A theoretical model for optimal control of banana Moko ( Musa AAB Simmonds).

A population simulation model with non-linear ordinary differential equations is presented, which interprets the dynamics of the banana Moko, with prevention of the disease and population of susceptible and infected plants over time. A crop with a variable population of plants and a logistic growth of replanting is assumed, taking into account the maximum capacity of plants in the delimited study area.

[Adjusted force of Dengue infection]. / Ajuste de la fuerza de infección del dengue.

OBJECTIVE: To propose a vector-host mathematical model for adjusting the force of infection with dengue in a variable population with logistic growth and a mosquito life cycle with periodic oviposition rate, considering immunity to a serotype. METHODS: The proposed mathematical model is represented by eight differential equations that calculate the force of infection through the scale-free network method. RESULTS: A simulation of the mathematical model solved by an algorithm implemented in MATLAB based on data obtained from the literature was obtained. The growth of the force of dengue infection over time can be observed in the simulation, and it varies as some parameters change. CONCLUSIONS: The force of infection increases over time, that is, the rate of new cases increases, which proves the need for prevention among inhabitants of high-risk areas through the use of metallic fabrics, mosquito nets, repellents, appropriate clothing, among other measures. Chemical control against the vector, such as larvicides and adulticides, as well as the elimination of breeding places to interrupt their life cycle are strongly advised.

OBJETIVO: Plantear un modelo matemático hospedero vector para el ajuste de la fuerza de infección del dengue en una población variable con crecimiento logístico y ciclo de vida del mosquito con tasa de ovoposición periódica, considerando inmunidad a un serotipo. MÉTODOS: El modelo matemático propuesto está representado por ocho ecuaciones diferenciales a las que se les calcula la fuerza de infección por el método de las redes libres de escala. RESULTADOS: Se presenta una simulación del modelo matemático resuelto mediante un algoritmo implementado en el software MATLAB con datos obtenidos de la literatura. En la simulación se puede observar el crecimiento de la fuerza de infección del dengue a través del tiempo, donde esta varía de acuerdo al cambio de algunos parámetros. CONCLUSIONES: La fuerza de la infección aumenta en el tiempo, es decir, la tasa de nuevos casos crece, mostrando la necesidad de prevención en las personas, mediante el uso de telas metálicas, mosquiteros, repelentes, ropa adecuada entre otras medidas; control químico como larvicidas y adulticidas contra el vector, como también la eliminación de criaderos que interrumpan su ciclo de vida.

[Mathematical modelling of an infectious disease in a prison setting and optimal preventative control strategies]. / Modelado matemático de una enfermedad infecciosa en un centro de reclusión y estrategias óptimas de control preventivo.

A mathematical model was constructed for modelling transmission dynamics and the evolution of an infectious disease in a prison setting, considering asymptomatic infectious people, symptomatic infectious people and isolated infectious people. The model was proposed as a nonlinear differential equation system for describing disease epidemiology. The model's stability was analysed for including a preventative control strategy which would enable finding a suitable basic reproduction number-based control protocol. A cost function related to the system of differential equations was formulated to minimise infectious populations and intervention costs; such function was minimised by using the Pontryagin maximum principle which determines optimum preventative control strategies by minimising both infectious populations and associated costs. A numerical analysis of the model was made, considering preventative control effectiveness levels and different control weighting constants. Conclusions were drawn. The basic reproduction number characterises system stability and leads to determining clear control criteria; a preventative control threshold was defined, based on the controlled basic reproduction number which enabled deducing that disease control requires uniform preventative control involving high rates of effectiveness.

