work is to investigate the optimal vaccine sharing between two susceptible, infected, removed (SIR) centres in the presence of migration fluxes of susceptibles and infected individuals during the mumps outbreak. Optimality of the vaccine allocation means the minimization of the total number of lost working days during the whole period of epidemic outbreak $[0,t_f]$, which can be described by the functional $Q=\int _0^{t_f}I(t)\,{\textrm{d}}t$, where $I(t)$ stands for the number of infectives at time $t$. We explain the behaviour of the optimal allocation, which depends on the model parameters and the amount of vaccine available $V$.

Association of infected probability of COVID-19 with ventilation rates in confined spaces: a Wells-Riley equation based investigation.. [DOI: 10.1101/2020.04.21.20072397]

Background

A growing number of epidemiological cases are proving the possibility of airborne transmission of coronavirus disease 2019 (COVID-19). Ensuring adequate ventilation rate is essential to reduce the risk of infection in confined spaces.

Methods

We obtained the quantum generation rate by a COVID-19 infector with a reproductive number based fitting approach, and then estimated the association between infected probability and ventilation rate with the Wells-Riley equation.

Results

The estimated quantum generation rate of COVID-19 is 14-48 /h. To ensure infected probabolity less than 1%, ventilation rate lareger than common values (100-350 m3/h and 1200-4000 m3/h for 15 minutes and 3 hours exposure, respectively) is required. If both the infector and susceptibles wear masks, the ventilation rate ensuring less than 1% infected probability is reduced to 50-180 m3/h and 600-2000 m3/h correspondingly, which is easier to be achieved by normal ventilation mode applied in some typical scenarios, including offices, classrooms, buses and aircraft cabins.

Interpretation

The risk of potential airborne transmission in confined spaces cannot be ignored. Strict preventive measures that have been widely adopted should be effective in reducing the risk of airborne transmitted infection.

A model of influenza infection and vaccination in children aged under 5 years in Beijing, China

BACKGROUND

Children aged under 5 years are particularly vulnerable to influenza infection. In this study, we aim to estimate the number and incidence of influenza among young children and estimate the impact of childhood vaccination in different scenarios from 2013/14 to 2016/17 seasons.

METHODS

The number and incidence rate of influenza infections among children aged under 5 years in Beijing was estimated by scaling up observed surveillance data. Then, we used a susceptible-exposed-infected-recovery (SEIR) model to reproduce the weekly number of influenza infections estimated in Beijing during the study seasons, and to estimate the number and proportion of influenza-attributed medically attended acute respiratory infections (I-MAARI) averted by vaccination in each season. Finally, we evaluated the impact of alternative childhood vaccination programs with different coverage and speed of vaccine distribution.

RESULTS

The estimated average annual incidence of influenza in children aged under 5 years was 33.9% (95% confidence interval (CI): 27.5%, 47.2%) during the study period. With the actual coverage during the included seasons at around 2.9%, an average of 3.9% (95%CI: 3.5%, 4.4%) I-MAARI was reduced compared to a no-vaccination scenario. Reaching 20%, 40%, 50%, 60%, 80% and 100% vaccine coverage would lead to an overall I-MAARI reduction of 25.3%, 42.7%, 51.9%, 57.0%, 65.3% and 71.2%. At 20% coverage scenario, an average of 28.8% I-MAARI will be prevented if intensive vaccination implemented in 2 months since the vaccine released.

CONCLUSION

In Beijing, the introduction of a program for vaccinating young children, even at relatively low vaccine coverage rates, would considerably reduce I-MAARI, particularly if the vaccines can be quickly delivered.

