Viable Control of an Epidemiological Model

In mathematical epidemiology, epidemic control often aims at driving the number of infected individuals to zero, asymptotically. However , during the transitory phase, the number of infected can peak at high values. In this paper, we consider mosquito vector control in the Ross-Macdonald epidemiological model, with the goal of capping the proportion of infected by dengue at the peak. We formulate this problem as one of control of a dynamical system under state constraint. We allow for time-dependent fumigation rates to reduce the population of mosquito vector, in order to maintain the proportion of infected individuals by dengue below a threshold for all times. The so-called viability kernel is the set of initial states (mosquitoes and infected individuals) for which such a fumigation control trajectory exists. Depending on whether the cap on the proportion of infected is low, high or medium, we provide different expressions of the viability kernel. We also characterize so-called viable policies that produce, at each time, a fumigation rate as a function of current proportions of infected humans and mosquitoes, such that the proportion of infected humans remains below a threshold for all times. We provide a numerical application in the case of control of a dengue outbreak in 2013 in Cali, Colombia

Optimal control of dengue epidemic outbreaks under limited resources

In this paper, we reflect upon control intervention practices habitually exerted by healthcare authorities in tropical areas that suffer from incidental outbreaks of dengue fever, in particular, the city of Cali, Colombia. Such control interventions, principally based on the insecticide spraying, are carried out sporadically in order to overcome an ongoing epidemic or at least to reduce its size. It is worth pointing out that control actions of this type do not usually account for sufficient budget because epidemic outbreaks are difficult to predict. In practical terms, these occasional control interventions are performed by spraying, as quickly as possible, all existing stock of insecticide (regardless of its lethality) and employing all available manpower. The goal of this paper is to design better strategies for insecticide-based control actions, which are capable of preventing more human infections at no additional cost, and to reveal the obsolescence of current vector eradication practices. Our approach relies on dynamic optimization, where the number of averted human infections is maximized under budget constraint and subject to a simple dengue transmission model amended with one control variable that stands for the insecticide spraying. As a result, we obtain structurally robust control intervention policies that demonstrate better performance and higher resilience to possible budget limitations than traditional modus operand

Dinámica del vih-sida en cali

Para el estudio de la dinámica del VIH-SIDA en la ciudad de Cali, se uso el clásico modelo SIR para la transmisión de enfermedades infecciosas, donde la población total es dividida en Susceptibles, Infectados y Removidos, con los supuestos de que la transmisión sexual es el único medio de contagio y los individuos no se recuperan  y mueren. Basándose en la información suministrada por la Secretaria Municipal de Salud de Santiago de Cali, DANE y Planeación Municipal, se estimaron los parámetros del modelo y del número reproductivo básico. Se realizaron simulaciones de algunos escenarios con el propósito de establecer tendencias a largo plazo de la enfermedad. Se estimaron los puntos de equilibrio del sistema y se analizaron las condiciones de estabilidad, encontrándose que actualmente el sistema posee dos puntos de equilibrio: E1: Libre de la enfermedad, de naturaleza inestable; E2: Endémico, de naturaleza asintóticamente estable. Tomando como condiciones iniciales la información del año 2008, se observo que la enfermedad tiende al equilibrio endémico después de 100 años. A través de simulaciones se logro establecer que al reducir la probabilidad de contagio entre susceptibles e infectados, la enfermedad tiende al equilibrio endémico de forma más lenta y se logra disminuir el  número máximo de infectados y removidos

Ross Macdonald: Un modelo para la dinámica del dengue en Cali, Colombia

El dengue es una infección transmitida por mosquitos que se presenta en todas las regiones tropicales y subtropicales del planeta. En años recientes, la transmisión ha aumentado de manera predominante en zonas urbanas y semiurbanas y se ha convertido en un importante problema de salud pública. El Instituto Nacional de Salud, INS, ubicó a Cali como el municipio con más casos de dengue en Colombia. De acuerdo con el INS, en la ciudad de Cali, hasta la semana epidemiológica 18 (del 28 de abril al 4 de mayo de 2013), se han notificado 5.134 casos de dengue y 171 de dengue grave.En este trabajo se presenta la descripción del modelo Ross-Macdonald, el análisis cualitativo de dicho modelo, y el análisis de sensibilidad del modelo a cambios en sus parámetros. Y base en el ajuste del modelo obtenido para los casos presentados en el 2010, se hacen y analizan simulaciones de posibles escenarios de brotes epidémicos en la ciudad de Cali

