Vaccination models and optimal control strategies to dengue

Vaccination for communicable endemic diseases: optimal allocation of initial and booster vaccine doses

For some communicable endemic diseases (e.g., influenza, COVID-19), vaccination is an effective means of preventing the spread of infection and reducing mortality, but must be augmented over time with vaccine booster doses. We consider the problem of optimally allocating a limited supply of vaccines over time between different subgroups of a population and between initial versus booster vaccine doses, allowing for multiple booster doses. We first consider an SIS model with interacting population groups and four different objectives: those of minimizing cumulative infections, deaths, life years lost, or quality-adjusted life years lost due to death. We solve the problem sequentially: for each time period, we approximate the system dynamics using Taylor series expansions, and reduce the problem to a piecewise linear convex optimization problem for which we derive intuitive closed-form solutions. We then extend the analysis to the case of an SEIS model. In both cases vaccines are allocated to groups based on their priority order until the vaccine supply is exhausted. Numerical simulations show that our analytical solutions achieve results that are close to optimal with objective function values significantly better than would be obtained using simple allocation rules such as allocation proportional to population group size. In addition to being accurate and interpretable, the solutions are easy to implement in practice. Interpretable models are particularly important in public health decision making.

Modeling and analysis of the effect of optimal virus control on the spread of HFMD

A within-host and between-host hand, foot and mouth disease (HFMD) mathematical model is established and the affect of optimal control in its within-host part on HFMD transmission is studied. Through define two basic reproduction numbers, by using the fast-slow system analysis method of time scale, the global stabilities of the between-host (slow) system and within-host (fast) system are researched, respectively. An optimal control problem with drug-treatment control on coupled within-host and between-host HFMD model is formulated and analysed theoretically. Finally, the purposed optimal control measures are applied to the actual HFMD epidemic analysis in Zhejiang Province, China from April 1, 2021 to June 30, 2021. The numerical results show that the drug control strategies can reduce the virus load per capita and can effectively prevent large-scale outbreaks of HFMD.

Modelling the transmission dynamics and optimal control strategies for HIV infection in China

In order to end the AIDS epidemic by 2030 that was put forward by the Joint United Nations Programme on HIV/AIDS in 2014, China needs to take more effective measures to achieve the three 90% goals (90-90-90). We establish a compartmental model to study the dynamics of HIV transmission with control strategies. The analytical results show the existence and stability of the disease-free equilibrium and endemic equilibrium. An optimal control model is constructed to evaluate the impacts of control measures. The simulation results show that the optimal control strategy proposed in this work can eradicate AIDS by 2030. The cost-effectiveness analysis indicates that the cost of the control strategy that combines screening for latent individuals and enhancing education for unaware infected individuals is the lowest. Our findings can provide guidance for public health authorities on effective mitigation strategies to achieve the goals proposed by the United Nations Program on HIV/AIDS.

Leveraging an epidemic-economic mathematical model to assess human responses to COVID-19 policies and disease progression

It is imperative that resources are channelled towards programs that are efficient and cost effective in combating the spread of COVID-19, the disease caused by the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2). This study proposed and analyzed control strategies for that purpose. We developed a mathematical disease model within an optimal control framework that allows us to investigate the best approach for curbing COVID-19 epidemic. We address the following research question: what is the role of community compliance as a measure for COVID-19 control? Analyzing the impact of community compliance of recommended guidelines by health authorities-examples, social distancing, face mask use, and sanitizing-coupled with efforts by health authorities in areas of vaccine provision and effective quarantine-showed that the best intervention in addition to implementing vaccination programs and effective quarantine measures, is the active incorporation of individuals' collective behaviours, and that resources should also be directed towards community campaigns on the importance of face mask use, social distancing, and frequent sanitizing, and any other collective activities. We also demonstrated that collective behavioral response of individuals influences the disease dynamics; implying that recommended health policy should be contextualized.

Multiobjective optimization to assess dengue control costs using a climate-dependent epidemiological model

Arboviruses, diseases transmitted by arthropods, have become a significant challenge for public health managers. The World Health Organization highlights dengue as responsible for millions of infections worldwide annually. As there is no specific treatment for the disease and no free-of-charge vaccine for mass use in Brazil, the best option is the measures to combat the vector, the Aedes aegypti mosquito. Therefore, we proposed an epidemiological model dependent on temperature, precipitation, and humidity, considering symptomatic and asymptomatic dengue infections. Through computer simulations, we aimed to minimize the amount of insecticides and the social cost demanded to treat patients. We proposed a case study in which our model is fitted with real data from symptomatic dengue-infected humans in an epidemic year in a Brazilian city. Our multiobjective optimization model considers an additional control using larvicide, adulticide, and ultra-low volume spraying. The work's main contribution is studying the monetary cost of the actions to combat the vector demand versus the hospital cost per confirmed infected, comparing approaches with and without additional control. Results showed that the additional vector control measures are cheaper than the hospital treatment without the vector control would be.

Assessing the impact of serostatus-dependent immunization on mitigating the spread of dengue virus

Dengue is the most rapidly spreading mosquito-borne disease that poses great threats to public health. We propose a compartmental model with primary and secondary infection and targeted vaccination to assess the impact of serostatus-dependent immunization on mitigating the spread of dengue virus. We derive the basic reproduction number and investigate the stability and bifurcations of the disease-free equilibrium and endemic equilibria. The existence of a backward bifurcation is proved and is used to explain the threshold dynamics of the transmission. We also carry out numerical simulations and present bifurcation diagrams to reveal rich dynamics of the model such as bi-stability of the equilibria, limit cycles, and chaos. We prove the uniform persistence and global stability of the model. Sensitivity analysis suggests that mosquito control and protection from mosquito bites are still the key measures of controlling the spread of dengue virus, though serostatus-dependent immunization is implemented. Our findings provide insightful information for public health in mitigating dengue epidemics through vaccination.

