Mathematical Modeling, Simulation, and Time Series Analysis of Seasonal Epidemics.

Seasonal and non-seasonal Susceptible-Exposed-Infective-Recovered-Susceptible (SEIRS) models are formulated and analyzed. It is proved that the disease-free steady state of the non-seasonal model is locally asymptotically stable if Rv \u3c 1, and disease invades if Rv \u3e 1. For the seasonal SEIRS model, it is shown that the disease-free periodic solution is locally asymptotically stable when R̅v \u3c 1, and I(t) is persistent with sustained oscillations when R̅v \u3e 1. Numerical simulations indicate that the orbit representing I(t) decays when R̅v \u3c 1 \u3c Rv. The seasonal SEIRS model with routine and pulse vaccination is simulated, and results depict an unsustained decrease in the maximum of prevalence of infectives upon the introduction of routine vaccination and a sustained decrease as pulse vaccination is introduced in the population. Mortality data of pneumonia and influenza is collected and analyzed. A decomposition of the data is analyzed, trend and seasonality effects ascertained, and a forecasting strategy proposed

Models Linking Epidemiology with Immunology and Ecology

Optimal control can be used to design intervention strategies for the control of infectious diseases and predator-prey systems. In this dissertation, we studied models encapsulating two relatively new areas of mathematical biology, which combine epidemiology with immunology and ecology. We formulated immuno-epidemiological models of coupled within-host model of ordinary differential equations and between-host model of ordinary differential equations and partial differential equations, using the Human Immunodeficiency Virus (HIV) for illustration, and set a framework for optimal control of immuno-epidemiological models. By constructing an iterative sequence from a representation formula for a solution to the linked model and using the fixed-point argument, existence and uniqueness of solution to the immuno-epidemiological model are obtained. An explicit expression for the basic reproduction number, R0 (R zero), of the linked model is derived, and local asymptotic and global stability results are obtained when R01, it is shown that the endemic equilibrium point is locally asymptotically stable. An optimal control problem with drug-treatment control on the within-host system is formulated and analyzed; these results are novel for optimal control of ODEs linked with such first order PDEs. Numerical simulations based on a forward-backward sweep method are obtained. Our analysis and control techniques give a new tool for investigating immuno-epidemiological models for other diseases. An eco-epidemiological model of predator and prey, motivated by cats and birds on the Marion Island, is formulated and analyzed. Basic and demographic reproduction numbers are obtained, and stability analysis of equilibria is investigated. An optimal control problem involving scalar and time-dependent controls is formulated and analyzed. Existence, characterization and uniqueness results are obtained. Numerical simulations based on a forward-backward sweep method illustrate the possibility of eradicating predators and conserving prey when a combination of control strategies are applied

Optimal insecticide-treated bed-net coverage and malaria treatment in a malaria-HIV co-infection model

We propose and study a mathematical model for malaria-HIV co-infection transmission and control, in which malaria treatment and insecticide-treated nets are incorporated. The existence of a backward bifurcation is established analytically, and the occurrence of such backward bifurcation is influenced by disease-induced mortality, insecticide-treated bed-net coverage and malaria treatment parameters. To further assess the impact of malaria treatment and insecticide-treated bed-net coverage, we formulate an optimal control problem with malaria treatment and insecticide-treated nets as control functions. Using reasonable parameter values, numerical simulations of the optimal control suggest the possibility of eliminating malaria and reducing HIV prevalence significantly, within a short time horizon

Optimal control of a multi-scale HIV-opioid model

In this study, we apply optimal control theory to an immuno-epidemiological model of HIV and opioid epidemics. For the multi-scale model, we used four controls: treating the opioid use, reducing HIV risk behaviour among opioid users, entry inhibiting antiviral therapy, and antiviral therapy which blocks the viral production. Two population-level controls are combined with two within-host-level controls. We prove the existence and uniqueness of an optimal control quadruple. Comparing the two population-level controls, we find that reducing the HIV risk of opioid users has a stronger impact on the population who is both HIV-infected and opioid-dependent than treating the opioid disorder. The within-host-level antiviral treatment has an effect not only on the co-affected population but also on the HIV-only infected population. Our findings suggest that the most effective strategy for managing the HIV and opioid epidemics is combining all controls at both within-host and between-host scales

Long-term neurological symptoms after acute COVID-19 illness requiring hospitalization in adult patients: insights from the ISARIC-COVID-19 follow-up study

in this study we aimed to characterize the type and prevalence of neurological symptoms related to neurological long-COVID-19 from a large international multicenter cohort of adults after discharge from hospital for acute COVID-19