Optimal control approach to dengue reduction and prevention in Cali, Colombia

The aim of this paper is to propose optimal strategies for dengue reduction and prevention in Cali, Colombia. For this purpose, we consider two variants of a simple dengue transmission model, epidemic and endemic, each of which is amended with two control variables. These variables express feasible control actions to be taken by an external decision‐maker. First control variable stands for the insecticide spraying and thus targets to suppress the vector population. The second one expresses the protective measures (such as use of repellents, mosquito nets, and insecticide‐treated clothes) that are destined to reduce the number of contacts (bites) between female mosquitoes (principal dengue transmitters) and human individuals. We use the Pontryagin's maximum principle in order to derive the optimal strategies for dengue control and then perform the cost‐effectiveness analysis of these strategies in order to choose the most sustainable one in terms of cost–benefit relationship. Copyright © 2016 John Wiley & Sons, Lt

Robust viability analysis of a controlled epidemiological model

Managing infectious diseases is a world public health issue, plagued by uncertainties. In this paper, we analyze the problem of viable control of a dengue outbreak under uncertainty. For this purpose, we develop a controlled Ross–Macdonald model with mosquito vector control by fumigation, and with uncertainties affecting the dynamics; both controls and uncertainties are supposed to change only once a day, then remain stationary during the day. The robust viability kernel is the set of all initial states such that there exists at least a strategy of insecticide spraying which guarantees that the number of infected individuals remains below a threshold, for all times, and whatever the sequences of uncertainties. Having chosen three nested subsets of uncertainties – a deterministic one (without uncertainty), a medium one and a large one – we can measure the incidence of the uncertainties on the size of the kernel, in particular on its reduction with respect to the deterministic case. The numerical results show that the viability kernel without uncertainties is highly sensitive to the variability of parameters — here the biting rate, the probability of infection to mosquitoes and humans, and the proportion of female mosquitoes per person. So, a robust viability analysis is a possible tool to reveal the importance of uncertainties regarding epidemics contro

The influenza AH1N1: a mathematical model with application of optimal control

Human influenza is transmitted directly from an ill person to a healthy one, by air, during the symptomatic period of the disease. The virulence and antigenicity of the virus, host immunity and environment, interact with each other affecting the transmission of the virus person-person. Since the alert in 2009 of the influenza due to the H1N1 virus, the number of new cases increased despite the control measures implement, such as wearing masks, and other recommendations made by the World Health Organization. In Colombia it was found that the first case of AH1N1 coincided with a person from Mexico. Therefore, and considering that all travelers from this country are a suspected case of the disease, it makes the transit from one country to another a possible route of transmission. Hence, the approach to the question: What has been the impact in Colombia of the outbreak caused by the migration of people from Mexico infected with the H1N1 influenza virus? The porpuse of this paper is, in a certain way, to respond this question using a mathematical model that studies the transmission of this disease in both immigrant and local populations. Population N is divided in NE immigrant populations and NL local population, where N = NE + NL, according to the natural history of H1N1. Each subpopulation is divided into three classes, susceptible S, infectious I and recovered R, resulting in the six compartments SE, IE,RE and SL, IL,RL. The equilibriums and their qualitative analysis were calculated. Besides, the basic reproductive number representing the classical measure of transmission of infectious diseases is estimated and, from a biological point of view, it is defined as the number of secondary cases produced by a typical infected individual when introduced in a fully susceptible host population, during its effective period of infectivity. If R0 1 an outbreak occurs. © 2016 Academic Publications, Lt

Predator-prey model for analysis of aedes aegypti population dynamics in Cali, Colombia

En este trabajo, presentamos un modelo depredador-presa para analizar la dinámica de la población del mosquito Aedes aegypti, al que se atribuye bastante la transmisión del dengue en Cali, Colombia. El modelo describe una interacción de una especie depredadora acuática introducida deliberadamente en el hábitat local de reproducción de mosquitos, donde los estadios inmaduros de Aedes aegypti (huevos, larvas y pupas) ya están presentes y constituyen la población de presas. El modelo también tiene en cuenta la población local de adultos.In this paper, we introduce a predator-prey model to analyze the population dynamics of the Aedes aegypti mosquito which is fairly blamed for the transmission of dengue in Cali, Colombia. The model describes an interaction of an aquatic predacious species deliberately introduced into local mosquito breeding habitat, where the Aedes aegypti immature stages (eggs, larvae, and pupae) are already present and constitute the prey population. The model also accounts for local population of adult female mosquitoes (or mature stage) emerging from the breeding site. This population is considered as a target for reduction by deploying an adequate predacious species since only female mosquitoes are held responsible for transmission of dengue and other vector-borne diseases. Having analyzed the model, we have derived the mosquito survival threshold with predation as a function of predator’s biological characteristics. The model’s parameters were adjusted to the average seasonal temperatures of Cali, Colombia and explicit conditions for biological characteristics of prospective efficient predators were established. Numerical simulation with introduction of an efficient predacious species in local mosquito breeding habitats revealed the possibility of eventual mosquito extinction in such localities. Finally, some particular biological species were proposed as potential candidates for efficient predator

