8 research outputs found
A contribution to the foundations of the theory of Quasifibration
The concept of quasifibration was invented by Dold and Thom (DT]. May (M2] approached quasifibrations from a new angle, making use of n-equivalences. This dissertation presents a study of the notion of n-equivalences and related types of maps. The first of our two main goals is to prove a result, Theorem 5.1, which generalizes the fundamental theorem (DT; Satz 2.2] by Dold and Thom on globalization of quasifibrations. Secondly we show that by means of adjunction or clutching constructions, this theorem enables us to retrieve the famous results of James (J2; Theorem 1.2 and Theorem 1.3] in his work on suspension of spheres. The results of James appear in the thesis as Theorem 13.8. For some of the applications we need a generalized version of n-equivalence. This generalization entails replacing, in the definition of n-equivalence, the isomorphisms by isomorphisms modulo a suitable Serre class [Se] of abelian groups. For the sake of having the thesis self-contained, we include a formal discussion of localization of 1-connected spaces and Serre classes of abelian groups. This summarizes the scope of the thesis. More detail on the content of the thesis will be given after we have sketched a historical perspective on quasifibrations
Mathematical analysis of TB model with vaccination and saturated incidence rate
The model system of ordinary differential equations considers two classes of latently infected individuals, with different risk of becoming infectious. The system has positive solutions. By constructing a Lyapunov function, it is proved that if the basic reproduction number is less than unity, then the disease-free equilibrium point is globally asymptotically stable. The Routh-Hurwitz criterion is used to prove the local stability of the endemic equilibrium when R0 > 1. The model is illustrated using parameters applicable to Ethiopia. A variety of numerical simulations are carried out to illustrate our main results
Control and elimination in an SEIR model for the disease dynamics of Covid-19 with vaccination
COVID-19 has become a serious pandemic affecting many countries around the world
since it was discovered in 2019. In this research, we present a compartmental model in ordinary
differential equations for COVID-19 with vaccination, inflow of infected and a generalized contact rate.
Existence of a unique global positive solution of the model is proved, followed by stability analysis of
the equilibrium points. A control problem is presented, with vaccination as well as reduction of the
contact rate by way of education, law enforcement or lockdown. In the last section, we use numerical
simulations with data applicable to South Africa, for supporting our theoretical results. The model
and application illustrate the interesting manner in which a diseased population can be perturbed from
within itself
Stochastic modeling of a mosquito-borne disease
We present and analyze a stochastic differential equation (SDE) model for the population dynamics of a mosquito-borne infectious disease. We prove the solutions to be almost surely positive and global. We introduce a numerical invariant R of the model with R<1 being a condition guaranteeing the almost sure stability of the disease-free equilibrium. We show that stochastic perturbations enhance the stability of the disease-free equilibrium of the underlying deterministic model. We illustrate the main stability theorem through simulations and show how to obtain interval estimates when making forward projections. We consulted a wide range of literature to find relevant numerical parameter values
A model for control of HIV/AIDS with parental care
In this study we investigate the HIV/AIDS epidemic in a population which experiences
a significant flow of immigrants. We derive and analyze a mathematical model that
describes the dynamics of HIV infection among the immigrant youths and how parental
care can minimize or prevent the spread of the disease in the population. We analyze
the model with both screening control and parental care, then investigate its stability
and sensitivity behavior. We also conduct both qualitative and quantitative analyses. It
is observed that in the absence of infected youths, disease-free equilibrium is achievable
and is globally asymptotically stable. We establish optimal strategies for the control
of the disease with screening and parental care, and provide numerical simulations to
illustrate the analytic results.Web of Scienc
A population model for the 2017/18 listeriosis outbreak in South Africa
We introduce a compartmental model of ordinary differential equations for the population dynamics of listeriosis, and we derive a model for analysing a listeriosis outbreak. The model explicitly accommodates neonatal infections. Similarly as is common in cholera modeling, we include a compartment to represent the reservoir of bacteria. We also include a compartment to represent the incubation phase. For the 2017/18 listeriosis outbreak that happened in South Africa, we calculate the time pattern and intensity of the force of infection, and we determine numerical values for some of the parameters in the model. The model is calibrated using South African data, together with existing data in the open literature not necessarily from South Africa. We make projections on the future outlook of the epidemiology of the disease and the possibility of eradication.South African Research Chairs Initiative - Department of
Science and Innovation and the South African
National Research Foundation - UID:6475
The fibre of a pinch map in a model category
In the category of pointed topological spaces, let F be the homotopy fibre of the
pinching map X ∪ CA → X ∪ CA/ X from the mapping cone on a cofibration A → X
onto the suspension of A. Gray (Proc Lond Math Soc (3) 26:497–520, 1973) proved
that F is weakly homotopy equivalent to the reduced product (X, A)∞. In this paper
we prove an analogue of this phenomenon in a model category, under suitable
conditions including a cube axiom.Web of Scienc