Texts in Biomathematics

From the Editors

Roumen Anguelov — 2018-07-23

This volume contains selected papers presented at the International Conference on Mathematical Methods and Models BIOMATH 2017, which took place from 25 to 30 June 2017 at Skukuza Camp in Kruger Park, South Africa (see www.biomath.bg/2017). The volume is available online at www.biomathforum.org/biomath/index.php/texts.

All papers included in the Proceedings of Biomath 2017 have been reviewed by at least two peer reviewers in a two-round review process. Total of 19 manuscripts were submitted for publication in the proceedings, of which 10 were accepted and included the present volume.

We would like to thank the referees for their professional work in ensuring the high quality of this publication. Further, we would like to thank all authors who contributed their results for publication in the Proceedings of Biomath 2017.

M. Lachowicz was supported by the National Science Centre Poland Grant 2017/25/B/ST1/00051. R. Anguelov was supported by the NRF/DST SARChI Chair in Mathematical Models and Methods in Bioengineering and Biosciences.

Foreword to BIOMATH 2017 Proceedings: Some comments on mathematical modelling and biomathematics.

Jacek Banasiak — 2018-07-23

Both biology and mathematics have existed as well established branches of science for hundreds of years and both, maybe not in a well defined way, have been with the humankind for a couple of thousands of years.В Though natureВ was studied by the ancient civilizations of Mesopotamia, Egypt, the Indian subcontinent and China, the origins of modern biology are typically traced back to the ancient Greece, where Aristotle (384-322 BC) contributed most extensively to its development. Similarly,В theВ ancient Babylonians were able to solve quadratic equation over four millennia ago and we can see the development of mathematical methods in all ancient civilisations, notably in China and on the Indian subcontinent. However, possibly again the Greeks were the first who studied mathematics for its own sake, as a collection of abstract objects and relations between them.В Nevertheless, despite the fact that the developmentВ of such a mathematics has not required any external stimuli, an amazing feature of the human mind is that a large number of abstract mathematical constructs has proved to be very well suited for describing natural phenomena.This prompted Eugene Wigner to write his famous article The Unreasonable Effectiveness of Mathematics in the Natural Sciences,В ...

FKPP equation with impulses on unbounded domain

Valaire Yatat — 2017-12-06

This paper deals with the problem of travelling wave solutions in a scalar impulsive FKPP-like equation. It is a first step of a more general study that aims to address existence of travelling wave solutions for systems of impulsive reaction-diffusion equations that model ecological systems dynamics such as fire-prone savannas. Using results on scalar recursion equations, we show existence of populated vs. extinction travelling waves invasion and compute an explicit expression of their spreading speed (characterized as the minimal speed of such travelling waves). In particular, we find that the spreading speed explicitly depends on the time between two successive impulses. In addition, we carry out a comparison with the case of time-continuous events. We also show that depending on the time between two successive impulses, the spreading speed with pulse events could be lower, equal or greater than the spreading speed in the case of time-continuous events. Finally, we apply our results to a model of fire-prone grasslands and show that pulse fires event may slow down the grassland vs. bare soil invasion speed.

Under the sea: Pulsing corals in ambient flow

Nicholas A. Battista — 2017-12-06

While many organisms filter feed and exchange heat or nutrients in flow, few benthic organisms also actively pulse to enhance feeding and exchange. One example is the pulsing soft coral (Alcyonacea: Xeniidae). Pulsing corals live in colonies, where each polyp actively pulses through contraction and relaxation of their tentacles. The pulses are typically out of phase and without a clear pattern. These corals live in lagoons and bays found in the Red Sea and Indian Ocean where they at times experience strong ambient flows. In this paper, 3D fluid-structure interaction simulations are used to quantify the effects of ambient flow on the exchange currents produced by the active contraction of pulsing corals. We find a complex interaction between the flows produced by the coral and the background flow. The dynamics can either enhance or reduce the upward jet generated in a quiescent medium. The pulsing behavior also slows the average horizontal flow near the polyp when there is a strong background flow. The dynamics of these flows have implications for particle capture and nutrient exchange.

Collapsing chaos

Pabel Shahrear — 2017-12-21

Genetic networks play a fundamental role in the regulation and control of the development and function of organisms. A simple model of gene networks assumes that a gene can be switched on or off as regulatory inputs to the gene cross critical thresholds. In studies of dynamics of such networks, we found unusual dynamical behavior in which phase plane trajectories display irregular dynamics that shrink over long times. This observation leads us to identify a type of dynamics, that can be described as collapsing chaos, in which the Lyapunov exponent is positive, but points on the trajectory approach the origin in the long time limit.

Stability analysis of a compartmental SEIHRD model for the Ebola virus disease

Diène Ngom — 2017-12-22

In this work, we perform a stability analysis of a compartmental SEIHRD model. This model is a simplified version of a previous approach. In this previous work, we proposed an original deterministic spatial-temporal model, called Be-CoDiS (Between-Countries Disease Spread), to study the evolution of human diseases within and between countries. This model was validated by considering data from the 2014-16 West African Ebola Virus Disease epidemic. Here, considering some simplification assumptions in Be-CODIS, our goal is to study the equilibria of the model and their stability using the basic reproduction ratio as a threshold parameter. Finally, we validate the obtained results by considering some numerical experiments based on data from the 2014-16 West African Ebola Virus Disease epidemic.

