Mathematical Modeling, Analysis and Control of Some Infectious Diseases

Mathematical models are useful in epidemiological research to describe, explain or predict the development and spread of infectious diseases. In addition they play a critical role in disease control, indicating intervention strategies concerning public health such as vaccinations, isolation, treatments and so on. The underlying theme of this Ph.D. thesis is the mathematical modeling and analysis of some aspects concerning infectious diseases, in the absence and presence of infection. In fact, the control of a possible epidemic can be carried out both by investigating aspects external to the infection, such as vaccination (as was done in chapter 1) or alternative prevention methods, such as controlling the life cycle of carriers of the virus (as presented in chapter 4) and by analyzing and studying models that describe the dynamics of the infection itself (as we will see in chapters 2 and 3). Chapter 1 models epidemiological aspects when the disease is not present in the population. Indeed it focuses mainly on the decision of individuals to be vaccinated or not, in the absence of infection and based on information they receive from the outside on the state of the disease and on the side effects of the vaccine. This information is modeled through the introduction of a spatial kernel that weighs the distance from which the information comes. The proposed new model is based on the theory of the "Imitation Game", in which the exchanges of opinions and contacts between individuals are taken into consideration. The model is parabolic, since the Fickian diffusion is assumed to be valid. In chapters 2 and 3, we investigate on the spread of two infectious diseases, which differ from each other via different transmission vehicles of the virus. In particular, chapter 2 deals with a new model for the transmission of cholera, a water-borne disease that involves host-environment interaction. The model extends works featured in the published literature on the subject, including the possibility of water exchange between two different aquatic environments, then the cohabitation of two populations of bacteria, different from each other due to the position occupied within the aquatic system, is incorporated in the model. Instead, chapter 3 is concerned with proposing a new model for host-vector diseases. The modeling carried out takes into account the possible vertical transmission in the vector population, in addition to the presence of two different types of host population. Furthermore, a maximum capacity is placed on the growth of the vector population. In order to investigate the spatial dynamics of disease, both models presented in chapters 2 and 3 are extended to a PDE systems, considering the diffusion of the populations described by Fick's law. For both models an estimate of the Basic Reproductive Number related to the spatial case is provided, comparing it later with the one corresponding to the non-spatial model. The conditions are given for which this quantity constitutes a threshold value for the eradication of the infection also in the spatial case. Furthermore, for the PDE model presented in the chapter 2, traveling waves are studied. Finally, chapter 4, in order to provide a possible prevention method against host-vector type diseases, proposes a new model for vector population dynamics, in which slow and fast diffusion processes coexist, assuming that the diffusion flux is consisting of a Fickian type part and a satisfactory part of an evolution equation of Maxwell-Cattaneo-Vernotte type. The analysis of the corresponding ODE model is carried out, and traveling wave solutions are investigated. A section of the chapter is devoted to the limit case analysis of the presented model, in which the Fickian diffusion is neglected, thus obtaining a strictly hyperbolic system. Numerical simulations compare the model characterized by the coexistence of both diffusion processes (slow and fast), the hyperbolic model characterized by slow diffusion only and the classical parabolic model, present in literature, characterized by fast diffusion only.