Parameter estimation of two mathematical models for the dynamics of dengue and its vector in Cali, Colombia

El dengue es una infección viral transmitida por la hembra del mosquitoAedes aegyptique se presenta en todas las regiones tropicales y subtro-picales del planeta. En Cali, Colombia, a pesar de los controles que lasautoridades de salud dicen estar haciendo, durante el año 2013, se reporta-ron más de 9.000 casos de dengue, de los cuales algunos han sido graves yotros han llegado a ser letales. Para la transmisión del virus del dengue, losmodelos matemáticos que simulan la dinámica de la población infectada,bien sea de humanos, de mosquitos o de ambos, permiten una buena com-prensión de la dinámica del virus, por lo que son una excelente herramienta para el seguimiento y control de la enfermedad causada por ellos. Sin em-bargo, para que esta herramienta sea realmente útil en casos concretos, losmodelos deben ser ajustados a las características particulares de la regióndonde se quieren utilizar. En este artículo, queremos presentar el ajuste dedos modelos matemáticos al área urbana de la ciudad de Cali, Colombia.Inicialmente, con base en el comportamiento natural del mosquitoAedesaegyptien una región como el área de interés, estimamos algunos de losparámetros de los modelos, teniendo en cuenta la literatura existente so-bre este tema. Posteriormente, estimamos el resto de parámetros como lasolución de mínimos cuadrados que mejor ajusta los resultados de los mo-delos a los datos de los casos reportados de dengue, según la SecretariaMunicipal de Salud de Cali, durante el año 2010Dengue is a viral infection transmitted by the femaleAedes aegyptimosquitothat lives in all tropical and subtropical regions of the planet. In Cali,Colombia, despite the controls that health officials say they are doing, thisyear, there have been more than 9,000 cases of dengue, of which somehave been serious and others have become lethal. For the case of denguevirus transmission, mathematical models to simulate the dynamics of theinfected population, either human or mosquito or both, allow a good un-derstanding of the dynamics of the virus, so they are a good tool for moni-toring and controlling the disease. However, for this tool to be really usefulin a specific case, the models must be tailored to the particular character-istics of the region where you want to use them. In this paper, we presentthe fitting of two mathematical models to the urban area of Cali, Colom-bia. Initially, based on the natural behavior of theAedes aegyptimosquitoin a region as the area of interest, we estimate some of the parametersof the models, taking into account the existing literature on this topic.Subsequently, we estimate the others parameters as the (non-linear) leastsquares solution that best fits the output of the models to the data of re-ported cases of dengue, according to the Municipal Secretary of Cali, in201

Ross-Macdonald: a model for the dengue dynamic in Cali, Colombia

El dengue es una infección transmitida por mosquitos que se presenta en todas las regiones tropicales y subtropicales del planeta. En años recientes, la transmisión ha aumentado de manera predominante en zonas urbanas y semiurbanas y se ha convertido en un importante problema de salud pública. El Instituto Nacional de Salud-INS, ubicó a Cali como el municipio con más casos de dengue en Colombia. De acuerdo con el INS, en la ciudad de Cali, hasta la semana epidemiológica 18 (del 28 de abril al 4 de mayo de 2013), se han notificado 5 134 casos de dengue y 171 de dengue grave. En este trabajo se presenta la descripción del modelo Ross McDonald, el análisis cualitativo de dicho modelo, y el análisis de sensibilidad del modelo a cambios en sus parámetros. Y base en el ajuste del modelo obtenido para los casos presentados en el 2010, se hacen y analizan simulaciones de posibles escenarios de brotes epidémicos en la ciudad de Cali.Dengue is an infection transmitted by mosquitoes and is present in all tropical and subtropical regions of the planet. In recent years, the transmission of the disease has increased, predominantly in urban and semi-urban areas, and has become an important public health problem. The National Health Institute (Instituto Nacional de Salud-INS) determined Cali to be the municipality with the most cases of dengue in Colombia. According to the INS, up to epidemiological week 18 (April 28 to May 4, 2013) 5 134 cases of dengue —and 171 cases of severe dengue— have been reported. This study presents a description of the Ross-Macdonald model, and qualitative analysis of this model, and an analysis of the sensitivity of the model to changes in its parameters. Based on the adjustment of the model obtained for cases that occurred in 2010, simulations of possible scenarios of epidemic outbreaks in the city of Cali are created and analyze