Optimal vaccine allocation for the control of sexually transmitted infections

The burden of sexually transmitted infections (STIs) poses a challenge due to its large negative impact on sexual and reproductive health worldwide. Besides simple prevention measures and available treatment efforts, prophylactic vaccination is a powerful tool for controlling some viral STIs and their associated diseases. Here, we investigate how prophylactic vaccines are best distributed to prevent and control STIs. We consider sex-specific differences in susceptibility to infection, as well as disease severity outcomes. Different vaccination strategies are compared assuming distinct budget constraints that mimic a scarce vaccine stockpile. Vaccination strategies are obtained as solutions to an optimal control problem subject to a two-sex Kermack-McKendrick-type model, where the control variables are the daily vaccination rates for females and males. One important aspect of our approach relies on conceptualizing a limited but specific vaccine stockpile via an isoperimetric constraint. We solve the optimal control problem via Pontryagin's Maximum Principle and obtain a numerical approximation for the solution using a modified version of the forward-backward sweep method that handles the isoperimetric budget constraint in our formulation. The results suggest that for a limited vaccine supply ([Formula: see text]-[Formula: see text] vaccination coverage), one-sex vaccination, prioritizing females, appears to be more beneficial than the inclusion of both sexes into the vaccination program. Whereas, if the vaccine supply is relatively large (enough to reach at least [Formula: see text] coverage), vaccinating both sexes, with a slightly higher rate for females, is optimal and provides an effective and faster approach to reducing the prevalence of the infection.

Qualitative Analysis of the Transmission Dynamics of Dengue with the Effect of Memory, Reinfection, and Vaccination

Dengue fever has a huge impact on people’s physical, social, and economic lives in low-income locations worldwide. Researchers use epidemic models to better understand the transmission patterns of dengue fever in order to recommend effective preventative measures and give data for vaccine and treatment development. We use fractional calculus to organise the transmission phenomena of dengue fever, including immunisation, reinfection, therapy, and asymptotic carriers. In addition, we focused our study on the dynamical behavior and qualitative approach of dengue infection. The existence and uniqueness of the solution of the suggested dengue dynamics are inspected through the fixed point theorems of Schaefer and Banach. The Ulam-Hyers stability of the suggested dengue model is established. To illustrate the contribution of the input factors on the system of dengue infection, the solution paths are studied using the Laplace Adomian decomposition approach. Furthermore, numerical simulations are used to show the effects of fractional-order, immunity loss, vaccination, asymptotic fraction, biting rate, and therapy. We have established that asymptomatic carriers, bite rates, and immunity loss rates are all important factors that might make controlling more challenging. The intensity of dengue fever may be controlled by reducing mosquito bite rates, whereas the asymptotic fraction is risky and can transmit the illness to noninfected regions. Vaccination, fractional order, index of memory, and medication can be employed as proper control parameters.

Sequential allocation of vaccine to control an infectious disease

The problem of optimally allocating a limited supply of vaccine to control a communicable disease has broad applications in public health and has received renewed attention during the COVID-19 pandemic. This allocation problem is highly complex and nonlinear. Decision makers need a practical, accurate, and interpretable method to guide vaccine allocation. In this paper we develop simple analytical conditions that can guide the allocation of vaccines over time. We consider four objectives: minimize new infections, minimize deaths, minimize life years lost, or minimize quality-adjusted life years lost due to death. We consider an SIR model with interacting population groups. We approximate the model using Taylor series expansions, and develop simple analytical conditions characterizing the optimal solution to the resulting problem for a single time period. We develop a solution approach in which we allocate vaccines using the analytical conditions in each time period based on the state of the epidemic at the start of the time period. We illustrate our method with an example of COVID-19 vaccination, calibrated to epidemic data from New York State. Using numerical simulations, we show that our method achieves near-optimal results over a wide range of vaccination scenarios. Our method provides a practical, intuitive, and accurate tool for decision makers as they allocate limited vaccines over time, and highlights the need for more interpretable models over complicated black box models to aid in decision making.

Mathematical analysis of a model of Ebola disease with control measures

The re-emergence of the Ebola virus disease has pushed researchers to investigate more on this highly deadly disease in order to better understand and control the outbreak and recurrence of epidemics. It is in this perspective that we formulate a realistic mathematical model for the dynamic transmission of Ebola virus disease, incorporating relevant control measures and factors such as ban on eating bush-meat, social distancing, observance of hygiene rules and containment, waning of the vaccine-induced, imperfect contact tracing and vaccine efficacy, quarantine, hospitalization and screening to fight against the spread of the disease. First, by considering the constant control parameters case, we thoroughly compute the control reproduction number [Formula: see text] from which the dynamics of the model is analyzed. The existence and stability of steady states are established under appropriate assumptions on [Formula: see text]. Also, the effect of all the control measures is investigated and the global sensitivity analysis of the control reproduction number is performed in order to determine the impact of parameters and their relative importance to disease transmission and prevalence. Second, in the time-dependent control parameters case, an optimal control problem is formulated to design optimal control strategies for eradicating the disease transmission. Using Pontryagin’s Maximum Principle, we derive necessary conditions for optimal control of the disease. The cost-effectiveness analysis of all combinations of the control measures is made by calculating the infection averted ratio and the incremental cost-effectiveness ratio. This reveals that combining the four restrictive measures conveyed through educational campaigns, screening, safe burial and the care of patients in health centers for better isolation is the most cost-effective among the strategies considered. Numerical simulations are performed to illustrate the theoretical results obtained.

Effect of a Vaccination against the Dengue Fever Epidemic in an Age Structure Population: From the Perspective of the Local and Global Stability Analysis

The effect of vaccination on the dengue fever epidemic described by an age structured modified SIR (Susceptible-Infected-Retired) model is studied using standard stability analysis. The chimeric yellow fever dengue tetravalent dengue vaccine (CYD-TDV™) is a vaccine recently developed to control this epidemic in several Southeast Asian countries. The dengue vaccination program requires a total of three injections, 6 months apart at 0, 6, and 12 months. The ages of the recipients are nine years and above. In this paper, we analyze the mathematical dynamics SIR transmission model of the epidemic. The stability of the model is established using Routh–Hurwitz criteria to see if a Hopf Bifurcation occurs and see when the equilibrium states are local asymptotically stable or global asymptotically stable. We have determined the efficiency of CYD-TDV by simulating the optimal numerical solution for each age range for this model. The numerical results showed the optimal age for vaccination and significantly reduced the severity and severity of the disease.

Analysis and dynamical behavior of a novel dengue model via fractional calculus

The infection of dengue is an intimidating vector-borne disease caused by a pathogenic agent that affects different temperature areas and brings many losses in human health and economy. Thus, it is valuable to identify the most influential parameters in the transmission process for the control of dengue to lessen these losses and to turn down the economic burden of dengue. In this research, we formulate the transmission phenomena of dengue infection with vaccination, treatment and reinfection via Atangana–Baleanu operator to thoroughly explore the intricate system of the disease. Furthermore, to come up with more realistic, dependable and valid results through fractional derivative rather than classical order derivative. The next-generation approach has been utilized to compute the basic reproduction number for the suggested fractional model, indicated by [Formula: see text]; moreover, we conducted sensitivity test of [Formula: see text] to recognize and point out the role of parameters on [Formula: see text]. Our numerical results predict that the reproduction number of the system of dengue infection can be controlled by controlling the index of memory. The uniqueness and existence result has been proved for the solution of the system. A novel numerical method is presented to highlight the time series of dengue system. Eventually, we get numerical results for different assumptions of [Formula: see text] with specifying factors to conceptualize the effect of [Formula: see text] on the dynamics. It has been noted that the fractional-order derivative offers realistic, clear-cut and valid information about the dynamics of dengue fever. Moreover, we note through our analysis that the input parameters’ index of memory, biting rate, transmission probability and recruitment rate of mosquitos can be used as control parameter to lower the level of infection.

