Mathematical modelling of an infectious disease in a prison setting and optimal preventative control strategies/Modelado matematico de una enfermedad infecciosa en un centro de reclusion y estrategias optimas 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. Key Words: Public health, population dynamics, basic reproduction number, communicable disease control (source: MeSH, NLM). Se construye un modelo matematico para la dinamica de transmision y evolucion de una enfermedad infecciosa en una carcel, considerando infecciosos asintomaticos, infecciosos sintomaticos e infecciosos aislados. El modelo se propone como un sistema de ecuaciones diferenciales no lineales que describe los aspectos epidemiologicos de la dinamica. Se realiza el analisis de estabilidad del modelo para posteriormente incluir en su formulacion una estrategia de control preventivo, que permite establecer un protocolo adecuado de control con base en el numero basico de reproduccion. Con el fin de minimizar las poblaciones infecciosas y los costos de aplicar la intervencion, se formula una funcional de costos ligada al sistema de ecuaciones diferenciales; esta funcional se minimiza mediante el principio del maximo de Pontryagin, lo que permite determinar estrategias optimas de control preventivo, haciendo minimas las poblaciones infecciosas y los costos de aplicar la intervencion. Luego, se hace un estudio numerico del modelo conside