# Biomath Coffee

Just to remind you to keep free the slot 15:15 to 16:00  every Tuesday for Biomath Coffee meetings.

## Highlights:

23 May 2017

Speaker:  Dr Joseph Malinzi

Co-authors: Innocenter Amima

Title: Mathematical analysis of a tumour-immune interaction model: A moving boundary problem

Abstract: We develop and analyse a spatio-temporal mathematical model, in the form of a moving boundary problem to explain cancer dormancy. Analysis of the model is carried out for both temporal and spatio-temporal cases. Stability analysis and numerical simulations of the temporal model replicate experimental observations of immune-induced tumour dormancy. Travelling wave solutions of the spatio-temporal model are determined using the hyperbolic tangent method and minimum wave speeds of invasion are calculated. Travelling wave analysis depicts that cell invasion dynamics are mainly driven by their motion and growth rates. A stability analysis of the spatio-temporal model shows a possibility of dynamical stabilization of the tumour-free steady state. Simulation results reveal that the tumour radius reduces to a dormant level but may regrow after a long time period. Our approach may lead to a deeper understanding of cancer dormancy and this may be helpful in the future development of better and effective therapeutic methods.

Keywords: Tumour cells, Immune cells, Tumour-immune interactions, Cancer dormancy, Travelling wave solutions, Hyperbolic tangent method, Tumour radius, Moving boundary

Presentation content: 45% Biology, 40% Applied Mathematical analysis, 15% Numerical Simulations

9 May 2017

Speaker: Roxanne Beauclair   Level of study: PhD

Supervisor: Prof. Wim Delva and Prof. Niel Hens

Title: Age-mixing and HIV transmission: A cluster randomised controlled trial in Malawi

Abstract: Age disparities in sexual relationships have been proposed as a key risk factor for HIV transmission in Sub-Saharan Africa, but evidence remains inconclusive. The SIHR study, a cluster randomised trial of a cash transfer programme in Malawi, found that young women in the intervention groups were less likely to have had a sexual partner aged 25 or older, and less likely to test positive for HIV and HSV-2 at follow-up compared to control groups. We conducted a secondary analysis of schoolgirls in the SIHR study aged 13-22 at baseline. We investigated the effects of study arm, trial stage and participant age on age differences in sexual relationships. We also explored relationship-level characteristics that affect HIV transmission risk. Girls receiving cash transfers, on average, had smaller age differences in relationships compared to controls. After the cash transfers had ended, the average age difference was larger than during the intervention for all groups. Larger age differences in relationships were associated with lower levels of condom use, more frequent sex, and longer relationship durations. Cash-transfer programmes may prevent HIV transmission in part by encouraging young women to form age-similar relationships, which are characterised by increased condom use, reduced sex frequency.

Keywords: Malawi, Sexual Risk Behaviour, Age-disparate Relationships, Age-Mixing, Southern Africa

Presentation content: 50% Epidemiology, 50% Applied Statistical Analysis

28 March 2017

Speaker: Ms Dumani                       Level of study: MSc

Supervisor(s): Dr Michael Chapwanya (University of Pretoria, South Africa)

Title: Simple model of rabies among wild dogs.

Abstract:

Background: Rabies is caused by a virus that affects the central nervous system, particularly causing inflammation in the brain. The virus is transmitted from an infected dog through a bite or by saliva of an infected animal touching an open wound of anuninfected animal. Rabies is the deadliest disease on earth with a 99.9% fatality rate. Dogs are the most common reservoir of the virus. There is no effective treatment or cure of rabies once symptoms show

Aim: The aim of this study is to understand the dynamics of rabies among dogs as well as seek to predict if the model proposed can help in preventing andcontrolling the disease from spreading uncontrollably.

Methods: We consider a model representing the dynamics of rabies in wild dogs. Infective dogs normally lose their senses and move randomly out of the packs.This has an effect of lowering the contact rate between the susceptive and infected dogs. The threshold quantity, R0, which in this case is the expected number of new infectious dogs introduced into a perfectly susceptible population by a single infected dog, obtained using the Next-Generation matrix method is used to investigate how the disease can be controlled. Numerical simulations are performed for different values of the basic reproduction number to investigate the effect of a change in the rate of contact between healthy and infected dogs.

Results: The numerical simulations show an existence of fluctuations of both populations, which biologically; we can think of the oscillations as outbreaks of the disease.

Discussions and conclusion: Our outcomes indicate that from the basic reproduction number, the virus can be controlled by minimising the contact rate.The spreading of the virus is controlled if R0 remains less than one.

Keywords: Basic reproduction number, saturation incidence.

Presentation content: 30% Biology -- 40% Mathematical Modeling and Analysis -- 30% Numerical analysis and Simulations.