Mathematical Modeling to Study Optimal Allocation of Vaccines against COVID-19 Using an Age-Structured Population

Vaccination against the coronavirus disease 2019 (COVID-19) started in early December of 2020 in the USA. The efficacy of the vaccines vary depending on the SARS-CoV-2 variant. Some countries have been able to deploy strong vaccination programs, and large proportions of their populations have been fully vaccinated. In other countries, low proportions of their populations have been vaccinated, due to different factors. For instance, countries such as Afghanistan, Cameroon, Ghana, Haiti and Syria have less than 10% of their populations fully vaccinated at this time. Implementing an optimal vaccination program is a very complex process due to a variety of variables that affect the programs. Besides, science, policy and ethics are all involved in the determination of the main objectives of the vaccination program. We present two nonlinear mathematical models that allow us to gain insight into the optimal vaccination strategy under different situations, taking into account the case fatality rate and age-structure of the population. We study scenarios with different availabilities and efficacies of the vaccines. The results of this study show that for most scenarios, the optimal allocation of vaccines is to first give the doses to people in the 55+ age group. However, in some situations the optimal strategy is to first allocate vaccines to the 15–54 age group. This situation occurs whenever the SARS-CoV-2 transmission rate is relatively high and the people in the 55+ age group have a transmission rate 50% or less that of those in the 15–54 age group. This study and similar ones can provide scientific recommendations for countries where the proportion of vaccinated individuals is relatively small or for future pandemics.

Analysis of deterministic models for dengue disease transmission dynamics with vaccination perspective in Johor, Malaysia

Dengue is a mosquito-borne disease which has continued to be a public health issue in Malaysia. This paper investigates the impact of singular use of vaccination and its combined effort with treatment and adulticide controls on the population dynamics of dengue in Johor, Malaysia. In a first step, a compartmental model capturing vaccination compartment with mass random vaccination distribution process is appropriately formulated. The model with or without imperfect vaccination exhibits backward bifurcation phenomenon. Using the available data and facts from the 2012 dengue outbreak in Johor, basic reproduction number for the outbreak is estimated. Sensitivity analysis is performed to investigate how the model parameters influence dengue disease transmission and spread in a population. In a second step, a new deterministic model incorporating vaccination as a control parameter of distinct constant rates with the efforts of treatment and adulticide controls is developed. Numerical simulations are carried out to evaluate the impact of the three control measures by implementing several control strategies. It is observed that the transmission of dengue can be curtailed using any of the control strategies analysed in this work. Efficiency analysis further reveals that a strategy that combines vaccination, treatment and adulticide controls is most efficient for dengue prevention and control in Johor, Malaysia.

Optimization methods for large-scale vaccine supply chains: a rapid review

Global vaccine revenues are projected at $59.2 billion, yet large-scale vaccine distribution remains challenging for many diseases in countries around the world. Poor management of the vaccine supply chain can lead to a disease outbreak, or at worst, a pandemic. Fortunately, a large number of those challenges, such as decision-making for optimal allocation of resources, vaccination strategy, inventory management, among others, can be improved through optimization approaches. This work aims to understand how optimization has been applied to vaccine supply chain and logistics. To achieve this, we conducted a rapid review and searched for peer-reviewed journal articles, published between 2009 and March 2020, in four scientific databases. The search resulted in 345 articles, of which 25 unique studies met our inclusion criteria. Our analysis focused on the identification of article characteristics such as research objectives, vaccine supply chain stage addressed, the optimization method used, whether outbreak scenarios were considered, among others. Approximately 64% of the studies dealt with vaccination strategy, and the remainder dealt with logistics and inventory management. Only one addressed market competition (4%). There were 14 different types of optimization methods used, but control theory, linear programming, mathematical model and mixed integer programming were the most common (12% each). Uncertainties were considered in the models of 44% of the studies. One resulting observation was the lack of studies using optimization for vaccine inventory management and logistics. The results provide an understanding of how optimization models have been used to address challenges in large-scale vaccine supply chains.

Optimal allocation of limited vaccine to minimize the effective reproduction number

We examine the problem of allocating a limited supply of vaccine for controlling an infectious disease with the goal of minimizing the effective reproduction number Re. We consider an SIR model with two interacting populations and develop an analytical expression that the optimal vaccine allocation must satisfy. With limited vaccine supplies, we find that an all-or-nothing approach is optimal. For certain special cases, we determine the conditions under which the optimal Re is below 1. We present an example of vaccine allocation for COVID-19 and show that it is optimal to vaccinate younger individuals before older individuals to minimize Re if less than 59% of the population can be vaccinated. The analytical conditions we develop provide a simple means of determining the optimal allocation of vaccine between two population groups to minimize Re.

Optimal Control of Dengue Transmission with Vaccination

Dengue disease is caused by four serotypes of the dengue virus: DEN-1, DEN-2, DEN-3, and DEN-4. The chimeric yellow fever dengue tetravalent dengue vaccine (CYD-TDV) is a vaccine currently used in Thailand. This research investigates what the optimal control is when only individuals having documented past dengue infection history are vaccinated. This is the present practice in Thailand and is the latest recommendation of the WHO. The model used is the Susceptible-Infected-Recovered (SIR) model in series configuration for the human population and the Susceptible-Infected (SI) model for the vector population. Both dynamical models for the two populations were recast as optimal control problems with two optimal control parameters. The analysis showed that the equilibrium states were locally asymptotically stable. The numerical solution of the control systems and conclusions are presented.