Modelling of activator-inhibitor dynamics via nonlocal integral operators

Roumen Anguelov — 2017-12-28

This paper proposes application of nonlocal operators to represent the biological pattern formation mechanism of self-activation and lateral inhibition. The blue-green algae Anabaena is discussed as a model example. The patterns are determined by the kernels of the integrals representing the nonlocal operators. The emergence of patters when varying the size of the support of the kernels is numerically investigated.

Analysis of a mathematical model of the dynamics of contagious bovine pleuropneumonia

Achamyelesh Amare Aligaz — 2017-12-28

Contagious bovine pleuropneumonia (CBPP) is a disease of cattle and water buffalo caused by Mycoplasma mycoides subspecies mycoides (Mmm). It attacks the lungs and the membranes that line the thoracic cavity. The disease is transmitted by inhaling droplets disseminated through coughing by infected cattle. In this paper a deterministic mathematical model for the transmission of Contagious Bovine plueropnemonia is presented. The model is a five compartmental model consisting of susceptible, Exposed, Infectious, Persistently infected and Recovered compartments. We derived a formula for the basic reproduction number R₀. For R₀ в‰¤ 1, the disease free equilibrium is globally asymptotically stable, thus CBPP dies out; whereas for R₀ > 1, the unique endemic equilibrium is globally asymptotically stable and hence the disease persists. Elasticity indices for R₀ with respect to different parameters are calculated; indicating parameters that are important for control strategies to bring R₀ below 1, the effective contact rateВ ОІ has the largest elasticity index. As the disease control options are associated to these parameters, for some values of these parameters, R₀ < 1, thus the disease can be controlled.

Analysis of model for the transmission dynamics of Zika with sterile insect technique

Usman A. Danbaba — 2018-01-19

A deterministic model for the transmission dynamics of Zika, that takes into account the aquatic and non-aquatic stages of mosquito development is constructed and rigorously analysed. The model with fraction of male mosquitoes being sterilized assumed direct (human-human) and indirect (human-mosquito-human) transmission. Stability analysis of the equilibria and sensitivity analysis of parameters associated with the computed reproduction number were presented. Numerical simulation were carried out to support the analysis.

Estimating the mean of a small sample under the two parameter lognormal distribution

Peter Hingley — 2018-03-03

Lognormally distributed variables are found in biological, economic and other systems. Here the sampling distributions of maximum likelihood estimates (MLE) for parameters are developed when data are lognormally distributed and estimation is carried out either by the correct lognormal model or by the mis-specified normal distribution. This is designed as an aid to experimental design when drawing a small sample under an assumption that the population follows a normal distribution while in fact it follows a lognormal distribution.В Distributions are derived analytically as far as possible by using a technique for estimator densities and are confirmed by simulations. For an independently and identically distributed lognormal sample, when a normal distribution is used for estimation then the distribution of the MLE of the mean is different to that for the MLE of the lognormal mean. The distribution is not known but can be well enough approximated by another lognormal. An analytic method for the distribution of the mis-specified normal variance uses computational convolution for a sample of size 2. The expected value of the mis-specified normal variance is also found as a way to give information about the effect of the model misspecification on inferences for the mean. The results are demonstrated on an example for a population distribution that is abstracted from a survey.

Numerical solutions of a 2D fluid problem coupled to a nonlinear non-local reaction-advection-diffusion problem for cell crawling migration in a discoidal domain

Christelle Etchegaray — 2018-03-25

In this work, we present a numerical scheme for the approximate solutions of a 2D crawling cell migration problem. The model, defined on a non-deformable discoidal domain, consists in a Darcy fluid problem coupled with a Poisson problem and a reaction-advection-diffusion problem. Moreover, the advection velocity depends on boundary values, making the problem nonlinear and non local. For a discoidal domain, numerical solutions can be obtained using the finite volume method on the polar formulation of the model. Simulations show that different migration behaviours can be captured.

Sensitivity analysis for a within-human-host immuno-pathogenesis dynamics of Plasmodium falciparum parasites

Woldegebriel A. Woldegerima — 2018-07-23

Sensitivity analysis has become increasingly useful in many fields of engineering and sciences. Researchers use sensitivity and uncertainty analysis in the mathematical modelling of biological phenomena because of its value in identifying essential parameters for model's output. Moreover, it can help in the process of experimental analysis, model order reduction, parameter estimation, decision making or development of recommendations for decision makers. Here, we demonstrate the use of local sensitivity analysis to understand the influence of different parameters on a threshold parameter, R_0^I, resulting from the analysis of a within human-host model for the dynamics of malaria parasites. %We highlight the different methods used in sensitivity analysis.
Our results reveal that the obtained R_0^I is most sensitive to the infection rate of healthy red blood cells (RBCs) by merozoites, the average number of merozoites released per bursting parasitized RBCs, the proportion of parasitized RBCs that continue asexual reproduction and the per capita natural death rate of merozoites.

ISBN

ISBN 978-619-7451-00-9 (print) — 2018-08-15