Optimal control of the transmission rate in compartmental epidemics

While SARS-CoV-2 vaccine distribution campaigns are underway across the world, communities face the challenge of a fair and effective distribution of limited supplies. We wonder whether suitable spatial allocation strategies might significantly improve a campaign's efficacy in averting damaging outcomes. To that end, we address the problem of optimal control of COVID-19 vaccinations in a country-wide geographic and epidemiological context characterized by strong spatial heterogeneities in transmission rate and disease history. We seek the vaccine allocation strategies in space and time that minimize the number of infections in a prescribed time horizon. We examine scenarios of unfolding disease transmission across the 107 provinces of Italy, from January to April 2021, generated by a spatially explicit compartmental COVID-19 model tailored to the Italian geographic and epidemiological context. We develop a novel optimal control framework to derive optimal vaccination strategies given the epidemiological projections and constraints on vaccine supply and distribution logistic. Optimal schemes significantly outperform simple alternative allocation strategies based on incidence, population distribution, or prevalence of susceptibles in each province. Our results suggest that the complex interplay between the mobility network and the spatial heterogeneities imply highly non-trivial prioritization of local vaccination campaigns. The extent of the overall improvements in the objectives grants further inquiry aimed at refining other possibly relevant factors so far neglected. Our work thus provides a proof-of-concept of the potential of optimal control for complex and heterogeneous epidemiological contexts at country, and possibly global, scales.

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Qualitative analysis and optimal control of an SIR model with logistic growth, non-monotonic incidence and saturated treatment

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2021004 ◽

2021 ◽

Author(s):

Jayanta Kumar Ghosh ◽

Prahlad Majumdar ◽

Uttam Ghosh

Keyword(s):

Optimal Control ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Transmission Rate ◽

Sir Model ◽

Logistic Growth ◽

Treatment Rate ◽

Globally Asymptotically Stable ◽

Susceptible Population ◽

Saturated Treatment

This paper describes an SIR model with logistic growth rate of susceptible population, non-monotonic incidence rate and saturated treatment rate. The existence and stability analysis of equilibria have been investigated. It has been shown that the disease free equilibrium point ( DFE ) is globally asymptotically stable if the basic reproduction number is less than unity and the transmission rate of infection less than some threshold. The system exhibits the transcritical bifurcation at DFE with respect to the cure rate. We have also found the condition for occurring the backward bifurcation, which implies the value of basic reproduction number less than unity is not enough to eradicate the disease. Stability or instability of different endemic equilibria has been shown analytically. The system also experiences the saddle-node and Hopf bifurcation. The existence of Bogdanov - Takens bifurcation ( BT ) of co-dimension 2 has been investigated which has also been shown through numerical simulations. Here we have used two control functions, one is vaccination control and other is treatment control. We have solved the optimal control problem both analytically and numerically. Finally, the efficiency analysis has been used to determine the best control strategy among vaccination and treatment.

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Optimal Allocation of Vaccine and Antiviral Drugs for Influenza Containment over Delayed Multiscale Epidemic Model considering Time-Dependent Transmission Rate

Computational and Mathematical Methods in Medicine ◽

10.1155/2021/4348910 ◽

2021 ◽

Vol 2021 ◽

pp. 1-27

Author(s):

Zohreh Abbasi ◽

Iman Zamani ◽

Amir Hossein Amiri Mehra ◽

Asier Ibeas ◽

Mohsen Shafieirad

Keyword(s):