Analysis of Delayed Vaccination Regimens: A Mathematical Modeling Approach

The first round of vaccination against coronavirus disease 2019 (COVID-19) began in early December of 2020 in a few countries. There are several vaccines, and each has a different efficacy and mechanism of action. Several countries, for example, the United Kingdom and the USA, have been able to develop consistent vaccination programs where a great percentage of the population has been vaccinated (May 2021). However, in other countries, a low percentage of the population has been vaccinated due to constraints related to vaccine supply and distribution capacity. Countries such as the USA and the UK have implemented different vaccination strategies, and some scholars have been debating the optimal strategy for vaccine campaigns. This problem is complex due to the great number of variables that affect the relevant outcomes. In this article, we study the impact of different vaccination regimens on main health outcomes such as deaths, hospitalizations, and the number of infected. We develop a mathematical model of COVID-19 transmission to focus on this important health policy issue. Thus, we are able to identify the optimal strategy regarding vaccination campaigns. We find that for vaccines with high efficacy (>70%) after the first dose, the optimal strategy is to delay inoculation with the second dose. On the other hand, for a low first dose vaccine efficacy, it is better to use the standard vaccination regimen of 4 weeks between doses. Thus, under the delayed second dose option, a campaign focus on generating a certain immunity in as great a number of people as fast as possible is preferable to having an almost perfect immunity in fewer people first. Therefore, based on these results, we suggest that the UK implemented a better vaccination campaign than that in the USA with regard to time between doses. The results presented here provide scientific guidelines for other countries where vaccination campaigns are just starting, or the percentage of vaccinated people is small.

Forecasting the Effects of the New SARS-CoV-2 Variant in Europe

Due to the emergence of a new SARS-CoV-2 variant, we use a previous model to simulate the behaviour of this new SARS-CoV-2 variant. The analysis and simulations are performed for Europe, in order to provide a global analysis of the pandemic. In this context, numerical results are obtained in the first 100 days of the pandemic assuming an infectivity of 70%, 56%, and 35%, respectively, higher for the new SAR-CoV-2 variant, as compared with the real data.

Fractional-calculus analysis of the transmission dynamics of the dengue infection

In this research paper, a novel approach in dengue modeling with the asymptomatic carrier and reinfection via the fractional derivative is suggested to deeply interrogate the comprehensive transmission phenomena of dengue infection. The proposed system of dengue infection is represented in the Liouville-Caputo fractional framework and investigated for basic properties, that is, uniqueness, positivity, and boundedness of the solution. We used the next-generation technique in order to determine the basic reproduction number R₀ for the suggested model of dengue infection; moreover, we conduct a sensitivity test of R₀ through a partial rank correlation coefficient technique to know the contribution of input factors on the output of R₀. We have shown that the infection-free equilibrium of dengue dynamics is globally asymptomatically stable for R₀<1 and unstable in other circumstances. The system of dengue infection is then structured in the Atangana-Baleanu framework to represent the dynamics of dengue with the non-singular and non-local kernel. The existence and uniqueness of the solution of the Atangana-Baleanu fractional system are interrogated through fixed-point theory. Finally, we present a novel numerical technique for the solution of our fractional-order system in the Atangana-Baleanu framework. We obtain numerical results for different values of fractional-order ϑ and input factors to highlight the consequences of fractional-order ϑ and input parameters on the system. On the basis of our analysis, we predict the most critical parameters in the system for the elimination of dengue infection.

Optimal allocation of limited vaccine to control an infectious disease: Simple analytical conditions

When allocating limited vaccines to control an infectious disease, policy makers frequently have goals relating to individual health benefits (e.g., reduced morbidity and mortality) as well as population-level health benefits (e.g., reduced transmission and possible disease eradication). We consider the optimal allocation of a limited supply of a preventive vaccine to control an infectious disease, and four different allocation objectives: minimize new infections, deaths, life years lost, or quality-adjusted life years (QALYs) lost due to death. We consider an SIR model with n interacting populations, and a single allocation of vaccine at time 0. We approximate the model dynamics to develop simple analytical conditions characterizing the optimal vaccine allocation for each objective. We instantiate the model for an epidemic similar to COVID-19 and consider n=2 population groups: one group (individuals under age 65) with high transmission but low mortality and the other group (individuals age 65 or older) with low transmission but high mortality. We find that it is optimal to vaccinate younger individuals to minimize new infections, whereas it is optimal to vaccinate older individuals to minimize deaths, life years lost, or QALYs lost due to death. Numerical simulations show that the allocations resulting from our conditions match those found using much more computationally expensive algorithms such as exhaustive search. Sensitivity analysis on key parameters indicates that the optimal allocation is robust to changes in parameter values. The simple conditions we develop provide a useful means of informing vaccine allocation decisions for communicable diseases.

Substantial impact of post-vaccination contacts on cumulative infections during viral epidemics

Background: The start of 2021 was marked by the initiation of a global vaccination campaign against the novel coronavirus SARS-CoV-2. Formulating an optimal distribution strategy under social and economic constraints is challenging. Optimal distribution is additionally constrained by the potential emergence of vaccine resistance. Analogous to chronic low-dose antibiotic exposure, recently inoculated individuals who are not yet immune play an outsized role in the emergence of resistance. Classical epidemiological modelling is well suited to explore how the behavior of the inoculated population impacts the total number of infections over the entirety of an epidemic. Methods: A deterministic model of epidemic evolution is analyzed, with seven compartments defined by their relationship to the emergence of vaccine-resistant mutants and representing three susceptible populations, three infected populations, and one recovered population. This minimally computationally intensive design enables simulation of epidemics across a broad parameter space. The results are used to identify conditions minimizing the cumulative number of infections. Results: When an escape variant is only modestly less infectious than the originating strain within a naïve population, the cumulative number of infections does not monotonically decrease with the rate of vaccine distribution. Analysis of the model also demonstrates that inoculated individuals play a major role in the mitigation or exacerbation of vaccine-resistant outbreaks. Modulating the rate of host-host contact for the inoculated population by less than an order of magnitude can alter the cumulative number of infections by more than 20%. Conclusions: Mathematical modeling shows that limiting post-vaccination contacts can perceptibly affect the course of an epidemic. The consideration of limitations on post-vaccination contacts remains relevant for the entire duration of any vaccination campaign including the current status of SARS-CoV-2 vaccination.

