An epidemic model for cholera with optimal control treatment

This research was supported by the Portuguese Foundation for Science and Technology (FCT) within projects UID/MAT/04106/2013 (CIDMA) and PTDC/EEI-AUT/2933/2014 (TOCCATA), funded by Project 3599 - Promover a Produção Científica e Desenvolvimento Tecnológico e a Constituição de Redes Temáticas and FEDER funds through COMPETE 2020, Programa Operacional Competitividade e Internacionalização (POCI). Lemos-Paião is also supported by the Ph.D. fellowship PD/BD/114184/2016; Silva by the post-doc fellowship SFRH/BPD/72061/2010.We propose a mathematical model for cholera with treatment through quarantine. The model is shown to be both epidemiologically and mathematically well posed. In particular, we prove that all solutions of the model are positive and bounded; and that every solution with initial conditions in a certain meaningful set remains in that set for all time. The existence of unique disease-free and endemic equilibrium points is proved and the basic reproduction number is computed. Then, we study the local asymptotic stability of these equilibrium points. An optimal control problem is proposed and analyzed, whose goal is to obtain a successful treatment through quarantine. We provide the optimal quarantine strategy for the minimization of the number of infectious individuals and bacteria concentration, as well as the costs associated with the quarantine. Finally, a numerical simulation of the cholera outbreak in the Department of Artibonite (Haiti), in 2010, is carried out, illustrating the usefulness of the model and its analysis. © 2016 Elsevier B.V

KONTROL ADAPTIF PADA MODEL PENYEBARAN KOLERA DENGAN ADANYA KETIDAKPASTIAN PARAMETER

In this paper the number of  humans infected with cholera was controlled under the uncertainty in cholera model parameters. The aim of this research is to design an adaptive control so that the number of infected humans decreases. To achieve this goal, an adaptive controller was proposed to a deterministic model for the transmission of cholera involving five state variables (susceptible humans, infected humans, quarantined humans, recovered humans, and bacterial concentration) and one input control variable, i.e, the proportion of quarantined humans. A control law was designed such that the number of infected humans was decreased tracking the given reference function. The tracking error convergence were analyzed by employing the  Lyapunov theorem. The performance of the proposed controller was evaluated through numerical simulations. The results show that the adaptive controller designed to the model ensures the tracking error convergence such that the number of infected humans has declined

OPTIMAL CONTROL ON CHOLERA DISEASE SPREADING MODEL WITH THREE VARIABLES CONTROL VARIATION

Cholera is an infection of the small intestine by some strains of the bacterium Vibrio Cholerae. This disease is a deadly disease that necessitates efficient prevention and control measures. In this research, the optimal control of the cholera spread model with variations of three control variables is discussed. There are four controls to minimize the spread of diseases such as sanitation, treatment consisting of quarantine, increased education, and chlorination. The dynamic system is formed with three controls variation. Then it is compared and analyzed for the most effective result. The optimal control solution is derived using the Pontryagin Minimum Principle and solved using the Runge-Kutta method

Optimal Control with Treatment and Water Sanitation for Cholera Epidemic Model

This paper proposes a mathematical model for cholera using optimal control of treatment through quarantine and water sanitation. Cholera is acute diarrhoea caused by Vibrio cholera bacteria infecting the intestinal tract. The analysis related to the spread of this disease is carried out through a mathematical approach. The constructed mathematical model is demonstrated epidemiologically. The proposed optimal control is the treatment of infected individuals during the quarantine period and sanitation, namely environmental hygiene, especially water. This strategy aims to suppress the number of infected individuals and reduce the concentration of bacteria due to cholera disease. To solve the optimal control problem, we employ the 4th-order forward-backward Runge-Kutta method. Based on the simulation results, the number of individuals infected by cholera and the concentration of bacteria decreased significantly. Moreover, the proposed method can transfer infected to recovered individuals faster than an optimal control treatment

Optimal Control Problem for Cholera Disease and Cost-Effectiveness Analysis

Cholera is a disease that continues to be a threat to public health globally and is an indicator of inequity and lack of social development in countries. For this reason, strategies for its control need to be investigated. In this work, an optimal control problem related to cholera disease was formulated by introducing personal protection, drug treatment and water sanitation as control strategies. First, the existence and characterization of controls to minimize the performance index or cost function was proved by using classic control theory. Then, the theoretical results were validated with numerical experiments by using data reported in the literature. Finally, the effectiveness and efficiency of the proposed controls were determined through a cost-effectiveness analysis. The results showed that the use of the three controls simultaneously is the cheapest and most effective strategy to control the disease

Fractional-order modelling and optimal control of cholera transmission

A Caputo-type fractional-order mathematical model for “metapopulation cholera transmission” was recently proposed in [Chaos Solitons Fractals 117 (2018), 37–49]. A sensitivity analysis of that model is done here to show the accuracy relevance of parameter estimation. Then, a fractional optimal control (FOC) problem is formulated and numerically solved. A cost-effectiveness analysis is performed to assess the relevance of studied control measures. Moreover, such analysis allows us to assess the cost and effectiveness of the control measures during intervention. We conclude that the FOC system is more effective only in part of the time interval. For this reason, we propose a system where the derivative order varies along the time interval, being fractional or classical when more advantageous. Such variable-order fractional model, that we call a FractInt system, shows to be the most effective in the control of the disease.publishe

