Special Issue "Optimal Control and Management of Infectious Diseases"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (30 November 2015).

Special Issue Editors

Prof. Bruno Buonomo
Website
Guest Editor
Department of Mathematics and Applications, University of Naples Federico II, I-80126 Naples, Italy
Interests: Qualitative Theory of Nonlinear Dynamical Systems, Optimal Control, Mathematics for Biology and Medicine, Mathematical Epidemiology, Population Dynamics
Prof. Andrea Di Liddo
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Guest Editor
Department of Economics, University of Foggia, 71121 Foggia, Italy
Interests: Game theory, Optimal Control, Mathematics for Economics and Finance, Epidemic Models
Prof. Leon A. Petrosjan
Website
Guest Editor
Faculty of Applied Mathematics and Control Processes, Saint Petersburg State University, 198504 Saint Petersburg, Russia
Interests: Operations Research, Theory of Games, Differential Games, Control Theory

Special Issue Information

Dear Colleagues,

The literature on mathematical models for the spread of epidemics is now huge and has been developed over many decades. Deterministic as well as stochastic approaches have been proposed by scholars and a variety of mathematical models (given by nonlinear equations) have been introduced, analyzed, and applied to real cases.  These models include, among others: ordinary and partial differential equations, integro-differential equations, and stochastic differential equations. More recently, much attention has also been paid to controlling the spread of infectious diseases by formulating suitable optimal control problems. One aspect not yet fully addressed in the control and management of epidemics is that of strategic interactions between public and private decision-makers. One framework in which these problems can be treated is the theory of dynamic games and, in particular, evolutionary games and differential games.

Both reviews and expository and original research articles dealing with the recent advances concerning the optimal control and management of epidemics, with or without strategic interactions, are invited for this Special Issue.

Prof. Leon A. Petrosjan
Prof. Andrea Di Liddo
Prof. Bruno Buonomo
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1200 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.


Keywords

  • epidemic models
  • ordinary differential equations
  • delay differential equations
  • partial differential equations
  • optimal control
  • dynamic games
  • evolutionary games
  • differential games
  • management

Published Papers (8 papers)