Substantial impact of post-vaccination contacts on cumulative infections during viral epidemics

Background: The start of 2021 was marked by the initiation of a global vaccination campaign against the novel coronavirus SARS-CoV-2. Formulating an optimal distribution strategy under social and economic constraints is challenging. Optimal distribution is additionally constrained by the potential emergence of vaccine resistance. Analogous to chronic low-dose antibiotic exposure, recently inoculated individuals who are not yet immune play an outsized role in the emergence of resistance. Classical epidemiological modelling is well suited to explore how the behavior of the inoculated population impacts the total number of infections over the entirety of an epidemic. Methods: A deterministic model of epidemic evolution is analyzed, with seven compartments defined by their relationship to the emergence of vaccine-resistant mutants and representing three susceptible populations, three infected populations, and one recovered population. This minimally computationally intensive design enables simulation of epidemics across a broad parameter space. The results are used to identify conditions minimizing the cumulative number of infections. Results: When an escape variant is only modestly less infectious than the originating strain within a naïve population, there exists an optimal rate of vaccine distribution. Exceeding this rate increases the cumulative number of infections due to vaccine escape. Analysis of the model also demonstrates that inoculated individuals play a major role in the mitigation or exacerbation of vaccine-resistant outbreaks. Modulating the rate of host-host contact for the inoculated population by less than an order of magnitude can alter the cumulative number of infections by more than 20%. Conclusions: Mathematical modeling shows that optimization of the vaccination rate and limiting post-vaccination contacts can perceptibly affect the course of an epidemic. The consideration of limitations on post-vaccination contacts remains relevant for the entire duration of any vaccination campaign including the current status of SARS-CoV-2 vaccination.

An optimal control model of the spread of the COVID-19 pandemic in Iraq: Deterministic and chance-constrained model

Many studies have attempted to understand the true nature of COVID-19 and the factors influencing the spread of the virus. This paper investigates the possible effect the COVID-19 pandemic spreading in Iraq considering certain factors, that include isolation and weather. A mathematical model of cases representing inpatients, recovery, and mortality was used in formulating the control variable in this study to describe the spread of COVID-19 through changing weather conditions between 17th March and 15th May, 2020. Two models having deterministic and an uncertain number of daily cases were used in which the solution for the model using the Pontryagin maximum principle (PMP) was derived. Additionally, an optimal control model for isolation and each factor of the weather factors was also achieved. The results simulated the reality of such an event in that the cases increased by 118%, with an increase in the number of people staying outside of their house by 25%. Further, the wind speed and temperature had an inverse effect on the spread of COVID-19 by 1.28% and 0.23%, respectively. The possible effect of the weather factors with the uncertain number of cases was higher than the deterministic number of cases. Accordingly, the model developed in this study could be applied in other countries using the same factors or by introducing other factors.

Dengue infection modeling and its optimal control analysis in East Java, Indonesia

In this study, we present a mathematical model of dengue fever transmission with hospitalization to describe the dynamics of the infection. We estimated the basic reproduction number for the infected cases in East Java Province for the year 2018 is R0≈1.1138. The parameters of the dengue model are estimated by using the confirmed notified cases of East Java province, Indonesia for the year 2018. We formulated the model for dengue with hospitalization and present its dynamics in details. Initially, we present the basic mathematical results and then show briefly the stability results for the model. Further, we formulate an optimal control problem with control functions and obtain the optimal control characterization. The optimal control problem is solved numerically and the results comprised of controls system for different strategies. The controls such as prevention and insecticide could use the best role in the disease eradication from the community. Our results suggest that the prevention of humans from the mosquitoes and the insecticide spray on mosquitoes can significantly reduce the infection of dengue fever and may reduce further spread of infection in the community.

An optimal control model of COVID-19 pandemic: a comparative study of five countries

This paper formulates an optimal control model of COVID-19 pandemic spreading. We discuss the health sector performance of Argentina, Hungary, Egypt, Malaysia, and Iraq. A mathematical model describes an actual case number of COVID-19. We investigate three strategies depend on recovery rate, death rate, and together (optimal). These strategies represent the percent of the health sector development. The explicit solution of the model using the Pontryagin maximum principle is derived. The results showed the ranking of countries based on the new percent of the recovery and death cases. A new percent as a result to the control variable value (health sector development). Also, the development percent of the health sector of each country, was determined. For example, 0.005 led to a significant reduce the death rates in Malaysia. Meanwhile, a half of death rates could reduce by this percent in Egypt.

Optimal Control of Mitigation Strategies for Dengue Virus Transmission

Dengue virus is transmitted by Aedes mosquitoes, posing threat to people's health and leading to great economic cost in many tropical and subtropical regions. We develop an ordinary differential equation model taking into account multiple strains of dengue virus. Using the model, we assess the effectiveness of human vaccination considering its waning and failure. We derive the lower bound and upper bound for the final size of the epidemic. Sensitivity analysis quantifies the impact of parameters on the basic reproduction number. Different scenarios of vaccinating humans show that it is better to vaccinate humans at early stages. We find that the cumulative number of infected humans is small when the vaccination rate is high or the waning rate is low for previously infected humans. We analyze the necessary conditions for implementing optimal control and derive the corresponding optimal solutions for mitigation dengue virus transmission by applying Pontryagin's Maximum Principle. Our findings may provide guidance for the public health authorities to implement human vaccination and other mitigation strategies.

Stability and optimal control strategies for a novel epidemic model of COVID-19

In this paper, a novel two-stage epidemic model with a dynamic control strategy is proposed to describe the spread of Corona Virus Disease 2019 (COVID-19) in China. Combined with local epidemic control policies, an epidemic model with a traceability process is established. We aim to investigate the appropriate control strategies to minimize the control cost and ensure the normal operation of society under the premise of containing the epidemic. This work mainly includes: (i) propose the concept about the first and the second waves of COVID-19, as well as study the case data and regularity of four cities; (ii) derive the existence and stability of the equilibrium, the parameter sensitivity of the model, and the existence of the optimal control strategy; (iii) carry out the numerical simulation associated with the theoretical results and construct a dynamic control strategy and verify its feasibility.

Substantial Impact of Post Vaccination Contacts on Cumulative Infections during Viral Epidemics

Background

The start of 2021 will be marked by a global vaccination campaign against the novel coronavirus SARS-CoV-2. Formulating an optimal distribution strategy under social and economic constraints is challenging. Optimal distribution is additionally constrained by the potential emergence of vaccine resistance. Analogous to chronic low-dose antibiotic exposure, recently inoculated individuals who are not yet immune play an outsized role in the emergence of resistance. Classical epidemiological modelling is well suited to explore how the behavior of the inoculated population impacts the total number of infections over the entirety of an epidemic.

Methods

A deterministic model of epidemic evolution is analyzed, with 7 compartments defined by their relationship to the emergence of vaccine-resistant mutants and representing three susceptible populations, three infected populations, and one recovered population. This minimally computationally intensive design enables simulation of epidemics across a broad parameter space. The results are used to identify conditions minimizing the cumulative number of infections.

Results

When an escape variant is only modestly less infectious than the originating strain within a naïve population, there exists an optimal rate of vaccine distribution. Exceeding this rate increases the cumulative number of infections due to vaccine escape. Analysis of the model also demonstrates that inoculated individuals play a major role in the mitigation or exacerbation of vaccine-resistant outbreaks. Modulating the rate of host-host contact for the inoculated population by less than an order of magnitude can alter the cumulative number of infections by more than 20%.