Optimal Control ◽

Time Delay ◽

Epidemic Model ◽

Disease Transmission ◽

Optimal Allocation ◽

Transmission Rate ◽

Antiviral Treatment ◽

Infected Individual ◽

Optimal Time ◽

Control Input

In this study, two types of epidemiological models called “within host” and “between hosts” have been studied. The within-host model represents the innate immune response, and the between-hosts model signifies the SEIR (susceptible, exposed, infected, and recovered) epidemic model. The major contribution of this paper is to break the chain of infectious disease transmission by reducing the number of susceptible and infected people via transferring them to the recovered people group with vaccination and antiviral treatment, respectively. Both transfers are considered with time delay. In the first step, optimal control theory is applied to calculate the optimal final time to control the disease within a host’s body with a cost function. To this end, the vaccination that represents the effort that converts healthy cells into resistant-to-infection cells in the susceptible individual’s body is used as the first control input to vaccinate the susceptible individual against the disease. Moreover, the next control input (antiviral treatment) is applied to eradicate the concentrations of the virus and convert healthy cells into resistant-to-infection cells simultaneously in the infected person’s body to treat the infected individual. The calculated optimal time in the first step is considered as the delay of vaccination and antiviral treatment in the SEIR dynamic model. Using Pontryagin’s maximum principle in the second step, an optimal control strategy is also applied to an SEIR mathematical model with a nonlinear transmission rate and time delay, which is computed as optimal time in the first step. Numerical results are consistent with the analytical ones and corroborate our theoretical results.

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Modelling the Periodic Outbreak of Measles in Mainland China

Mathematical Problems in Engineering ◽

10.1155/2020/3631923 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13

Author(s):

Yuyi Xue ◽

Xiaoe Ruan ◽

Yanni Xiao

Keyword(s):

Optimal Control ◽

Numerical Simulations ◽

Basic Reproduction Number ◽

Reproduction Number ◽

Transmission Rate ◽

Control Strategies ◽

Mainland China ◽

Asymptomatic Infection ◽

Protection Measures ◽

The Basic Reproduction Number

In mainland China, measles infection reached the lowest level in 2012 but resurged again after that with a seasonally fluctuating pattern. To investigate the phenomenon of periodic outbreak and identify the crucial parameters that play in the transmission dynamics of measles, we formulate a mathematical model incorporating periodic transmission rate and asymptomatic infection with waning immunity. We define the basic reproduction number as the threshold value to govern whether measles infection dies out or not. Fitting the reported measles cases from 2013 to 2016 to our proposed model, we estimate the basic reproduction number R0 with immunization to be 1.0077. From numerical simulations, we conclude asymptomatic infection does not cause much new infections and the key parameters affecting the transmission of measles are vaccination rate, transmission rate, and recovery rate, which suggests the public to enhance vaccination and protection measures to reduce effective contacts between susceptible and infective individuals and treat infected individuals timely. To minimize the number of infected individuals at a minimal cost, we formulate an optimal control system to design optimal control strategies. Numerical simulations show the effectiveness of optimal control strategies and recommend us to implement the control strategies as soon as possible. In particular, enhancing vaccination is especially effective in lowering the initial outbreak and making disease recurrence less likely.

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MODELING ZIKA TRANSMISSION DYNAMICS: PREVENTION AND CONTROL

Journal of Biological System ◽

10.1142/s021833902050014x ◽

2020 ◽

Vol 28 (03) ◽

pp. 719-749

Author(s):

PARIMITA ROY ◽

RANJIT KUMAR UPADHYAY ◽

JASMINE CAUR

Keyword(s):

Optimal Control ◽

Zika Virus ◽

Deterministic Model ◽

Transmission Rate ◽

Spatial Diversity ◽

Transmission Dynamics ◽

Model Parameters ◽

Human Populations ◽

Mosquito Vector ◽

The Impact

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.

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Measles dynamics on network models with optimal control strategies

Advances in Difference Equations ◽

10.1186/s13662-021-03306-y ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Yuyi Xue ◽

Xiaoe Ruan ◽

Yanni Xiao

Keyword(s):

Optimal Control ◽

Transmission Rate ◽

Control Strategies ◽

Random Network ◽

Network Models ◽

Control Measures ◽

Effective Control ◽

Threshold Dynamics ◽

Heterogeneous Model ◽

Strong Control

AbstractTo investigate the influences of heterogeneity and waning immunity on measles transmission, we formulate a network model with periodic transmission rate, and theoretically examine the threshold dynamics. We numerically find that the waning of immunity can lead to an increase in the basic reproduction number $R_{0}$ R 0 and the density of infected individuals. Moreover, there exists a critical level for average degree above which $R_{0}$ R 0 increases quicker in the scale-free network than in the random network. To design the effective control strategies for the subpopulations with different activities, we examine the optimal control problem of the heterogeneous model. Numerical studies suggest us no matter what the network is, we should implement control measures as soon as possible once the outbreak takes off, and particularly, the subpopulation with high connectivity should require high intensity of interventions. However, with delayed initiation of controls, relatively strong control measures should be given to groups with medium degrees. Furthermore, the allocation of costs (or resources) should coincide with their contact patterns.