Association of the infection probability of COVID-19 with ventilation rates in confined spaces

A growing number of cases have proved the possibility of airborne transmission of the coronavirus disease 2019 (COVID-19). Ensuring an adequate ventilation rate is essential to reduce the risk of infection in confined spaces. In this study, we estimated the association between the infection probability and ventilation rates with the Wells-Riley equation, where the quantum generation rate (q) by a COVID-19 infector was obtained using a reproductive number-based fitting approach. The estimated q value of COVID-19 is 14-48 h^-1. To ensure an infection probability of less than 1%, a ventilation rate larger than common values (100-350 m³/h per infector and 1200-4000 m³/h per infector for 0.25 h and 3 h of exposure, respectively) is required. If the infector and susceptible person wear masks, then the ventilation rate ensuring a less than 1% infection probability can be reduced to a quarter respectively, which is easier to achieve by the normal ventilation mode applied in typical scenarios, including offices, classrooms, buses, and aircraft cabins. Strict preventive measures (e.g., wearing masks and preventing asymptomatic infectors from entering public spaces using tests) that have been widely adopted should be effective in reducing the risk of infection in confined spaces.

Prioritization of vaccine strategy using an age-dependent mathematical model for 2009 A/H1N1 influenza in the Republic of Korea

We developed a mathematical model of the 2009 A/H1N1 influenza epidemic in the Republic of Korea by considering five age groups and suggested the best way to prioritize an age-dependent vaccination strategy for mitigating the epidemic. An age-structured SEIAR influenza model was constructed based on the laboratory confirmed data obtained from the Korea Centers for Disease Control and Prevention (KCDC). The estimated transmission matrix captured one of the main characteristics of the 2009 A/H1N1 influenza, the transmission rate of which is high among young people, unlike that of seasonal influenza. We investigated the impact of age-dependent vaccination priority on the transmission dynamics of the 2009 A/H1N1 influenza and evaluated the Korean government vaccination policy when the vaccination started being administered 90 days (or 120 days) after the onset of the outbreak. We found that the government's age priority vaccination policy (Group 2, Group 1, Group 5, Group 4, and Group 3 in order) was a good strategy for reducing 62.06% of the cumulative cases when the vaccination was applied 90 days after the onset of the outbreak, while the proposed model's best suggestion (Group 2, Group 1, Group 3, Group 4, and Group 5 in order) showed 64.52% reduction. Furthermore, we studied the region-specific vaccination policy. For instance, the best age-priority of vaccination in Gwangwon province showed a different order (Group 3, Group 1, Group 2, Group 4, and Group 5 in order) and it reduced the incidence by 58.1%, which is 5.54% higher than that of the 2009 Korean government policy.

Optimal time-profiles of public health intervention to shape voluntary vaccination for childhood diseases

In order to seek the optimal time-profiles of public health systems (PHS) Intervention to favor vaccine propensity, we apply optimal control (OC) to a SIR model with voluntary vaccination and PHS intervention. We focus on short-term horizons, and on both continuous control strategies resulting from the forward-backward sweep deterministic algorithm, and piecewise-constant strategies (which are closer to the PHS way of working) investigated by the simulated annealing (SA) stochastic algorithm. For childhood diseases, where disease costs are much larger than vaccination costs, the OC solution sets at its maximum for most of the policy horizon, meaning that the PHS cannot further improve perceptions about the net benefit of immunization. Thus, the subsequent dynamics of vaccine uptake stems entirely from the declining perceived risk of infection (due to declining prevalence) which is communicated by direct contacts among parents, and unavoidably yields a future decline in vaccine uptake. We find that for relatively low communication costs, the piecewise control is close to the continuous control. For large communication costs the SA algorithm converges towards a non-monotone OC that can have oscillations.

Assessment of optimal strategies in a two-patch dengue transmission model with seasonality

Emerging and re-emerging dengue fever has posed serious problems to public health officials in many tropical and subtropical countries. Continuous traveling in seasonally varying areas makes it more difficult to control the spread of dengue fever. In this work, we consider a two-patch dengue model that can capture the movement of host individuals between and within patches using a residence-time matrix. A previous two-patch dengue model without seasonality is extended by adding host demographics and seasonal forcing in the transmission rates. We investigate the effects of human movement and seasonality on the two-patch dengue transmission dynamics. Motivated by the recent Peruvian dengue data in jungle/rural areas and coast/urban areas, our model mimics the seasonal patterns of dengue outbreaks in two patches. The roles of seasonality and residence-time configurations are highlighted in terms of the seasonal reproduction number and cumulative incidence. Moreover, optimal control theory is employed to identify and evaluate patch-specific control measures aimed at reducing dengue prevalence in the presence of seasonality. Our findings demonstrate that optimal patch-specific control strategies are sensitive to seasonality and residence-time scenarios. Targeting only the jungle (or endemic) is as effective as controlling both patches under weak coupling or symmetric mobility. However, focusing on intervention for the city (or high density areas) turns out to be optimal when two patches are strongly coupled with asymmetric mobility.