The risk of contagion spreading and its optimal control in the economy

The global crisis of 2008 provoked a heightened interest among scientists to study the phenomenon, its propagation and negative consequences. The process of modelling the spread of a virus is commonly used in epidemiology. Conceptually, the spread of a disease among a population is similar to the contagion process in economy. This similarity allows considering the contagion in the world financial system using the same mathematical model of infection spread that is often used in epidemiology. Our research focuses on the dynamic behaviour of contagion spreading in the global financial network. The effect of infection by a systemic spread of risks in the network of national banking systems of countries is tested. An optimal control problem is then formulated to simulate a control that may avoid significant financial losses. The results show that the proposed approach describes well the reality of the world economy, and emphasizes the importance of international relations between countries on the financial stability.publishe

Parameter estimation, sensitivity analysis and optimal control of a periodic epidemic model with application to HRSV in Florida

A state wide Human Respiratory Syncytial Virus (HRSV) surveillance system was implemented in Florida in 1999 to support clinical decision-making for prophylaxis of premature infants. The research presented in this paper addresses the problem of fitting real data collected by the Florida HRSV surveillance system by using a periodic SEIRS mathematical model. A sensitivity and cost-effectiveness analysis of the model is done and an optimal control problem is formulated and solved with treatment as the control variable.publishe

New algorithms for solving high-dimensional time-dependent optimal control problems and their applications in infectious disease models

Doctor of PhilosophyDepartment of Industrial & Manufacturing Systems EngineeringChih-Hang 'John' WuInfectious diseases have been the primary cause of human death worldwide nowadays. The optimal control strategy for infectious disease has attracted increasing attention, becoming a significant issue in the healthcare domain. Optimal control of diseases can affect the progression of diseases and achieve high-quality healthcare. In previous studies, massive efforts on the optimal control of diseases have been made. However, some infectious diseases' mortality is still high and even developed into the second highest cause of mortality in the US. According to the limitations in existing research, this research aims to study the optimal control strategy via some industrial engineering techniques such as mathematical modeling, optimization algorithm, analysis, and numerical simulation. To better understand the optimal control strategy, two infectious disease models (epidemic disease, sepsis) are studied. Complex nonlinear time-series and high-dimensional infectious disease control models are developed to study the transmission and optimal control of deterministic SEIR or stochastic SIS epidemic diseases. In addition, a stochastic sepsis control model is introduced to study the progression and optimal control for sepsis system considering possible medical measurement errors or system uncertainty. Moreover, an improved complex nonlinear sepsis model is presented to more accurately study the sepsis progression and optimal control for sepsis system. In this dissertation, some analysis methods such as stability analysis, bifurcation analysis, and sensitivity analysis are utilized to help reader better understand the model behavior and the effectiveness of the optimal control. The significant contributions of this dissertation are developing or improving nonlinear complex disease optimal control models and proposing several effective and efficient optimization algorithms to solve the optimal control in those researched disease models, such as an optimization algorithm combining machine learning (EBOC), an improved Bayesian Optimization algorithm (IBO algorithm), a novel high-dimensional Bayesian Optimization algorithm combining dimension reduction and dimension fill-in (DR-DF BO algorithm), and a high-dimensional Bayesian Optimization algorithm combining Recurrent Neural Network (RNN-BO algorithm). Those algorithms can solve the optimal control solution for complex nonlinear time-series and high-dimensional systems. On top of that, numerical simulation is used to demonstrate the effectiveness and efficiency of the proposed algorithms

A framework to support the decision-making process for modelling of communicable diseases