14 March 2017

Speaker: Mr Patrick Shabangu     Level of study: PhD

Supervisor: Prof. M Banda and Dr M Chapwanya

Title: Modelling of a blood flow problem

Abstract:

Background: Blood clotting is a natural response by the vascular system through hemostasis triggered when there is leakage on blood vessel wall. Significant progress has been made in literature to understand this process. However, a major challenge in understanding of hemostasis is to better integrate the various sub-processes involved during clotting and thrombus development. The ability to predict how simultaneous variation of multiple hemostasis factors affects thrombus development would be of significant biomedical value.

Aim: This presentation will focus on how blood lose through a vessel wall leakage can be blocked.

Discussions: Different ideas will be discussed with the aim to formulate mathematical models representing this process.

Presentation content: 70% Biology -- 30% Mathematical Modeling and Analysis

7 March 2017

Speaker: Ms Dumani                            Level of study: MSc

Supervisor(s): Dr Michael Chapwanya (University of Pretoria, South Africa)

Title: Simple model of rabies among wild dogs.

Abstract:

Background: Rabies is caused by a virus that affects the central nervous system, particularly causing inflammation in the brain. The virus is transmitted from an infected dog through a bite or by saliva of an infected animal touching an open wound of an uninfected animal. Rabies is the deadliest disease on earth with a 99.9% fatality rate. Dogs are the most common reservoir of the virus. There is no effective treatment or cure of rabies once symptoms show.

Aim: The aim of this study is to understand the dynamics of rabies among dogs as well as seek to predict if the model proposed can help in preventing and controlling the disease from spreading uncontrollably.

Methods: We consider a model representing the dynamics of rabies in wild dogs. Infective dogs normally lose their senses and move randomly out of the packs.This has an e ect of lowering the contact rate between t ff he susceptive and infected

dogs. The threshold quantity, R0, which in this case is the expected number of new infectious dogs introduced into a perfectly susceptible population by a single infected dog, obtained using the Next-Generation matrix method is used to investigate how the disease can be controlled. Numerical simulations are performed for different values of the basic reproduction number to investigate the effect of a change in the rate of contact between healthy and infected dogs.

Results: The numerical simulations show an existence of fluctuations of both populations, which biologically; we can think of the oscillations as outbreaks ofthe disease.

Discussions and conclusion: Our outcomes indicate that from the basic reproduction number, the virus can be controlled by minimising the contact rate. The spreading of the virus is controlled if R0 remains less than one.

Keywords: Basic reproduction number, saturation incidence.

Presentation content: 30% Biology -- 40% Mathematical Modeling and Analysis -- 30% Numerical analysis and Simulations.

28 February 2017

Speaker: Dr R. Ouifki

Title: Studying the effects of density dependency and temperature change on the dynamics of tsetse flies

Abstract:

Background: You may be an espresso lover, or prefer cappuccino or a fancy latté. Maybe you are someone who can be just as happy with a mug of old fashioned filter coffee. Join us at the Biomath Coffee. Here we like our coffee, we love it even more when combined with a dash of mathematics and a hint of biology. The end product is a well roasted coffee with a spicy taste of mathematics combined and a strong aroma of simulations and data analysis.

Aim: Discuss the problems and prospects associated with modelling the population dynamics of tsetse flies (Glossina spp) and the disease (trypanosomiasis) that they transmit in Africa to game animals, domestic livestock and humans. The main focus will be on analysing the simultaneous impact of density dependency, index of vegetation and temperature change on the dynamics of tsetse flies (and on the disease itself).

Methods: We propose a mathematical model that incorporates both pupae and adults tsetse flies. The model accounts for temperature change and density-dependent mortality, and uses the normalised index of vegetation NDVI to explore the impact of vegetation density on the fly survival.

The proposed model is fitted to data on G. m. morsitans from Rekomitjie in Zimbabwe using Markov Chain Monte Carlo method.

Results: Our analysis reveals the importance of density dependency in capturing the scale of the distribution of the tsetse population while the shape is more governed by temperature change. Finally NDVI seems to have an impact though it is insignificant compared to aforementioned factors. .

Discussions: The findings clearly highlight the importance of temperature as key factor in shaping the dynamics of tsetse flies. Density depend is equally important and should be taken into account when modelling the transmission dynamics of the trypanosomiasis.

The fitted model will later be incorporated into SIR type models for the transmission of trypanosomiasis with temporal variability, to assess the impact of temperature on the disease spread in both humans and livestock. A comprehensive cost analysis will be carried out to investigate the impact of various interventions in the light of temperature change.

Keywords: tsetse, model, density dependent mortality, temperature, NDVI.

Presentation content: 30% Biology -- 30% Data analysis -- 10% Mathematical Modeling and Analysis -- 30% Numerical analysis and Simulations.

