Sterile insect technology for control of Anopheles mosquito: a mathematical feasibility study

Anopheles mosquito is a vector responsible for the transmission of diseases like Malaria which a_ect many people. Hence its control is a major prevention strategy. Sterile Insect Technology (SIT) is a nonpolluting method of insect control that relies on the release of sterile males. Mating of the released sterile males with wild females leads to non hatching eggs. Thus, if sterile males are released in su_cient numbers or over a su_cient period of time, it can leads to the local reduction or elimination of the wild population. We study the e_ectiveness of the application of SIT for control of Anopheles mosquito via mathematical modeling. Our main result is that there exists a threshold release rate ^_ depending only on the basic o_spring number R and the wild mosquito equilibrium for males such that a release rate higher than ^_ results in elimination of the mosquito population irrespective of its initial size. A release rate _ which is lower than ^_ eliminates the mosquito populations only if it is su_ciently small. If the population is at the wild equilibrium it is reduced by a percentage depending on _ and R only. (Résumé d'auteur

Weighted Sobolev spaces and regularity for polyhedral domains

We prove a regularity result for the Poisson problem $-\Delta u = f$, u |\_{\pa \PP} = g on a polyhedral domain \PP \subset \RR^3 using the \BK\ spaces \Kond{m}{a}(\PP). These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges \cite{Babu70, Kondratiev67}. In particular, we show that there is no loss of \Kond{m}{a}--regularity for solutions of strongly elliptic systems with smooth coefficients. We also establish a "trace theorem" for the restriction to the boundary of the functions in \Kond{m}{a}(\PP)

Stability analysis and dynamics preserving non-standard finite difference schemes for a Malaria model

We extend the results in [2] by proving the GAS of the DFE and specifying the region of possible bakward bifurcation. Futhermore, we design a nonstandard finite difference (NSFD) scheme, which is dynamically consistent with the continuous model. (Résumé d'auteur

Forward invariant set preservation in discrete dynamical systems and numerical schemes for ODEs: application in biosciences

We present two results on the analysis of discrete dynamical systems and finite difference discretizations of continuous dynamical systems, which preserve their dynamics and essential properties. The first result provides a sufficient condition for forward invariance of a set under discrete dynamical systems of specific type, namely time-reversible ones. The condition involves only the boundary of the set. It is a discrete analog of the widely used tangent condition for continuous systems (viz. the vector field points either inwards or is tangent to the boundary of the set). The second result is nonstandard finite difference (NSFD) scheme for dynamical systems defined by systems of ordinary differential equations. The NSFD scheme preserves the hyperbolic equilibria of the continuous system as well as their stability. Further, the scheme is time reversible and, through the first result, inherits from the continuous model the forward invariance of the domain. We show that the scheme is of second order, thereby solving a pending problem on the construction of higher-order nonstandard schemes without spurious solutions. It is shown that the new scheme applies directly for mass action-based models of biological and chemical processes. The application of these results, including some numerical simulations for invariant sets, is exemplified on a general Susceptible-Infective-Recovered/Removed (SIR)-type epidemiological model, which may have arbitrary large number of infective or recovered/removed compartments.DSI/NRF SARChI Chair on Mathematical Models and Methods in Bioengineering and Biosciences at the University of Pretoria. The Competitive Programme for Rated Researchers (CPRR). The University of the Witwatersrand under the Science Faculty Start-up Funds for Research.https://advancesincontinuousanddiscretemodels.springeropen.comMathematics and Applied Mathematic

A nonstandard Volterra difference equation for the SIS epidemiological model

By considering the contact rate as a function of infective individuals and by using a general distribution of the infective period, the SIS-model extends to a Volterra integral equation that exhibits complex behaviour such as the backward bifurcation phenomenon.We design a nonstandard finite difference (NSFD) scheme, which is reliable in replicating this complex dynamics. It is shown that the NSFD scheme has no spurious fixed-points compared to the equilibria of the continuous model. Furthermore, there exist two threshold parameters Rc 0 andRm0 , Rc 0 ≤ 1 ≤ Rm0 , such that the disease-free fixed-point is globally asymptotically stable (GAS) for R0, the basic reproduction number, less than Rc 0 and unstable for R0 > 1, while it is locally asymptotically stable (LAS) and coexists with a LAS endemic fixed-point forRc 0 Rm0 andRm0 < ∞. Numerical experiments that support the theory are provided.DST/NRF SARChI Chair in Mathematical Models and Methods in Bioengineering and Biosciences.http://www.thelancet.com/2016-09-30hb201

Analysis of a time implicit scheme for the Oseen model driven by nonlinear slip boundary conditions

This work is concerned with the time discrete analysis of the Oseen system of equations driven by nonlinear slip boundary conditions of friction type. We study the existence of solutions of the time discrete model and derive several a priori estimates needed to recover the solution of the continuous problem by means of weak compactness. Moreover, for the difference between the exact and approximate solutions, we obtainhttp://link.springer.com/journal/212017-12-31hb2016Mathematics and Applied Mathematic

Positivity-preserving nonstandard finite difference schemes for cross-diffusion equations in biosciences

We design nonstandard finite difference (NSFD) schemes which are unconditionally dynamically consistent with respect to the positivity property of solutions of cross-diffusion equations in biosciences. This settles a problem that was open for quite some time. The study is done in the setting of three concrete and highly relevant cross-diffusion systems regarding tumor growth, convective predator–prey pursuit and evasion model and reaction–diffusion–chemotaxis model. It is shown that NSFD schemes used for classical reaction–diffusion equations, such as the Fisher equation, for which the solutions enjoy the positivity property, are not appropriate for cross-diffusion systems. The reliable NSFD schemes are therefore obtained by considering a suitable implementation on the crossdiffusive term of Mickens’ rule of nonlocal approximation of nonlinear terms, apart from his rule of complex denominator function of discrete derivatives. We provide numerical experiments that support the theory as well as the power of the NSFD schemes over the standard ones. In the case of the cancer growth model, we demonstrate computationally that our NSFD schemes replicate the property of traveling wave solutions of developing shocks observed in Marchant et al. (2000).South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation : SARChI Chair in Mathematical Models and Methods in Bioengineering and Biosciences.http://www.elsevier.com/locate/camwa2015-11-30hb201

Global stability of a two-patch cholera model with fast and slow transmissions

Please read abstract in the article.The Abdus Salam International Center for Theoretical Physics (ICTP) in Trieste-Italy under the Associateship Scheme, the African Center of Excellence in Information and Communication Technologies (CETIC) in Cameroon and the South African Research Chairs Initiative (SARChI Chair), in Mathematical Models and Methods in Bioengineering and Biosciences.http://www.elsevier.com/locate/matcomhj2021Mathematics and Applied Mathematic

Analysis and dynamically consistent nonstandard discretization for a rabies model in humans and dogs

Rabies is a fatal disease in dogs as well as in humans. A possible model to represent rabies transmission dynamics in human and dog populations is presented. The next generation matrix operator is used to determine the threshold parameter R0, that is the average number of new infective individuals produced by one infective individual intro- duced into a completely susceptible population. If R0 < 1, the disease-free equilibrium is globally asymptotically stable, while it is unstable and there exists a locally asymptot- ically stable endemic equilibrium when R0 > 1. A nonstandard nite di erence scheme that replicates the dynamics of the continuous model is proposed. Numerical tests to support the theoretical analysis are provided.DST/NRF SARChI Chair in Mathematics Models and Methods in Bioengineering and Biosciences.http://link.springer.com/journal/133982017-09-30hb2016Mathematics and Applied Mathematic