Thesis (MEng)--Stellenbosch University, 2019.ENGLISH ABSTRACT: Infectious disease outbreaks have the potential to disrupt and strain the global health care system, even more so when a localised disease outbreak propagates rapidly to a large area. Such a disease outbreak is referred to as a pandemic disease outbreak. Pandemic outbreaks often inspire global collaboration between researchers and modelling practitioners with a view to devise strategies, disease propagation models and actions on how to address the outbreak. Modelling of infectious disease is a complex endeavour. The literature on the available modelling approaches and general application to disease modelling is well documented in the literature. What is, however, less evident, especially to a modelling practitioner with less rigorous modelling experience, is the selection and consideration of modelling considerations based on the specific context of the disease outbreak. To address this challenge, a modelling support framework is designed in this research project, with a view to formalise the most salient universal modelling steps and assist novice modelling practitioners in the consideration and selection of appropriate approaches for modelling infectious diseases. The research consists of three phases, namely the design and execution of a structured literature review, analysis of the findings of the literature review, and the construction of a modelling support and guidance framework. During the first phase of the research, the chain of infection is used as an overarching metaphor to guide the process in identifying relevant considerations, disease characteristics and contextual factors which may potentially affect disease propagation, and this is used as the basis for determining the scope of the structured literature review. The review is designed to construct a sufficiently detailed dataset which is well representative of the various modelling approaches as applied in literature. The 283 identified literature pieces are methodically analysed and the relevant modelling considerations, disease characteristics and contextual factors from each of the pieces are captured to the dataset. During the second phase of the research the dataset is analysed. The modelling considerations are analysed in relation to the disease transmission mode, and the relationship between modelling considerations are also analysed. In general, the selection of modelling approaches and considerations were not reducible to a single factor. This suggests that numerous factors must be considered in the model decision making process, and additionally, it highlights the importance of contextualising the disease outbreak. The third phase of the research consists of the framework construction. Both the first and the second phases of the research are used to inform and guide the framework construction. The framework is constructed with two goals in mind, namely to inform modelling considerations from a holistic viewpoint and to aid in the selection of the relevant modelling considerations. The framework use is verified with an illustrative case study and validated with semi-structured interviews that are conducted with external subject matter experts with a background in engineering and health care modelling.AFRIKAANSE OPSOMMING: Die uitbreek van ’n aansteeklike siekte het die potensiaal om die globale gesondheidsorgsisteem te ontwrig en onder geweldige druk te plaas, des te meer wanneer so ’n gelokaliseerde uitbreking spoedig na ’n groter area versprei. Sulke siekte-uitbrekings staan bekend as pandemiese siektes. Die ontstaan van pandemiese uitbrekings van siektes lei tipies tot wêreldwye samewerking tussen navorsers en modelleerders. Die doel van samewerking hou verband met die skep van strategieë, modelle wat siekte-oordrag modelleer en aksieplanne om die uitbreking te bestuur. Die modellering van aansteeklike siektes is ’n komplekse onderneming. Beskikbare modellerings-benaderings en die generiese gebruik daarvan om siektes te modelleer is goed opgeteken in die literatuur. Wat minder ooglopend is van hierdie benaderings, veral vir die modelleerder met elementêre modelleringskennis, is die oorweging en selektering van modelleringelemente gebaseer op die spesifieke kontekstuele omstandighede van die siekte-uitbreking. Om hierdie uitdaging aan te pak word daar in hierdie navorsingsprojek ’n ondersteuningsraamwerk vir modellering geskep. Die doel hiervan is die formalisering van die belangrikste modellerings-stappe en om onervare modelleerders te ondersteun in die oorweging en selektering van toepaslike benaderings om aansteeklike siektes te modelleer. Die navorsing bestaan uit drie fases, naamlik die ontwerp en uitvoering van ’n gestruktureerde literatuuroorsig, ’n analise van die bevindinge van die literatuuroorsig, en die opstel van ’n raamwerk wat ondersteuning en raadgewing ten opsigte van modellering bied. As deel van die eerste fase van die navorsing, word die ketting van infeksie as ’n oorhoofse metafoor gebruik. Hierdie metafoor word gebruik om relevante oorwegings, siekte-eienskappe en kontekstuele faktore te identifiseer wat die potensiaal het om die verspreiding van siektes te beïnvloed. Dit word ook as die basis gebruik om die bestek van die gestruktureerde literatuuroorsig te bepaal. Die gestruktureerde literatuuroorsig is ontwerp om ’n gedetailleerde datastel op te stel wat ’n goeie verteenwoordiging is van die verskeie modelleringsbenaderings soos dit in die literatuur toegepas is. Die geïdentifiseerde 283 literatuurstukke is stapsgewys geanaliseer en die relevante modelleringsbenaderings, siekte-eienskappe en kontekstuele faktore van die literatuurstukke is in die datastel opgeneem. As deel van die tweede fase van die navorsing word die datastel geanaliseer. Die modelleringsoorwegings is geanaliseer met betrekking tot die siekte-oordragsmetode en die verhoudings tussen ander modelleringsoorwegings. Oor die algemeen is daar bevind dat die keuse van ’n modelleringsbenadering of -oorweging nie reduseerbaar is tot die oorweging van ’n enkele faktor nie. Die afleiding is dus dat verskeie faktore in ag geneem moet word in die seleksieproses van ’n modelleringsbenadering, en dat die belangrikheid van die kontekstualisering van ’n siekte-uitbreking benadruk moet word. As deel van die derde fase van die navorsing is die raamwerk opgestel. Beide die eerste en tweede fases van die navorsing is gebruik om die opstelproses van die raamwerk te lei en die opstelkeuses in te lig. Die raamwerk is opgestel met twee verwagte uitkomstes, naamlik om die modellerings-oorwegings vanuit ’n holistiese oogpunt in te lig, sowel as om die selektering van relevante modelleringsoorwegings te ondersteun. Die gebruik van die raamwerk is geverifieer met behulp van ’n verduidelikende gevallestudie. Die validasie is voltooi met behulp van semi-gestruktureerde onderhoude met eksterne vakgebied-kenners met ’n agtergrond in die ingenieurswese en gesondheidssorg-modelleringsvelde