Conclusions

Mathematical modeling shows that optimization of the vaccination rate and limiting post-vaccination contacts can affect the course of an epidemic. Given the relatively short window between inoculation and the acquisition of immunity, these results might merit consideration for an immediate, practical public health response.

Optimal vaccination strategy for dengue transmission in Kupang city, Indonesia

Dengue is a public health problem with around 390 million cases annually and is caused by four distinct serotypes. Infection by one of the serotypes provides lifelong immunity to that serotype but have a higher chance of attracting the more dangerous forms of dengue in subsequent infections. Therefore, a perfect strategy against dengue is required. Dengue vaccine with 42-80% efficacy level has been licensed for the use in reducing disease transmission. However, this may increase the likelihood of obtaining the dangerous forms of dengue. In this paper, we have developed single and two-serotype dengue mathematical models to investigate the effects of vaccination on dengue transmission dynamics. The model is validated against dengue data from Kupang city, Indonesia. We investigate the effects of vaccination on seronegative and seropositive individuals and perform a global sensitivity analysis to determine the most influential parameters of the model. A sensitivity analysis suggests that the vaccination rate, the transmission probability and the biting rate have greater effects on the reduction of the proportion of dengue cases. Interestingly, with vaccine implementation, the mosquito-related parameters do not have significant impact on the reduction in the proportion of dengue cases. If the vaccination is implemented on seronegative individuals only, it may increase the likelihood of obtaining the severe dengue. To reduce the proportion of severe dengue cases, it is better to vaccinate seropositive individuals. In the context of Kupang City where the majority of individuals have been infected by at least one dengue serotype, the implementation of vaccination strategy is possible. However, understanding the serotype-specific differences is required to optimise the delivery of the intervention.

Model predictive control to mitigate the COVID-19 outbreak in a multi-region scenario

The COVID-19 outbreak is deeply influencing the global social and economic framework, due to restrictive measures adopted worldwide by governments to counteract the pandemic contagion. In multi-region areas such as Italy, where the contagion peak has been reached, it is crucial to find targeted and coordinated optimal exit and restarting strategies on a regional basis to effectively cope with possible onset of further epidemic waves, while efficiently returning the economic activities to their standard level of intensity. Differently from the related literature, where modeling and controlling the pandemic contagion is typically addressed on a national basis, this paper proposes an optimal control approach that supports governments in defining the most effective strategies to be adopted during post-lockdown mitigation phases in a multi-region scenario. Based on the joint use of a non-linear Model Predictive Control scheme and a modified Susceptible-Infected-Recovered (SIR)-based epidemiological model, the approach is aimed at minimizing the cost of the so-called non-pharmaceutical interventions (that is, mitigation strategies), while ensuring that the capacity of the network of regional healthcare systems is not violated. In addition, the proposed approach supports policy makers in taking targeted intervention decisions on different regions by an integrated and structured model, thus both respecting the specific regional health systems characteristics and improving the system-wide performance by avoiding uncoordinated actions of the regions. The methodology is tested on the COVID-19 outbreak data related to the network of Italian regions, showing its effectiveness in properly supporting the definition of effective regional strategies for managing the COVID-19 diffusion.

MODELING ZIKA TRANSMISSION DYNAMICS: PREVENTION AND CONTROL

The Zika virus (ZIKV) epidemic is depicted to have high spatial diversity and slow growth, attributable to the dynamics of the mosquito vector and mobility of the human populations. In an effort to understand the transmission dynamics of Zika virus, we formulate a new compartmental epidemic model with a system of seven differential equations and 11 parameters incorporating the decaying transmission rate and study the impact of protection measure on basic public health. We do not fit the model to the observed pattern of spread, rather we use parameter values estimated in the past and examine the extent to which the designed model prediction agrees with the pattern of spread seen in Brazil, via reaction–diffusion modeling. Our work includes estimation of key epidemiological parameters such as basic reproduction number ([Formula: see text], and gives a rough estimate of how many individuals can be typically infected during an outbreak if it occurs in India. We used partial rank correlation coefficient method for global sensitivity analysis to identify the most influential model parameters. Using optimal control theory and Pontryagin’s maximum principle, a control model has been proposed and conditions for the optimal control are determined for the deterministic model of Zika virus. The control functions for the strategies (i) vector-to-human contact reduction and (ii) vector elimination are introduced into the system. Numerical simulations are also performed. This work aimed at understanding the potential extent and timing of the ZIKV epidemic can be used as a template for the analysis of future mosquito-borne epidemics.

Optimization of the Controls against the Spread of Zika Virus in Populations

In this paper, we study and explore two control strategies to decrease the spread of Zika virus in the human and mosquito populations. The control strategies that we consider in this study are awareness and spraying campaigns. We solve several optimal control problems relying on a mathematical epidemic model of Zika that considers both human and mosquito populations. The first control strategy is broad and includes using information campaigns, encouraging people to use bednetting, wear long-sleeve shirts, or similar protection actions. The second control is more specific and relies on spraying insecticides. The control system relies on a Zika mathematical model with control functions. To develop the optimal control problem, we use Pontryagins’ maximum principle, which is numerically solved as a boundary value problem. For the mathematical model of the Zika epidemic, we use parameter values extracted from real data from an outbreak in Colombia. We study the effect of the costs related to the controls and infected populations. These costs are important in real life since they can change the outcomes and recommendations for health authorities dramatically. Finally, we explore different options regarding which control measures are more cost-efficient for society.

Optimal public health intervention in a behavioural vaccination model: the interplay between seasonality, behaviour and latency period

Hesitancy and refusal of vaccines preventing childhood diseases are spreading due to 'pseudo-rational' behaviours: parents overweigh real and imaginary side effects of vaccines. Nonetheless, the 'Public Health System' (PHS) may enact public campaigns to favour vaccine uptake. To determine the optimal time profiles for such campaigns, we apply the optimal control theory to an extension of the susceptible-infectious-removed (SIR)-based behavioural vaccination model by d'Onofrio et al. (2012, PLoS ONE, 7, e45653). The new model is of susceptible-exposed-infectious-removed (SEIR) type under seasonal fluctuations of the transmission rate. Our objective is to minimize the total costs of the disease: the disease burden, the vaccination costs and a less usual cost: the economic burden to enact the PHS campaigns. We apply the Pontryagin minimum principle and numerically explore the impact of seasonality, human behaviour and latency rate on the control and spread of the target disease. We focus on two noteworthy case studies: the low (resp. intermediate) relative perceived risk of vaccine side effects and relatively low (resp. very low) speed of imitation. One general result is that seasonality may produce a remarkable impact on PHS campaigns aimed at controlling, via an increase of the vaccination uptake, the spread of a target infectious disease. In particular, a higher amplitude of the seasonal variation produces a higher effort and this, in turn, beneficially impacts the induced vaccine uptake since the larger is the strength of seasonality, the longer the vaccine propensity remains large. However, such increased effort is not able to fully compensate the action of seasonality on the prevalence.