Controlling infectious disease outbreaks: A deterministic allocation-scheduling model with multiple discrete resources

Infectious disease outbreaks occurred many times in the past and are more likely to happen in the future. In this paper the problem of allocating and scheduling limited multiple, identical or non-identical, resources employed in parallel, when there are several infected areas, is considered. A heuristic algorithm, based on Shih's (1974) and Pappis and Rachaniotis' (2010) algorithms, is proposed as the solution methodology. A numerical example implementing the proposed methodology in the context of a specific disease outbreak, namely influenza, is presented. The proposed methodology could be of significant value to those drafting contingency plans and healthcare policy agendas.

Mathematical model of transmission dynamics and optimal control strategies for 2009 A/H1N1 influenza in the Republic of Korea

A mathematical model for the transmission dynamics of the 2009 A/H1N1 influenza epidemic in the Republic of Korea is developed. The simulation period is separated into three consecutive periods based on the government's intervention strategies: the nonpharmaceutical strategy is used during Period 1. The nonpharmaceutical and antiviral strategies are executed during Period 2 and the vaccine strategy is added during Period 3. During Period 1, we estimate the reduction in the transmission rate due to the government's intervention policies as a difference between the data-fitted and uncontrolled transmission rate that is derived from the basic reproductive number, R₀, of the model without intervention. This quantified reduced transmission rate is used as an upperbound of the nonpharmaceutical control for studying optimal control strategies, which is a new approach for determining the realistic upperbound of control. In this study, we also explore the real-time prediction of incidence using the mathematical model during the early stage of the epidemic. We investigate the impact of vaccination coverage and timing with respect to the cumulative incidence. The result implies that early vaccination plays a significant role for preventing the epidemic.

Constrained optimal control applied to vaccination for influenza

The efficient time schedule and prioritization of vaccine supplies are important in mitigating impact of an influenza pandemic. In practice, there are restrictions associated with limited vaccination coverage and the maximum daily vaccine administration. We extend previous work on optimal control for influenza to reflect these realistic restrictions using mixed constraints on state and control variables. An optimal control problem is formulated with the aim of minimizing the number of infected individuals while considering intervention costs. Time-dependent vaccination is computed and analysed using a model incorporating heterogeneity in population structure under different settings of transmissibility levels, vaccine coverages, and time delays.

Efficient Vaccine Distribution Based on a Hybrid Compartmental Model

To effectively and efficiently reduce the morbidity and mortality that may be caused by outbreaks of emerging infectious diseases, it is very important for public health agencies to make informed decisions for controlling the spread of the disease. Such decisions must incorporate various kinds of intervention strategies, such as vaccinations, school closures and border restrictions. Recently, researchers have paid increased attention to searching for effective vaccine distribution strategies for reducing the effects of pandemic outbreaks when resources are limited. Most of the existing research work has been focused on how to design an effective age-structured epidemic model and to select a suitable vaccine distribution strategy to prevent the propagation of an infectious virus. Models that evaluate age structure effects are common, but models that additionally evaluate geographical effects are less common. In this paper, we propose a new SEIR (susceptible-exposed-infectious šC recovered) model, named the hybrid SEIR-V model (HSEIR-V), which considers not only the dynamics of infection prevalence in several age-specific host populations, but also seeks to characterize the dynamics by which a virus spreads in various geographic districts. Several vaccination strategies such as different kinds of vaccine coverage, different vaccine releasing times and different vaccine deployment methods are incorporated into the HSEIR-V compartmental model. We also design four hybrid vaccination distribution strategies (based on population size, contact pattern matrix, infection rate and infectious risk) for controlling the spread of viral infections. Based on data from the 2009-2010 H1N1 influenza epidemic, we evaluate the effectiveness of our proposed HSEIR-V model and study the effects of different types of human behaviour in responding to epidemics.

