Mathematical modelling of some diseases related to water

Abstract

This thesis dissertation focusses on the study of some infectious diseases dynamics from a double point of view: modelization and control. Our main aim is to formulate new mathematical models and combining them with existing ones in order to analyze the dynamics of diseases related to water. We consider compartmental models described by ordinary di↵erential equations and perform rigorous qualitative and quantitative techniques for acquiring insights into the dynamics of these models. This thesis discusses some recent knowledge and investigation on the transmission dynamics of Ebola disease, Zika disease, Japanese encephalitis disease as well as COVID-19. The following are the main topics: (1) The Ebola virus disease is a severe viral haemorrhagic fever syndrome caused by Ebola virus. This disease is transmitted by direct contact with the body fluids of an infected person and objects contaminated with virus or infected animals, with a death rate close to 90% in humans. Recently, some mathematical models have been presented to analyse the spread of the 2014 Ebola outbreak in West Africa. For this disease, we introduce vaccination of the susceptible population with the aim of controlling the spread of the disease and analyze two optimal control problems related with the transmission of Ebola disease with vaccination. Firstly, we consider the case where the total number of available vaccines in a fixed period of time is limited. Secondly, we analyze the situation where there is a limited supply of vaccines at each instant of time for a fixed interval of time. The optimal control problems have been solved analytically. Finally, we have performed a number of numerical simulations in order to compare the models with vaccination and the model without vaccination, which has recently been shown to fit the real data. Three vaccination scenarios have been considered for our numerical simulations, namely: unlimited supply of vaccines; limited total number of vaccines; and limited supply of vaccines at each instant of time. (2) We propose a compartmental mathematical model for the spread of the COVID-19 disease with special focus on the transmissibility of super-spreaders individuals. We compute the basic reproduction number threshold, we study the local stability of the disease free equilibrium in terms of the basic reproduction number, and we investigate the sensitivity of the model with respect to the variation of each one of its parameters. Numerical simulations show the suitability of the proposed COVID-19 model for the outbreak that occurred in Wuhan, China. (3) We propose a new mathematical model for the spread of Zika virus. Special attention is paid to the transmission of microcephaly. Numerical simulations show the accuracy of the model with respect to the Zika outbreak occurred in Brazil. (4) Also, we propose a mathematical model for the spread of Japanese encephalitis, with emphasis on environmental e↵ects on the aquatic phase of mosquitoes. The model is shown to be biologically well-posed and to have a biologically and ecologically meaningful disease free equilibrium point. Local stability is analyzed in terms of the basic reproduction number and numerical simulations presented and discussed.