Ajuste de la fuerza de infección del dengue / Adjusted force of Dengue infection

RESUMEN Objetivo Plantear un modelo matemático hospedero vector para el ajuste de la fuerza de infección del dengue en una población variable con crecimiento logístico y ciclo de vida del mosquito con tasa de ovoposición periódica, considerando inmunidad a un serotipo. Métodos El modelo matemático propuesto está representado por ocho ecuaciones diferenciales a las que se les calcula la fuerza de infección por el método de las redes libres de escala. Resultados Se presenta una simulación del modelo matemático resuelto mediante un algoritmo implementado en el software MATLAB con datos obtenidos de la literatura. En la simulación se puede observar el crecimiento de la fuerza de infección del dengue a través del tiempo, donde esta varía de acuerdo al cambio de algunos parámetros. Conclusiones La fuerza de la infección aumenta en el tiempo, es decir, la tasa de nuevos casos crece, mostrando la necesidad de prevención en las personas, mediante el uso de telas metálicas, mosquiteros, repelentes, ropa adecuada entre otras medidas; control químico como larvicidas y adulticidas contra el vector, como también la eliminación de criaderos que interrumpan su ciclo de vida.(AU)

ABSTRACT Objective To propose a vector-host mathematical model for adjusting the force of infection with dengue in a variable population with logistic growth and a mosquito life cycle with periodic oviposition rate, considering immunity to a serotype. Methods The proposed mathematical model is represented by eight differential equations that calculate the force of infection through the scale-free network method. Results A simulation of the mathematical model solved by an algorithm implemented in MATLAB based on data obtained from the literature was obtained. The growth of the force of dengue infection over time can be observed in the simulation, and it varies as some parameters change. Conclusions The force of infection increases over time, that is, the rate of new cases increases, which proves the need for prevention among inhabitants of high-risk areas through the use of metallic fabrics, mosquito nets, repellents, appropriate clothing, among other measures. Chemical control against the vector, such as larvicides and adulticides, as well as the elimination of breeding places to interrupt their life cycle are strongly advised.(AU)

[A mathematical model for controlling the spread of dengue]. / Modelo matemático para el control de la transmisión del Dengue.

OBJECTIVE: A mathematical model is presented in this paper showing the dynamics of dengue transmission. The goal was to studyAedes aegyptipopulation behaviour and that of affected people to scientifically support the decision-making involved in surveillance and control programmes. METHODS: The proposed mathematical model involved eight differential equations having constant delays; each represented each population's variation either in humans or the mosquito vector. RESULTS: Two of the mathematical model's simulation scenarios are presented; they were solved by means of an algorithm implemented in MATLAB software. The data was obtained from the Colombian Statistics Department (DANE), the World Health Organisation (WHO) and from a review of the pertinent literature. The data regarding human and vector populations was analysed (with and without using controls). CONCLUSIONS: The proposed mathematical model was able to simulate the dynamics of dengue transmission; it simulated the population-related behaviour ofAedes aegyptiand the affected people. This model could be a tool for scientifically supporting surveillance and control programmes' decision-making.

[A mathematical model for the chemical control of Aedes aegypti (Diptera: Culicidae) having acquired chemical resistance]. / Modelo matemático para el control químico con resistencia del Aedes aegypti (Diptera: Culicidae).

Dengue fever is a common vector-borne disease in tropical and subtropical areas. It is transmitted to humans by the bite of an infected female Aedes mosquito. Since no vaccines are currently available which can protect against infection, disease control relies on controlling the mosquito population. This work was aimed at modelling such mosquito's population dynamics regarding chemical control of the adult population and its acquired resistance to chemicals. The model was analysed by using classical dynamic system theory techniques and mosquito growth threshold was determined as this establishes when a particular population may prosper in the environment or when it is likely to disappear. A suitable chemical control strategy was developed from such threshold. Simulations were made in control and non-control scenarios; this determined the degree of control application effectiveness against different levels of acquired resistance.

[A mathematical model representing HIV/AIDS transmission dynamics in a sexually-active population]. / Modelo matemático para la dinámica de transmisión del VIH/SIDA en una población sexualmente activa.

This article presents a new model explaining acquired immunodeficiency syndrome (AIDS) transmission dynamics amongst heterosexually active couples. It covers the assumptions made, the variables analysed, the model's sensitivity and the ordinary differential equations and control strategies used. The information was obtained from the Colombian state Statistics Department (DANE) and applied to different simulations in the system (with and without control), using the MAPLE programme code. Simulation results led to concluding that control using condoms was relevant in the model. It was not important if control were applied in women or men, nor was the percentage of sexually-active men and women.