Administration of Defective Virus Inhibits Dengue Transmission into Mosquitoes

The host-vector shuttle and the bottleneck in dengue transmission is a significant aspect with regard to the study of dengue outbreaks. As mosquitoes require 100–1000 times more virus to become infected than human, the transmission of dengue virus from human to mosquito is a vulnerability that can be targeted to improve disease control. In order to capture the heterogeneity in the infectiousness of an infected patient population towards the mosquito population, we calibrate a population of host-to-vector virus transmission models based on an experimentally quantified infected fraction of a mosquito population. Once the population of models is well-calibrated, we deploy a population of controls that helps to inhibit the human-to-mosquito transmission of the dengue virus indirectly by reducing the viral load in the patient body fluid. We use an optimal bang-bang control on the administration of the defective virus (transmissible interfering particles (TIPs)) to symptomatic patients in the course of their febrile period and observe the dynamics in successful reduction of dengue spread into mosquitoes.

Modelling the Use of Vaccine and Wolbachia on Dengue Transmission Dynamics

The use of vaccine and Wolbachia has been proposed as strategies against dengue. Research showed that the Wolbachia intervention is highly effective in areas with low to moderate transmission levels. On the other hand, the use of vaccine is strongly effective when it is implemented on seropositive individuals and areas with high transmission levels. The question that arises is could the combination of both strategies result in higher reduction in the number of dengue cases? This paper seeks to answer the aforementioned question by the use of a mathematical model. A deterministic model in the presence of vaccine and Wolbachia has been developed and analysed. Numerical simulations were presented and public health implications were discussed. The results showed that the performance of Wolbachia in reducing the number of dengue cases is better than that of vaccination if the vaccine efficacy is low, otherwise, the use of vaccine is sufficient to reduce dengue incidence and hence the combination of Wolbachia and vaccine is not necessary.

Transmission Dynamics and Control Mechanisms of Vector-Borne Diseases with Active and Passive Movements Between Urban and Satellite Cities

A metapopulation model which explicitly integrates vector-borne and sexual transmission of an epidemic disease with passive and active movements between an urban city and a satellite city is formulated and analysed. The basic reproduction number of the disease is explicitly determined as a combination of sexual and vector-borne transmission parameters. The sensitivity analysis reveals that the disease is primarily transmitted via the vector-borne mode, rather than via sexual transmission, and that sexual transmission by itself may not initiate or sustain an outbreak. Also, increasing the population movements from one city to the other leads to an increase in the basic reproduction number of the later city but a decrease in the basic reproduction number of the former city. The influence of other significant parameters is also investigated via the analysis of suitable partial rank correlation coefficients. After gauging the effects of mobility, we explore the potential effects of optimal control strategies relying upon several distinct restrictions on population movement.

A population of bang-bang switches of defective interfering particles makes within-host dynamics of dengue virus controllable

The titre of virus in a dengue patient and the duration of this viraemia has a profound effect on whether or not a mosquito will become infected when it feeds on the patient and this, in turn, is a key driver of the magnitude of a dengue outbreak. The assessment of the heterogeneity of viral dynamics in dengue-infected patients and its precise treatment are still uncertain. Infection onset, patient physiology and immune response are thought to play major roles in the development of the viral load. Research has explored the interference and spontaneous generation of defective virus particles, but have not examined both the antibody and defective particles during natural infection. We explore the intrinsic variability in the within-host dynamics of viraemias for a population of patients using the method of population of models (POMs). A dataset from 208 patients is used to initially calibrate 20,000 models for the infection kinetics for each of the four dengue virus serotypes. The calibrated POMs suggests that naturally generated defective particles may interfere with the viraemia, but the generated defective virus particles are not adequate to reduce high fever and viraemia duration. The effect of adding excess defective dengue virus interfering particles to patients as a therapeutic is evaluated using the calibrated POMs in a bang-bang (on-off or two-step) optimal control setting. Bang-bang control is a class of binary feedback control that turns either 'ON' or 'OFF' at different time points, determined by the system feedback. Here, the bang-bang control estimates the mathematically optimal dose and duration of the intervention for each model in the POM set.

Bézier curve parametrisation and echo state network methods for solving optimal control problems of SIR model

In this work, we introduce an optimal control problem with two control variables of the SIR (susceptible-infected-recovered) epidemic model to minimise the infective and susceptible individuals. To solve the control problem, we use direct Bernstein-Bézier parametrisation of the control variables and metaheuristic optimisation methods of an objective function, and indirect methods, adaptive critic design (ACD) with echo state networks (ESNs) and Pontryagin's maximum principle. We propose Bernstein-Bézier parametrisation and ESNs based algorithms to solve optimal control problems for systems governed by differential equations. We use ACD with ESNs to approximate co-state equations. Our simulations have shown that the proposed two methods are able to solve optimal control problems.

Modeling Public Health Campaigns for Sexually Transmitted Infections via Optimal and Feedback Control

Control of sexually transmitted infections (STIs) poses important challenges to public health authorities. Obstacles for STIs' control include low priority in public health programs and disease transmission mechanisms. This work uses a compartmental pair model to explore different public health strategies on the evolution of STIs. Optimal control and feedback control are used to model realistic strategies for reducing the prevalence of these infections. Feedback control is proposed to model the reaction of public health authorities relative to an alert level. Optimal control is used to model the optimization of available resources for implementing strategies. Numerical simulations are performed using trichomoniasis, gonorrhea, chlamydia and human papillomavirus (HPV) as study cases. HPV is non-curable, and it is analyzed only under transmission control such as condom promotion campaigns. Trichomoniasis, gonorrhea and chlamydia are curable STIs that are modeled here additionally under treatment control. Increased cost-effectiveness ratio is employed as a criterion to measure control strategies performance. The features and drawbacks of control strategies under the pair formation process are discussed.

