Abdul-Aziz Yakubu

Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory

New directions in the mathematics of infectious disease * Fred Brauer * Kenneth L. Cooke * Basic ideas of mathematical epidemiology * Extensions of the basic models * New vaccination strategies for pertussis * Time delay in epidemic models * Nonlocal response in a simple epidemiological model * Discrete-time S-I-S models with simple and complex population dynamics * Intraspecific competition, dispersal, and disease dynamics in discrete-time patchy environments * The impact of long-range dispersal on the rate of spread in population and epidemic models * Endemicity, persistence, and quasi-stationarity * On the computation of Ro and its role in global stability * Nonlinear mating models for populations with discrete generations * Center manifolds and normal forms in epidemic models * Remarks on modeling host-viral dynamics and treatment * A multiple compartment model for the evolution of HIV-1 after highly active antiretroviral therapy * Modeling cancer as an infectious disease * Frequency dependent risk of infection and the spread of infectious diseases * Long-term dynamics and emergence of tuberculosis

Discrete-Time SIS EpidemicModel in a Seasonal Environment

We study the combined effects of seasonal trends and diseases on the extinction and persistence of discretely reproducing populations. We introduce the epidemic threshold parameter, R0 , for predicting disease dynamics in periodic environments. Typically, in periodic environments, R0 > 1 implies disease persistence on a cyclic attractor, while R0 < 1 implies disease extinction. We also explore the relationship between the demographic equation and the epidemic process. In particular, we show that in periodic environments, it is possible for the infective population to be on a chaotic attractor while the demographic dynamics is nonchaotic.

Allee effects in a discrete-time SIS epidemic model with infected newborns

We extend the framework of Rios-Soto et al. (Contemporary Mathematics, 2006, 410, 297) to include both compensatory (contest competition) and overcompensatory (scramble competition) population dynamics with and without the Allee effect. We compute the basic reproductive number ℛ0, and use it to predict the (uniform) persistence or extinction of the infective population, where the population dynamics are compensatory and the Allee effect is either present or absent. We also explore the relationship between the demographic equation and the epidemic process, where the total population dynamics are overcompensatory. In particular, we show that the demographic dynamics drive both the susceptible and infective dynamics. This is in contrast to the recent observations of Franke and Yakubu, that the demographic dynamics can be chaotic while the infective dynamics are oscillatory and non-chaotic in periodically-forced SIS epidemic models (Mathematical Biosciences, 2006, 204, 68).

Global attractions in competitive systems

We establish the ecological principle of mutual exclusion as a mathematical theorem in a discrete system modeling the interactions of pioneer species

Disease-induced mortality in density-dependent discrete-time S-I-S epidemic models

The dynamics of simple discrete-time epidemic models without disease-induced mortality are typically characterized by global transcritical bifurcation. We prove that in corresponding models with disease-induced mortality a tiny number of infectious individuals can drive an otherwise persistent population to extinction. Our model with disease-induced mortality supports multiple attractors. In addition, we use a Ricker recruitment function in an SIS model and obtained a three component discrete Hopf (Neimark–Sacker) cycle attractor coexisting with a fixed point attractor. The basin boundaries of the coexisting attractors are fractal in nature, and the example exhibits sensitive dependence of the long-term disease dynamics on initial conditions. Furthermore, we show that in contrast to corresponding models without disease-induced mortality, the disease-free state dynamics do not drive the disease dynamics.

Population models with periodic recruitment functions and survival rates

We study the combined effects of periodically varying carrying capacity and survival rate on populations. We show that our populations with constant recruitment functions do not experience either resonance or attenuance when either only the carrying capacity or the survival rate is fluctuating. However, when both carrying capacity and survival rate are fluctuating the populations experience either attenuance or resonance, depending on parameter regimes. In addition, we show that our populations with Beverton–Holt recruitment functions experience attenuance when only the carrying capacity is fluctuating.

Fatal disease and demographic Allee effect: population persistence and extinction

If a healthy stable host population at the disease-free equilibrium is subject to the Allee effect, can a small number of infected individuals with a fatal disease cause the host population to go extinct? That is, does the Allee effect matter at high densities? To answer this question, we use a susceptible–infected epidemic model to obtain model parameters that lead to host population persistence (with or without infected individuals) and to host extinction. We prove that the presence of an Allee effect in host demographics matters even at large population densities. We show that a small perturbation to the disease-free equilibrium can eventually lead to host population extinction. In addition, we prove that additional deaths due to a fatal infectious disease effectively increase the Allee threshold of the host population demographics.

Multiple attractors via CUSP bifurcation in periodically varying environments

Periodically forced (non-autonomous) single species population models support multiple attractors via tangent bifurcations, where the corresponding autonomous models support single attractors. Elaydi and Sacker obtained conditions for the existence of single attractors in periodically forced discrete-time models. In this paper, the Cusp Bifurcation Theorem is used to provide a general framework for the occurrence of multiple attractors in such periodic dynamical systems.

Periodic dynamical systems in unidirectional metapopulation models

In periodically varying environments, population models generate periodic dynamical systems. To understand the effects of unidirectional dispersal on local patch dynamics in fluctuating environments, dynamical systems theory is used to study the resulting periodic dynamical systems. In particular, a unidirectional dispersal linked two patch nonautonomous metapopulation model is constructed and used to explain the qualitative dynamics of linked versus unlinked independent patches. As in single-patch, single-species population models, unidirectional nonautonomous models support multiple attractors where local population models support single attractors.

Population Models in Almost Periodic Environments

We establish the basic theory of almost periodic sequences on ℤ+. Dichotomy techniques are then utilized to find sufficient conditions for the existence of a globally attracting almost periodic solution of a semilinear system of difference equations. These existence results are, subsequently, applied to discretely reproducing populations with and without overlapping generations. Furthermore, we access evidence for attenuance and resonance in almost periodically forced population models.