[Classical dengue transmission dynamics involving mechanical control and prophylaxis]. / Dinámica de transmisión del Dengue clásico con control mecánico y profilaxis.

Dengue fever transmission dynamics were studied in an endemic region considering the use of preventative measures and mechanical control in reducing transmission of the disease. A system of ordinary differential equations was proposed, describing the dynamics and their evolution as determined by numerical simulation. Different mechanical control and prophylaxis strategies were compared to the situation without control. The basic reproduction number R0 was determined R0 to show that if R0 > 1 there would be a risk of an epidemic and otherwise the disease would have low impact levels. The basic reproduction number helps determine the dynamics' future pattern and contrast the results so obtained with those obtained numerically. It was concluded that although prophylaxis and mechanical control alone provide effective results in controlling the disease, if both controls are combined then infection levels become significantly reduced. Around 60 % mechanical control and prevention levels are needed to provide suitable results in controlling dengue outbreaks.

Modelo matemático para el control de la transmisión del Dengue / A mathematical model for controlling the spread of dengue

Objetivo: En este trabajo se presenta un modelo matemático que muestra la dinámica de transmisión del dengue, con el objetivo de estudiar el comportamiento de las poblaciones delAedes aegyptiy de las personas afectadas. Este modelo puede ser tenido en cuenta por los programas de vigilancia y control a la hora de tomar decisiones. Métodos: El modelo matemático propuesto está representado por ocho ecuaciones diferenciales con retardos constantes. Cada ecuación representa la variación de cada subpoblación tanto en los humanos como en el mosquito transmisor. Resultados: Se presentan dos escenarios de simulación del modelo matemático resueltos mediante un algoritmo implementado en el software MATLAB, con datos obtenidos del Departamento Nacional de Estadísticas de Colombia (DANE), la Organización Mundial de la Salud (OMS) y la revisión de literatura. En cada escenario se analizan tanto la población humana como la del mosquito, con la utilización o no de controles. Conclusiones: El modelo matemático propuesto es capaz de simular la dinámica de transmisión del dengue, muestra el comportamiento de las poblaciones delAedes aegyptiy de las personas afectadas y puede ser una herramienta a tener en cuenta para apoyar de forma científica la toma de decisiones en los programas de vigilancia y control.

Objective: A mathematical model is presented in this paper showing the dynamics of dengue transmission. The goal was to studyAedes aegyptipopulation behaviour and that of affected people to scientifically support the decision-making involved in surveillance and control programmes. Methods: The proposed mathematical model involved eight differential equations having constant delays; each represented each population's variation either in humans or the mosquito vector. Results: Two of the mathematical model's simulation scenarios are presented; they were solved by means of an algorithm implemented in MATLAB software. The data was obtained from the Colombian Statistics Department (DANE), the World Health Organisation (WHO) and from a review of the pertinent literature. The data regarding human and vector populations was analysed (with and without using controls). Conclusions: The proposed mathematical model was able to simulate the dynamics of dengue transmission; it simulated the population-related behaviour ofAedes aegyptiand the affected people. This model could be a tool for scientifically supporting surveillance and control programmes' decision-making.

Modelo matemático para la dinámica de transmisión del VIH/SIDA en una población sexualmente activa / A mathematical model representing HIV/AIDS transmission dynamics in a sexually-active population

Se presenta un nuevo modelo que explica la dinámica de transmisión de la enfermedad del Síndrome de Inmunodeficiencia Adquirida entre parejas heterosexualmente activas. Se incluyen los supuestos, las variables, las ecuaciones diferenciales ordinarias, el análisis de sensibilidad del modelo y las estrategias de control. Los datos fueron obtenidos del Departamento Nacional de Estadísticas y se llevaron a cabo diferentes simulaciones sin control y con control en el sistema, utilizando el código del programa MAPLE. Los resultados obtenidos en las simulaciones permiten concluir que el control con preservativos es relevante en el modelo, así se lleve a cabo sobre la población de hombres o mujeres y sin importar la proporción de hombres y mujeres sexualmente activa.