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Research

Open AccessArticle
Optimal Control and Treatment of Infectious Diseases. The Case of Huge Treatment Costs
Mathematics 2016, 4(2), 21; https://doi.org/10.3390/math4020021 - 01 Apr 2016
Cited by 6
Abstract
The representation of the cost of a therapy is a key element in the formulation of the optimal control problem for the treatment of infectious diseases. The cost of the treatment is usually modeled by a function of the price and quantity of [...] Read more.
The representation of the cost of a therapy is a key element in the formulation of the optimal control problem for the treatment of infectious diseases. The cost of the treatment is usually modeled by a function of the price and quantity of drugs administered; this function should be the cost as subjectively perceived by the decision-maker. Nevertheless, in literature, the choice of the cost function is often simply done to make the problem more tractable. A specific problem is also given by very expensive therapies in the presence of a very high number of patients to be treated. Firstly, we investigate the optimal treatment of infectious diseases in the simplest case of a two-class population (susceptible and infectious people) and compare the results coming from five different shapes of cost functions. Finally, a model for the treatment of the HCV virus using the blowing-up cost function is investigated. Some numerical simulations are also given. Full article
(This article belongs to the Special Issue Optimal Control and Management of Infectious Diseases)
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Open AccessArticle
Cost Effectiveness Analysis of Optimal Malaria Control Strategies in Kenya
Mathematics 2016, 4(1), 14; https://doi.org/10.3390/math4010014 - 09 Mar 2016
Cited by 6
Abstract
Malaria remains a leading cause of mortality and morbidity among the children under five and pregnant women in sub-Saharan Africa, but it is preventable and controllable provided current recommended interventions are properly implemented. Better utilization of malaria intervention strategies will ensure the gain [...] Read more.
Malaria remains a leading cause of mortality and morbidity among the children under five and pregnant women in sub-Saharan Africa, but it is preventable and controllable provided current recommended interventions are properly implemented. Better utilization of malaria intervention strategies will ensure the gain for the value for money and producing health improvements in the most cost effective way. The purpose of the value for money drive is to develop a better understanding (and better articulation) of costs and results so that more informed, evidence-based choices could be made. Cost effectiveness analysis is carried out to inform decision makers on how to determine where to allocate resources for malaria interventions. This study carries out cost effective analysis of one or all possible combinations of the optimal malaria control strategies (Insecticide Treated Bednets—ITNs, Treatment, Indoor Residual Spray—IRS and Intermittent Preventive Treatment for Pregnant Women—IPTp) for the four different transmission settings in order to assess the extent to which the intervention strategies are beneficial and cost effective. For the four different transmission settings in Kenya the optimal solution for the 15 strategies and their associated effectiveness are computed. Cost-effective analysis using Incremental Cost Effectiveness Ratio (ICER) was done after ranking the strategies in order of the increasing effectiveness (total infections averted). The findings shows that for the endemic regions the combination of ITNs, IRS, and IPTp was the most cost-effective of all the combined strategies developed in this study for malaria disease control and prevention; for the epidemic prone areas is the combination of the treatment and IRS; for seasonal areas is the use of ITNs plus treatment; and for the low risk areas is the use of treatment only. Malaria transmission in Kenya can be minimized through tailor-made intervention strategies for malaria control which produces health improvements in the most cost effective way for different epidemiological zones. This offers the good value for money for the public health programs and can guide in the allocation of malaria control resources for the post-2015 malaria eradication strategies and the achievement of the Sustainable Development Goals. Full article
(This article belongs to the Special Issue Optimal Control and Management of Infectious Diseases)
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Open AccessArticle
Modeling ITNs Usage: Optimal Promotion Programs Versus Pure Voluntary Adoptions
Mathematics 2015, 3(4), 1241-1254; https://doi.org/10.3390/math3041241 - 11 Dec 2015
Cited by 6
Abstract
We consider a mosquito-borne epidemic model, where the adoption by individuals of insecticide–treated bed–nets (ITNs) is taken into account. Motivated by the well documented strong influence of behavioral factors in ITNs usage, we propose a mathematical approach based on the idea of information–dependent [...] Read more.
We consider a mosquito-borne epidemic model, where the adoption by individuals of insecticide–treated bed–nets (ITNs) is taken into account. Motivated by the well documented strong influence of behavioral factors in ITNs usage, we propose a mathematical approach based on the idea of information–dependent epidemic models. We consider the feedback produced by the actions taken by individuals as a consequence of: (i) the information available on the status of the disease in the community where they live; (ii) an optimal health-promotion campaign aimed at encouraging people to use ITNs. The effects on the epidemic dynamics of each of these feedback are assessed and compared with the output of classical models. We show that behavioral changes of individuals may sensibly affect the epidemic dynamics. Full article
(This article belongs to the Special Issue Optimal Control and Management of Infectious Diseases)
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Open AccessArticle
HIV vs. the Immune System: A Differential Game
Mathematics 2015, 3(4), 1139-1170; https://doi.org/10.3390/math3041139 - 03 Dec 2015
Cited by 2
Abstract
A differential game is formulated in order to model the interaction between the immune system and the HIV virus. One player is represented by the immune system of a patient subject to a therapeutic treatment and the other player is the HIV virus. [...] Read more.
A differential game is formulated in order to model the interaction between the immune system and the HIV virus. One player is represented by the immune system of a patient subject to a therapeutic treatment and the other player is the HIV virus. The aim of our study is to determine the optimal therapy that allows to prevent viral replication inside the body, so as to reduce the damage caused to the immune system, and allow greater survival and quality of life. We propose a model that considers all the most common classes of antiretroviral drugs taking into account different immune cells dynamics. We validate the model with numerical simulations, and determine optimal structured treatment interruption (STI) schedules for medications. Full article