Optimal control of a multi-patch Dengue model under the influence of Wolbachia bacterium

In this work, a multi-patch model for dengue transmission dynamics including the bacterium Wolbachia is studied and by that the control efforts to minimize the disease spread by host and vector control are investigated. The multi-patch system models the host movement within the patches which coupled via a residence-time budgeting matrix P. Numerical results confirm that the control mechanism embedded in incidence rates of the disease transmission, effectively reduce the spread of the disease.

Optimal control of vaccination in a vector-borne reaction-diffusion model applied to Zika virus

Zika virus has acquired worldwide concern after a recent outbreak in Latin America that started in Brazil, with associated neurological conditions such as microcephaly in newborns from infected mothers. The virus is transmitted mainly by Aedes aegypti mosquitoes, but direct (sexual) transmission has been documented. We formulate a reaction diffusion model that considers spatial movement of humans and vectors, with local contact transmission of Zika virus. Vaccination is introduced as a control variable, giving immunity to susceptible humans, in order to characterize an optimal vaccination strategy that minimizes the costs associated with infections and vaccines. The optimal control characterization is obtained in terms of state and adjoint equations. Parameter estimation and numerical simulations are carried out using data for the initial 2015 Zika outbreak in the state of Rio Grande do Norte in Brazil. Several scenarios are considered and analyzed in terms of number of new infections and costs, showing that the optimal control application is successful, significantly reducing these quantities.

Optimal Impulse Vaccination Approach for an SIR Control Model with Short-Term Immunity

Vaccines are not administered on a continuous basis, but injections are practically introduced at discrete times often separated by an important number of time units, and this differs depending on the nature of the epidemic and its associated vaccine. In addition, especially when it comes to vaccination, most optimization approaches in the literature and those that have been subject to epidemic models have focused on treating problems that led to continuous vaccination schedules but their applicability remains debatable. In search of a more realistic methodology to resolve this issue, a control modeling design, where the control can be characterized analytically and then optimized, can definitely help to find an optimal regimen of pulsed vaccinations. Therefore, we propose a susceptible-infected-removed (SIR) hybrid epidemic model with impulse vaccination control and a compartment that represents the number of vaccinated individuals supposed to not acquire sufficient immunity to become permanently recovered due to the short-term effect of vaccines. A basic reproduction number, when the control is defined as a constant parameter, is calculated. Since we also need to find the optimal values of this impulse control when it is defined as a function of time, we start by stating a general form of an impulse version of Pontryagin’s maximum principle that can be adapted to our case, and then we apply it to our model. Finally, we provide our numerical simulations that are obtained via an impulse progressive-regressive iterative scheme with fixed intervals between impulse times (theoretical example of an impulse at each week), and we conclude with a discussion of our results.

The Role of Vertical Transmission in the Control of Dengue Fever

In this work, a two-strain dengue model with vertical transmission in the mosquito population is considered. Although vertical transmission is often ignored in models of dengue fever, we show that effective control of an outbreak of dengue can depend on whether or not the vertical transmission is a significant mode of disease transmission. We model the effect of a control strategy aimed at reducing human-mosquito transmissions in an optimal control framework. As the likelihood of vertical transmission increases, outbreaks become more difficult and expensive to control. However, even for low levels of vertical transmission, the additional, uncontrolled, transmission from infected mosquito to eggs may undercut the effectiveness of any control function. This is of particular importance in regions where existing control policies may be effective and the endemic strain does not exhibit vertical transmission. If a novel strain that does exhibit vertical transmission invades, then existing, formerly effective, control policies may no longer be sufficient. Therefore, public health officials should pay more attention to the role of vertical transmission for more effective interventions and policy.

Mathematical modeling of dengue epidemic: control methods and vaccination strategies

Dengue is, in terms of death and economic cost, one of the most important infectious diseases in the world. So, its mathematical modeling can be a valuable tool to help us to understand the dynamics of the disease and to infer about its spreading by the proposition of control methods. In this paper, control strategies, which aim to eliminate the Aedes aegypti mosquito, as well as proposals for the vaccination campaign are evaluated. In our mathematical model, the mechanical control is accomplished through the environmental support capacity affected by a discrete function that represents the removal of breedings. Chemical control is carried out using insecticide and larvicide. The efficiency of vaccination is studied through the transfer of a fraction of individuals, proportional to the vaccination rate, from the susceptible to the recovered compartments. Our major find is that the dengue fever epidemic is only eradicated with the use of an immunizing vaccine because control measures, directed against its vector, are not enough to halt the disease spreading. Even when the infected mosquitoes are eliminated from the system, the susceptible ones are still present, and infected humans cause dengue fever to reappear in the human population.

Multiple Equilibria in a Non-smooth Epidemic Model with Medical-Resource Constraints

The issue of medical-resource constraints has the potential to dramatically affect disease management, especially in developing countries. We analyze a non-smooth epidemic model with nonlinear incidence rate and resource constraints, which defines a vaccination program with vaccination rate proportional to the number of susceptible individuals when this number is below the threshold level and constant otherwise. To better understand the impact of this non-smooth vaccination policy, we provide a comprehensive qualitative analysis of global dynamics for the whole parameter space. As the threshold value varies, the target model admits multistability of three regular equilibria, bistability of two regular equilibria, that of one disease-free equilibrium and one generalized endemic equilibria, and that of one disease-free equilibrium and one crossing cycle. The steady-state regimes include healthy, low epidemic and high epidemic. This suggests the key role of the threshold value, as well as the initial infection condition in disease control. Our findings demonstrate that the case number can be contained at a satisfactorily controllable level or range if eradicating it proves to be impossible.

Time Needed to Control an Epidemic with Restricted Resources in SIR Model with Short-Term Controlled Population: A Fixed Point Method for a Free Isoperimetric Optimal Control Problem

In this paper, we attempt to determine the optimal duration of an anti-epidemic control strategy which targets susceptible people, under the isoperimetric condition that we could not control all individuals of this category due to restricted health resources. We state and prove the local and global stability conditions of free and endemic equilibria of a simple epidemic compartmental model devised in the form of four ordinary differential equations which describe the dynamics of susceptible-controlled-infected-removed populations and where it is taken into account that the controlled people cannot acquire long-lived immunity to move towards the removed compartment due to the temporary effect of the control parameter. Thereafter, we characterize the sought optimal control and we show the effectiveness of this limited control policy along with the research of the optimal duration that is needed to reduce the size of the infected population. The isoperimetric constraint is defined over a fixed horizon, while the objective function is defined over a free horizon present under a quadratic form in the payoff term. The complexity of this optimal control problem requires the execution of three numerical methods all combined together at the same time, namely, the forward–backward sweep method to generate the optimal state and control functions, the secant method adapted to the isoperimetric restriction, and, finally, the fixed point method to obtain the optimal final time.