14 Febryary 2017

Speaker: Mr. Mahasa                Level of study: PhD

Supervisors: Dr. Rachid Ouifki (University of Pretoria, South Africa) and Dr. Amina Eladdadi (The College of Saint Rose, Albany, NY, USA)

# Title: Mathematical modeling of oncolytic potency and reduced virus tumor-specificity in virotherapy

Coauthors: Dr. Rachid Ouifki, Dr. Amina Eladdadi and Lisette de Pillis (Harvey Mudd College, Claremont, CA, USA)

Abstract:

Background: Oncolytic virotherapy is an emerging cancer treatment modality that uses naturally occurring or genetically engineered viruses to destroy cancerous cells. This therapeutic approach, however, faces many challenges including the immune system’s response to the virus and/or infected cells, which might impede the success of therapy. Additionally, clinical evidence indicates that some oncolytic viruses have the ability to infect and replicate within normal cells as well. While this could be seen as another challenge to virotherapy, it could also be used to increase viral potency, as long as the replication within normal cells is well understood and controlled.

Aim: The aim of this study is to understand the interplay between virus tumor-specificity and the anti-viral immune response in virotherapy and investigate conditions that maximize tumor reduction and minimize losses in normal cells.

Methods: We formulate a mathematical model describing the interactions between the oncolytic virus, anti-tumoral and antiviral immune responses. The model consists of a system of delayed differential equations with one (discrete) delay. We derive the model’s basic reproductive number within tumor and normal cell populations and use their ratio as a metric for virus tumor-specificity. Numerical simulations are performed for different values of the basic reproduction numbers and their ratios to investigate potential trade-offs between tumor reduction and normal cells losses.

Results: A fundamental feature unravelled by the model simulations is its great sensitivity to parameters that account for most variation in the early or late stages of oncolytic virotherapy. Moreover, our findings indicate that when infected tissues can be regenerated, oncolytic viral infection of normal cells could improve cancer treatment.

Discussions: From a clinical point of view, our findings indicate that designing an oncolytic virus that is not 100% tumor-specific can increase virus particles, which in turn, can further infect tumor cells. In conclusion, our mathematical model shows that viral infections on normal cells can indeed augment oncolytic virotherapy if the virus replicates fast within the infected cells. Our results may be useful in the discovery of new oncolytic viruses or attenuation of known wild viral strains.

Keywords: Oncolytic virotherapy, tumor-specificity.

Presentation content: 30% Biology -- 30% Mathematical Modeling and Analysis -- 40% Numerical analysis and Simulations.

15 November 2016

### Performance of some finite difference methods for a 3D advection-diffusion equation

Hagos Gidey (Department of Mathematics and Applied Mathematics, UP)

Abstract: In this work, a new finite difference scheme is presented to discretize a 3D advection-diffusion equation following the work of Dehghan (2005). We then use this scheme and two existing schemes namely Crank-Nicolson and Implicit Chapeau function to solve a 3D advection-diffusion equation with given initial and boundary conditions. We compare the performance of the methods by computing $L_2$- error, $L_\infty$-error, dispersion error, dissipation error, total mean square error and some performance indices such as mass distribution ratio (MDR), mass conservation ratio (MCR), total mass and $R^2$ which is a measure of total variation in particle distribution. We also compute the rate of convergence to validate the order of accuracy of the numerical methods. We then use optimization techniques to improve the results from the numerical methods.

8 November 2016

### Some basic models for the population dynamics of honeybee colonies

Mataeli B. Lerata (Department of Mathematics and Applied Mathematics, UP)

View abstract here

1st November 2016

### A glimpse of the center manifold theory.

Kenneth Dukusa(Department of Mathematics and Applied Mathematics, UP)

25 October 2016

### Structural stability and bifurcation of flows and maps.

Roumen Anguelov (Department of Mathematics and Applied Mathematics, UP)

(Click on the image to view the article)

06 September 2016

### Assessing the impact of cross-immunity on the transmission dynamics of two strains of dengue.

Salisu Garba (Department of Mathematics and Applied Mathematics, UP)

Abstract: In this talk, the impact of cross-immunity on the transmission dynamics of two strains of dengue disease is presented. It shown that the model, consisting of mutually exclusive compartments representing the human and vector dynamics, can have infinitely many co-existence equilibria if infection with one strain confers complete cross-immunity against the other strain and the associated reproduction number of each strain exceeds unity. Phenomena such as competitive exclusion or co-existence of the two strains can occur under certain conditions. The effect of seasonality on the transmission dynamics of dengue is explored using numerical simulations.

