Date of Award
2017
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematical Sciences
Committee Chair
Jia Li
Committee Member
Dongsheng Wu
Committee Member
Ellen Weisskopf
Committee Member
David Halpern
Committee Member
John Mayer
Subject(s)
Malaria--Transmission--Mathematical models., Mosquitoes as carriers of disease., Discrete-time systems.
Abstract
Mosquito-borne diseases, including malaria, transmitted between human beings by blood-feeding mosquitoes, have been big concerns for the public health. No vaccines are available. An effective way to prevent such diseases is to control the amount of mosquitoes. The Sterile Insect Technique (SIT) is indeed a method of biological control. Mathematical models have proven useful in gaining insights into challenging questions in population dynamics and epidemiology. The objectives in this dissertation is to formulate new models for interactive wild and sterile mosquitoes so that the dynamics are relatively simpler and mathematically more tractable, but the fundamental model features are snatched. Instead of the Ricker-type of nonlinearity for the survival functions, we assume Beverton-Holt-type survival functions. We first formulate models with the assumption that there are no generation overlaps in the mosquito population. Then the models based on the assumption of overlapped generations are considered. We consider three different strategies for the releases of sterile mosquitoes and investigate the model dynamics. Threshold values for the releases of sterile mosquitoes are established for all of the models that determine whether the wild mosquitoes are wiped out or coexist with the sterile mosquitoes. We also formulate stage-structured interactive models. Detailed analysis is carried out. Threshold values for the existence and stability of positive fixed points are derived, respectively. When the positive fixed point is unstable, a 2-cycle is bifurcated. To incorporate the interactive mosquitoes into malaria transmissions, we formulate susceptible-exposed-infective-recovered (SEIR) compartmental discrete-time models for malaria, which are of high dimensions, and then include the interactive mosquito models into these disease models. We derive formulas for the reproductive number $R_0$ of infection for the malaria models with or without sterile mosquitoes and explore the existence of endemic fixed points as well. We then study the impact of sterile mosquitoes releases on the disease transmissions by investigating the effects of varying the releases of sterile mosquitoes. We use numerical simulations to verify our results for all cases and finally give brief discussions of our findings and future study.
Recommended Citation
Li, Yang, "Discrete time structured models and their dynamics for interactive wild and sterile mosquitoes and malaria transmissions" (2017). Dissertations. 135.
https://louis.uah.edu/uah-dissertations/135