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Bifurcation and Stabillity Analysis of HIV Transmission Model with Optimal Control

Journal of Mathematics ◽

10.1155/2021/7471290 ◽

2021 ◽

Vol 2021 ◽

pp. 1-14

Author(s):

Kumama Regassa Cheneke ◽

Koya Purnachandra Rao ◽

Geremew Kenassa Edessa

Keyword(s):

Optimal Control ◽

Cost Effectiveness ◽

Hiv Transmission ◽

Reproduction Number ◽

Transmission Rate ◽

Control Measures ◽

Transmission Model ◽

Progression Rate ◽

Asymptotically Stable ◽

The Cost

A mathematical model of HIV transmission is built and studied in this paper. The system’s equilibrium is calculated. A next-generation matrix is used to calculate the reproduction number. The novel method is used to examine the developed model’s bifurcation and equilibrium stability. The stability analysis result shows that the disease-free equilibrium is locally asymptotically stable if 0 < R 0 < 1 but unstable if R 0 > 1 . However, the endemic equilibrium is locally and globally asymptotically stable if R 0 > 1 and unstable otherwise. The sensitivity analysis shows that the most sensitive parameter that contributes to increasing of the reproduction number is the transmission rate β 2 of HIV transmission from HIV individuals to susceptible individuals and the parameter that contributes to the decreasing of the reproduction number is identified as progression rate η of HIV-infected individuals to AIDS individuals. Furthermore, it is observed that as we change η from 0.1 to 1 , the reproduction number value decreases from 1.205 to 1.189, where the constant value of β 2 = 0.1 . On the other hand, as we change the value of β 2 from 0.1 to 1 , the value of the reproduction number increases from 0.205 to 1.347, where the constant value of η = 0.1 . Further, the developed model is extended to the optimal control model of HIV/AIDS transmission, and the cost-effectiveness of the control strategy is analyzed. Pontraygin’s Maximum Principle (PMP) is applied in the construction of the Hamiltonian function. Moreover, the optimal system is solved using forward and backward Runge–Kutta fourth-order methods. The numerical simulation depicts the number of newly infected HIV individuals and the number of individuals at the AIDS stage reduced as a result of taking control measures. The cost-effectiveness study demonstrates that when combined and used, the preventative and treatment control measures are effective. MATLAB is used to run numerical simulations.

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Optimal Control Strategies for Virus Spreading in Inhomogeneous Epidemic Dynamics

Canadian Mathematical Bulletin ◽

10.4153/cmb-2012-007-2 ◽

2013 ◽

Vol 56 (3) ◽

pp. 621-629 ◽

Cited By ~ 11

Author(s):

Yilun Shang

Keyword(s):

Optimal Control ◽

Cyber Security ◽

Transmission Rate ◽

Control Strategies ◽

High Reliability ◽

Threshold Function ◽

Epidemic Dynamics ◽

Sis Epidemic Model ◽

The Cost ◽

Virus Spreading

Abstract.In this paper, we study the spread of virus/worm in computer networks with a view to addressing cyber security problems. Epidemic models have been applied extensively to model the propagation of computer viruses, which characterize the fact that infected machines may spread malware to other hosts connected to the network. In our framework, the dynamics of hosts evolves according to a modified inhomogeneous Susceptible-Infectious-Susceptible (SIS) epidemic model with time-varying transmission rate and recovery rate. The infection of computers is subject to direct attack as well as propagation among hosts. Based on optimal control theory, optimal attack strategies are provided by minimizing the cost (equivalently maximizing the profit) of the attacker. We present a threshold function of the fraction of infectious hosts, which captures the dynamically evolving strategies of the attacker and reflects the persistence of virus spreading. Moreover, our results indicate that if the infectivity of a computer worm is low and the computers are installed with antivirus software with high reliability, the intensity of attacks incurred will likely be low. This agrees with our intuition.

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