(This article belongs to the Special Issue Optimal Control and Management of Infectious Diseases)
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Open AccessArticle
The San Francisco MSM Epidemic: A Retrospective Analysis
Mathematics 2015, 3(4), 1083-1094; https://doi.org/10.3390/math3041083 - 24 Nov 2015
Abstract
We investigate various scenarios for ending the San Francisco MSM (men having sex with men) HIV/AIDS epidemic (1978–1984). We use our previously developed model and explore changes due to prevention strategies such as testing, treatment and reduction of the number of contacts. Here [...] Read more.
We investigate various scenarios for ending the San Francisco MSM (men having sex with men) HIV/AIDS epidemic (1978–1984). We use our previously developed model and explore changes due to prevention strategies such as testing, treatment and reduction of the number of contacts. Here we consider a “what-if” scenario, by comparing different treatment strategies, to determine which factor has the greatest impact on reducing the HIV/AIDS epidemic. The factor determining the future of the epidemic is the reproduction number R0; if R0 < 1, the epidemic is stopped. We show that treatment significantly reduces the total number of infected people. We also investigate the effect a reduction in the number of contacts after seven years, when the HIV/AIDS threat became known, would have had in the population. Both reduction of contacts and treatment alone, however, would not have been enough to bring R0 below one; but when combined, we show that the effective R0 becomes less than one, and therefore the epidemic would have been eradicated. Full article
(This article belongs to the Special Issue Optimal Control and Management of Infectious Diseases)
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Open AccessArticle
Optimal Intervention Strategies for a SEIR Control Model of Ebola Epidemics
Mathematics 2015, 3(4), 961-983; https://doi.org/10.3390/math3040961 - 21 Oct 2015
Cited by 6
Abstract
A SEIR control model describing the Ebola epidemic in a population of a constant size is considered over a given time interval. It contains two intervention control functions reflecting efforts to protect susceptible individuals from infected and exposed individuals. For this model, the [...] Read more.
A SEIR control model describing the Ebola epidemic in a population of a constant size is considered over a given time interval. It contains two intervention control functions reflecting efforts to protect susceptible individuals from infected and exposed individuals. For this model, the problem of minimizing the weighted sum of total fractions of infected and exposed individuals and total costs of intervention control constraints at a given time interval is stated. For the analysis of the corresponding optimal controls, the Pontryagin maximum principle is used. According to it, these controls are bang-bang, and are determined using the same switching function. A linear non-autonomous system of differential equations, to which this function satisfies together with its corresponding auxiliary functions, is found. In order to estimate the number of zeroes of the switching function, the matrix of the linear non-autonomous system is transformed to an upper triangular form on the entire time interval and the generalized Rolle’s theorem is applied to the converted system of differential equations. It is found that the optimal controls of the original problem have at most two switchings. This fact allows the reduction of the original complex optimal control problem to the solution of a much simpler problem of conditional minimization of a function of two variables. Results of the numerical solution to this problem and their detailed analysis are provided. Full article
(This article belongs to the Special Issue Optimal Control and Management of Infectious Diseases)
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Open AccessArticle
Understanding Visceral Leishmaniasis Disease Transmission and its Control—A Study Based on Mathematical Modeling
Mathematics 2015, 3(3), 913-944; https://doi.org/10.3390/math3030913 - 23 Sep 2015
Cited by 4
Abstract
Understanding the transmission and control of visceral leishmaniasis, a neglected tropical disease that manifests in human and animals, still remains a challenging problem globally. To study the nature of disease spread, we have developed a compartment-based mathematical model of zoonotic visceral leishmaniasis transmission [...] Read more.
Understanding the transmission and control of visceral leishmaniasis, a neglected tropical disease that manifests in human and animals, still remains a challenging problem globally. To study the nature of disease spread, we have developed a compartment-based mathematical model of zoonotic visceral leishmaniasis transmission among three different populations—human, animal and sandfly; dividing the human class into asymptomatic, symptomatic, post-kala-azar dermal leishmaniasis and transiently infected. We analyzed this large model for positivity, boundedness and stability around steady states in different diseased and disease-free scenarios and derived the analytical expression for basic reproduction number (R0). Sensitive parameters for each infected population were identified and varied to observe their effects on the steady state. Epidemic threshold R0 was calculated for every parameter variation. Animal population was identified to play a protective role in absorbing infection, thereby controlling the disease spread in human. To test the predictive ability of the model, seasonal fluctuation was incorporated in the birth rate of the sandflies to compare the model predictions with real data. Control scenarios on this real population data were created to predict the degree of control that can be exerted on the sensitive parameters so as to effectively reduce the infected populations. Full article
(This article belongs to the Special Issue Optimal Control and Management of Infectious Diseases)
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Open AccessArticle
A Note on Necessary Optimality Conditions for a Model with Differential Infectivity in a Closed Population
Mathematics 2015, 3(3), 880-890; https://doi.org/10.3390/math3030880 - 21 Aug 2015
Abstract
The aim of this note is to present the necessary optimality conditions for a model (in closed population) of an immunizing disease similar to hepatitis B following. We study the impact of medical tests and controls involved in curing this kind of immunizing [...] Read more.
The aim of this note is to present the necessary optimality conditions for a model (in closed population) of an immunizing disease similar to hepatitis B following. We study the impact of medical tests and controls involved in curing this kind of immunizing disease and deduced a well posed adjoint system if there exists an optimal control. Full article
(This article belongs to the Special Issue Optimal Control and Management of Infectious Diseases)
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