Multi-patch Dengue Models

This thesis considers a few dengue epidemic models which describe the human and mosquito interactions. This study is mainly designed to observe the connection between human dispersal and the disease spread. In this scenario, few of the existing single patch models are extended and adapted using the metapopulation concept. These interactions can happen due to the movements of humans as well as mosquitoes. But, in this work, it is assumed with acceptable pieces of evidence that the mosquitoes will hardly move long distances. However, the main concentration is on the short term movements of the humans within the areas. The long term movements are omitted in this study as migrations are negligible when compared to the short term movements. These patches are coupled with the respective mathematical models by using the concept of a residence time budgeting matrix P. The entries of this matrix will represent the rates of the human dispersal from one area to the other. Three different multi patch deterministic models are described such as the vector-host dengue model SIRUV, vector host dengue model with Wolbachia-bacterium, and vector host dengue model with two strains of dengue virus. There is no proper treatments or medicine to cure dengue disease, more awareness is about vector control mechanisms. Therefore, specific optimal control problems are described to reduce the infected individuals of both host and mosquito populations. Theories of optimal control are used to evaluate and identify the patch specific control measures. Also, the aim is on reducing the dengue spread at a minimal cost. The reaction-diffusion epidemic models describe the spatial spread of the diseases. Epidemic SI and SIRUV models are adapted with space fractional reaction-diffusion concept to observe non-local movements of the humans. Here, the space derivatives of the Laplacian are replaced by fractional order derivatives of order between 1 and 2.