This article presents a new model explaining acquired immunodeficiency syndrome (AIDS) transmission dynamics amongst heterosexually active couples. It covers the assumptions made, the variables analysed, the model's sensitivity and the ordinary differential equations and control strategies used. The information was obtained from the Colombian state Statistics Department (DANE) and applied to different simulations in the system (with and without control), using the MAPLE programme code. Simulation results led to concluding that control using condoms was relevant in the model. It was not important if control were applied in women or men, nor was the percentage of sexually-active men and women.

Dinámica de transmisión del Dengue clásico con control mecánico y profilaxis / Classical dengue transmission dynamics involving mechanical control and prophylaxis

Se modela la dinámica de transmisión del dengue clásico en una región endémica considerando el uso de medidas preventivas y de control mecánico en la reducción de la transmisión de la enfermedad. Se plantea un sistema de ecuaciones diferenciales ordinarias que describe la dinámica y mediante simulación numérica se determina su evolución en el tiempo. Se comparan diferentes estrategias de control mecánico y profilaxis con la situación sin control. Se determina el número básico de reproducción R0, mostrando que si R0 > 1 hay un alto riesgo de epidemia y que en caso contrario la enfermedad se mantiene en niveles de bajo impacto; estos resultados se contratan con los obtenidos numéricamente. Se concluye que si bien la profilaxis y el control mecánico por si solos brindan resultados efectivos en el control de la enfermedad, cuando se combinan ambos controles los niveles de infección se ven reducidos significativamente. Niveles de control mecánico y profilaxis cercanos al 60 por ciento son los que brindan resultados adecuados en el control del brote de dengue.

Dengue fever transmission dynamics were studied in an endemic region considering the use of preventative measures and mechanical control in reducing transmission of the disease. A system of ordinary differential equations was proposed, describing the dynamics and their evolution as determined by numerical simulation. Different mechanical control and prophylaxis strategies were compared to the situation without control. The basic reproduction number R0 was determined R0 to show that if R0 > 1 there would be a risk of an epidemic and otherwise the disease would have low impact levels. The basic reproduction number helps determine the dynamics' future pattern and contrast the results so obtained with those obtained numerically. It was concluded that although prophylaxis and mechanical control alone provide effective results in controlling the disease, if both controls are combined then infection levels become significantly reduced. Around 60 percent mechanical control and prevention levels are needed to provide suitable results in controlling dengue outbreaks.

Modelo matemático para el control químico con resistencia del Aedes aegypti (Diptera: Culicidae) / A mathematical model for the chemical control of Aedes aegypti (Diptera: Culicidae) having acquired chemical resistance

El dengue es una enfermedad viral común en zonas tropicales y subtropicales transmitida por mosquitos del género Aedes. El virus es transmitido a los humanos por la picadura de un mosquito hembra infectado. Ya que no existen vacunas que protejan contra la infección, el control de la enfermedad se hace controlando la población adulta o inmadura del mosquito. En este trabajo se modela la dinámica de crecimiento del mosquito sometido a control adulticida y con resistencia al químico. Se hace el análisis del modelo mediante análisis clásico de estabilidad local de sistemas dinámicos, lo que permite determinar el umbral de crecimiento del mosquito y a partir de éste establecer una estrategia adecuada de control químico. Se incluye la simulación numérica para diferentes escenarios con el fin de evaluar si hay diferencias en el comportamiento del sistema cuando la resistencia está presente y cuando no lo está.

Dengue fever is a common vector-borne disease in tropical and subtropical areas. It is transmitted to humans by the bite of an infected female Aedes mosquito. Since no vaccines are currently available which can protect against infection, disease control relies on controlling the mosquito population. This work was aimed at modelling such mosquito's population dynamics regarding chemical control of the adult population and its acquired resistance to chemicals. The model was analysed by using classical dynamic system theory techniques and mosquito growth threshold was determined as this establishes when a particular population may prosper in the environment or when it is likely to disappear. A suitable chemical control strategy was developed from such threshold. Simulations were made in control and non-control scenarios; this determined the degree of control application effectiveness against different levels of acquired resistance.