Modeling optimal treatment strategies in a heterogeneous mixing model

Background

Many mathematical models assume random or homogeneous mixing for various infectious diseases. Homogeneous mixing can be generalized to mathematical models with multi-patches or age structure by incorporating contact matrices to capture the dynamics of the heterogeneously mixing populations. Contact or mixing patterns are difficult to measure in many infectious diseases including influenza. Mixing patterns are considered to be one of the critical factors for infectious disease modeling.

Methods

A two-group influenza model is considered to evaluate the impact of heterogeneous mixing on the influenza transmission dynamics. Heterogeneous mixing between two groups with two different activity levels includes proportionate mixing, preferred mixing and like-with-like mixing. Furthermore, the optimal control problem is formulated in this two-group influenza model to identify the group-specific optimal treatment strategies at a minimal cost. We investigate group-specific optimal treatment strategies under various mixing scenarios.

Results

The characteristics of the two-group influenza dynamics have been investigated in terms of the basic reproduction number and the final epidemic size under various mixing scenarios. As the mixing patterns become proportionate mixing, the basic reproduction number becomes smaller; however, the final epidemic size becomes larger. This is due to the fact that the number of infected people increases only slightly in the higher activity level group, while the number of infected people increases more significantly in the lower activity level group. Our results indicate that more intensive treatment of both groups at the early stage is the most effective treatment regardless of the mixing scenario. However, proportionate mixing requires more treated cases for all combinations of different group activity levels and group population sizes.

Conclusions

Mixing patterns can play a critical role in the effectiveness of optimal treatments. As the mixing becomes more like-with-like mixing, treating the higher activity group in the population is almost as effective as treating the entire populations since it reduces the number of disease cases effectively but only requires similar treatments. The gain becomes more pronounced as the basic reproduction number increases. This can be a critical issue which must be considered for future pandemic influenza interventions, especially when there are limited resources available.

The role of residence times in two-patch dengue transmission dynamics and optimal strategies

The reemergence and geographical dispersal of vector-borne diseases challenge global health experts around the world and in particular, dengue poses increasing difficulties in the Americas, due in part to explosive urban and semi-urban growth, increases of within and between region mobility, the absence of a vaccine, and the limited resources available for public health services. In this work, a simple deterministic two-patch model is introduced to assess the impact of dengue transmission dynamics in heterogeneous environments. The two-patch system models the movement (e.g. urban versus rural areas residence times) of individuals between and within patches/environments using residence-time matrices with entries that budget within and between host patch relative residence times, under the assumption that only the human budgets their residence time across regions. Three scenarios are considered: (i) resident hosts in Patch i visit patch j, where i≠j but not the other way around, a scenario referred to as unidirectional motion; (ii) symmetric bi-directional motion; and (iii) asymmetric bi-directional motion. Optimal control theory is used to identify and evaluate patch-specific control measures aimed at reducing dengue prevalence in humans and vectors at a minimal cost. Optimal policies are computed under different residence-matrix configurations mentioned above as well as transmissibility scenarios characterized by the magnitude of the basic reproduction number. Optimal patch-specific polices can ameliorate the impact of epidemic outbreaks substantially when the basic reproduction number is moderate. The final patch-specific epidemic size variation increases as the residence time matrix moves away from the symmetric case (asymmetry). As expected, the patch where individuals spend most of their time or in the patch where transmissibility is higher tend to support larger patch-specific final epidemic sizes. Hence, focusing on intervention that target areas where individuals spend "most" time or where transmissibility is higher turn out to be optimal. Therefore, reducing traffic is likely to take a host-vector system into the world of manageable outbreaks.

