Thesis Summary:

The main objective of this thesis is to gain insight into the spatio-temporal dynamics of infectious diseases described by systems of convection-diffusion equations that on one hand, represent spatial extensions of well-knowm temporal compartmental epidemics, and on the other hand involve recently developed models of directed and undirected biological movement. Due to the involved nonlinear and nonlocal nature of the governing model, we aim to obtain insight by the implementation of advanced numerical methods. The choice of scenarios is motivated by the existing connection between dynamic systems of ordinary differential equations and spatio-temporal systems of partial differential equations. Roughly speaking, dynamic systems determine the variation of quantities with respect to the time variable. Considering that these quantities correspond to the number of individuals of a population which move in a certain environment, this motivates adding one or more spatial variables, from which a system of partial differential equations is obtained. The dynamic systems to be presented correspond to models used mainly in the study of infectious diseases such as in uenza AH1N1, malaria, hantavirus, Ebola, and sexually transmitted diseases. For the study of these diseases the starting point is to consider a certain population of individuals and classify them in two or more states acording to a particular disease. An important role is played by the basic reproductive R0, which represents the average number of secondary cases that an infectious can cause during the infectious period. This number is determined by constant values that are obtained empirically for each particular disease model. If R0 < 1, then the disease does not generate an outbreak, and if R0 > 1, then an epidemic occurs. Other important concepts in epidemiology are those of an agent (person, animal or microorganism), a vector (any agent that transports and transmits an infectious pathogen in another living organism) and a host (any agent that hosts the infectious but does not transmit it). When in a dynamic system there is a vector that carries the disease to a host then we talk about an infectious disease transmitted by a vector. In partial differential equations involving temporal and spatial derivatives, two important effects associated with spatial derivatives are the phenomenon of diffusion, which in this case is interpreted as how individuals tend to move away from each other and the phenomenon of convolution, which is related to the way in which some individuals have the ability to locate others, and to move in their direction. One zoonotic infectious disease present in Chile is the Hantavirus, which has two major manifestations in humans, a hemorrhagic fever with renal syndrome and the other as a cardiopulmonary syndrome. This disease that affects humans and can be fatal, it is transmited by contact with the long-tailed mouse Oligoryzomys longicaudatus or with the feces or urine of the animal. In the first instance, we work with a system that deals with the spread of Hantavirus infection in rodents, which is described by a spatio-temporal compartmental model susceptible-exposed-infectious-recovered (SEIR) that distinguishes between male and female subpopulations. It is assumed in this case that both subpopulations differ in their movement with respect to the local variations of their respective own densities and of the opposite gender group; three alternative models for the movement of the masculine individuals are examined. In some cases, the movement is not only directed by the gradient of a density (as in the standard diffusive case), but also by a non-local convolution of density values. An efficient numerical method is then proposed for the reaction-diffusion convection model of the resulting partial differential equations system. This method involves essentially non-oscillatory weighted reconstruction techniques (WENO) in combination with explicit implicit methods of Runge-Kutta (IMEX-RK) for time stepping. The second part of this work is based on another type of dynamic systems, known as predator-prey models, which are often studied in the context of ecology, as they are used to predict the number of prey and their predator. An interesting phenomenon in nature is the formation of patterns, which can be described through systems of partial differential equations associated with spatio-temporal extensions of predatorprey models that under certain conditions give rise to solutions that stabilize over time. A spatio-temporal eco-epidemiological model is formulated by combining an available non-spatial model for predator-prey dynamics with infected prey with a spatiotemporal susceptible-infective (SI)-type epidemic model of pattern formation due to diffusion. It is assumed that predators exclusively eat infected prey, in agreement with the hypothesis that the infection weakens the prey and increases its susceptibility to predation. In addition, the movement of predators is described by a non-local convolution of the density of infected prey. In this part the objective is to observe the effect of convolution, the patterns formed by the prey change once the predator appears.

ISI Publications from the Thesis

Raimund BüRGER, Gerardo CHOWELL, Elvis GAVILáN, Pep MULET, Luis M. VILLADA: Numerical solution of a spatio-temporal predator-prey model with infected prey. Mathematical Biosciences and Engineering, vol. 16, 1, pp. 438-473, (2019).

Raimund BüRGER, Gerardo CHOWELL, Elvis GAVILáN, Pep MULET, Luis M. VILLADA: Numerical solution of a spatio-temporal gender-structured model for hantavirus infection in rodents. Mathematical Biosciences and Engineering, vol. 15, pp. 95-123, (2018).

Other Publications (ISI)

Raimund BüRGER, Elvis GAVILáN, Daniel INZUNZA, Pep MULET, Luis M. VILLADA: Exploring a convection-diffusion-reaction model of the propagation of forest fires: computation of risk maps for heterogeneous environments. Mathematics, vol. 8, 10, article: 1674 (20pp), (2020).

Raimund BüRGER, Elvis GAVILáN, Daniel INZUNZA, Pep MULET, Luis M. VILLADA: Implicit-explicit methods for a convection-diffusion-reaction model of the propagation of forest fires. Mathematics, vol. 8, article: 1034 (21pp), (2020).