Assessment of intensive vaccination and antiviral treatment in 2009 influenza pandemic in Korea

OBJECTIVES

We characterized and assessed public health measures, including intensive vaccination and antiviral treatment, implemented during the 2009 influenza pandemic in the Republic of Korea.

METHODS

A mathematical model for the 2009 influenza pandemic is formulated. The transmission rate, the vaccination rate, the antiviral treatment rate, and the hospitalized rate are estimated using the least-squares method for the 2009 data of the incidence curves of the infected, vaccinated, treated, and hospitalized.

RESULTS

The cumulative number of infected cases has reduced significantly following the implementation of the intensive vaccination and antiviral treatment. In particular, the intensive vaccination was the most critical factor that prevented severe outbreak.

CONCLUSION

We have found that the total infected proportion would increase by approximately six times under the half of vaccination rates.

Assessment of the Intensive Countermeasures in the 2009 Pandemic Influenza in Korea

Objectives

Methods

Results

Conclusion

Optimal strategies of social distancing and vaccination against seasonal influenza

Optimal control strategies for controlling seasonal influenza transmission in the US are of high interest, because of the significant epidemiological and economic burden of influenza. To evaluate optimal strategies of vaccination and social distancing, we used an age-structured dynamic model of seasonal influenza. We applied optimal control theory to identify the best way of reducing morbidity and mortality at a minimal cost. In combination with the Pontryagins maximum principle, we calculated time-dependent optimal policies of vaccination and social distancing to minimize the epidemiological and economic burden associated with seasonal influenza. We computed optimal age-specific intervention strategies and analyze them under various costs of interventions and disease transmissibility. Our results show that combined strategies have a stronger impact on the reduction of the final epidemic size. Our results also suggest that the optimal vaccination can be achieved by allocating most vaccines to preschool-age children (age under five) followed by young adults (age 20-39) and school age children (age 6-19). We find that the optimal vaccination rates for all age groups are highest at the beginning of the outbreak, requiring intense effort at the early phase of an epidemic. On the other hand, optimal social distancing of clinical cases tends to last the entire duration of an outbreak, and its intensity is relatively equal for all age groups. Furthermore, with higher transmissibility of the influenza virus (i.e. higher R0), the optimal control strategy needs to include more efforts to increase vaccination rates rather than efforts to encourage social distancing. Taken together, public health agencies need to consider both the transmissibility of the virus and ways to encourage early vaccination as well as voluntary social distancing of symptomatic cases in order to determine optimal intervention strategies against seasonal influenza.

Optimal vaccine allocation for the early mitigation of pandemic influenza

With new cases of avian influenza H5N1 (H5N1AV) arising frequently, the threat of a new influenza pandemic remains a challenge for public health. Several vaccines have been developed specifically targeting H5N1AV, but their production is limited and only a few million doses are readily available. Because there is an important time lag between the emergence of new pandemic strain and the development and distribution of a vaccine, shortage of vaccine is very likely at the beginning of a pandemic. We coupled a mathematical model with a genetic algorithm to optimally and dynamically distribute vaccine in a network of cities, connected by the airline transportation network. By minimizing the illness attack rate (i.e., the percentage of people in the population who become infected and ill), we focus on optimizing vaccine allocation in a network of 16 cities in Southeast Asia when only a few million doses are available. In our base case, we assume the vaccine is well-matched and vaccination occurs 5 to 10 days after the beginning of the epidemic. The effectiveness of all the vaccination strategies drops off as the timing is delayed or the vaccine is less well-matched. Under the best assumptions, optimal vaccination strategies substantially reduced the illness attack rate, with a maximal reduction in the attack rate of 85%. Furthermore, our results suggest that cooperative strategies where the resources are optimally distributed among the cities perform much better than the strategies where the vaccine is equally distributed among the network, yielding an illness attack rate 17% lower. We show that it is possible to significantly mitigate a more global epidemic with limited quantities of vaccine, provided that the vaccination campaign is extremely fast and it occurs within the first weeks of transmission.