16 August 2016

### Model of viral diffusion through tumour tissue: attempts to determine analytical and numerical solutions.

Joseph Malinzi (Department of Mathematics and Applied Mathematics, UP)

26 July 2016

### Diffusion in soil borne plant pathogen models.

Rebecca Bekker (Department of Mathematics and Applied Mathematics, UP)

24 May 2016

### Catching a drone with a drone: using quadcopters for honeybee population census

Yusuf Abdullahi (Department of Zoology and Entomology, UP)

View abstract here.

17 May 2016

### Multi-scale models in mathematical biosciences II.

Jacek Banasiak (Department of Mathematics and Applied Mathematics, UP)

03 May 2016

### Transport-Equilibrium scheme for computing nonclassical solutions of hyperbolic conservation laws.

Koffi Messan Agbavon (Department of Mathematics and Applied Mathematics, UP)

View abstract here.

26 April 2016

### Multi-scale models in mathematical biosciences.

Jacek Banasiak (Department of Mathematics and Applied Mathematics, UP)

19 April 2016

### Mathematical model of malaria in children under five years in an endemic area.

Baaba Ghansah (Department of Mathematics, Statistics and Computer Sciences, UKZN)

05 April 2016

### Modeling the spread of a vector-borne disease in a crop

#### 3rd UP-UNISA workshop on Theoretical and Mathematical Epidemiology (27 February - 05 March 2016) ​​

Michael Chapwanya (Department of Mathematics and Applied Mathematics, UP)

15 March 2016

### Mathematical modeling and analysis of Zika virus

#### 3rd UP-UNISA workshop on Theoretical and Mathematical Epidemiology (27 February - 05 March 2016) ​​

Yibelta Terefe (Department of Mathematics and Applied Mathematics, UP)

08 March 2016

### Development of a Human Risk Assessment Model for Underground Coal mine Dust Exposure ​​

Julize van Niekerk

Abstract: Chronic exposure to coal mine dust might result in pneumoconiosis or massive pulmonary fibrosis (and various other diseases). It is generally believed that these diseases are caused by the respirable fraction of the coal dust inhaled, with current focus given to the quartz content of the dust concentration. There are however, various factors that could contribute to the health risk; e.g. Mining Methodology,Mining Equipment used, Shift Patterns, Particle size, Mineral Composition, as well as Human Genetics. These factors however also have dynamic interactions that will impact disease formation in different ways. In my study I will evaluate various coal dust characteristics and incorporate those into a human health risk assessment model to predict a combined risk rating for coal mine workers underground. The chosen Mathematical/Statistical framework to be chosen that will give the best output of these variable needs to be determined. Due to the various parameters to be considered the weightings will have to allocated by some multivariate analysis. Current thoughts goes around using Factor Analysis (Structural Equation Modelling) or Fuzzy Logic (Neural Networks), however, any inputs and suggestions will be highly appreciated.

16 February 2016

10 November 2015

20 October 2015

### ​​A Mathematical Contribution to the Prevention of AVL-HIV Co-endemicity in South Africa

Nafiu Hussaini (Department of Mathematics and Applied Mathematics, UP)

The abstract is available here.

08 September 2015

### ​​Mathematical Model of Morphogen Gradient Formation

Americo Matusse (Department of Mathematics and Applied Mathematics, UP)

Abstract: Morphogen are signalling molecules that emanate from a restricted region of a tissue and spread away from their source to form concentration gradient. The concentration gradients of different morphogen-receptor complexes are known to be responsible for cell differentiation and patterning of biological organism. In this work, we present the two main theories to explain the formation of the gradients morphogen (Diffusion theory and Positional theory) as well as their proposed models.

1st September 2015

### ​​Nonstandard finite difference scheme for Predator-Prey model.

Jejeniwa Ayodeji (Department of Mathematics and Applied Mathematics, UP)

Abstract: In this work, nonstandard finite difference scheme which preserves positivity and stability/instability for predator-Prey model is presented and analyzed. The proposed scheme is formulated based on a non-local modeling of the growth-rate function and a nonstandard discretization of the time derivative. The techniques lead to significant improvements in the qualitative behaviour of the numerical solution.

18 August 2015

### ​​MAT model for the control of Bactrocera invadens.

Claire Dufourd (Department of Mathematics and Applied Mathematics, UP)

Abstract: Work in progress on the development of a mathematical approach to model the control of the invasive fruit fly, Bactrocera invadens, using Male Annihilation Technique (MAT).

11 August 2015

### ​​Morphological features of erythrocytes and platelets as a complication seen in rheumatoid arthritis patients

Oore-ofe Akeredolu (Department of Physiology, UP)

Abstract: We assessed properties such as cellular structure, dimensions and organelles from these patients and compared with those from healthy individuals. This is connected to potential physiological (functional) implications.

04 August 2015

### ​​Bioremediation.

Michael Chapwanya (Department of Mathematics and Applied Mathematics, UP)

Abstract: In this presentation we present a model for wastewater treatment based on the Autothermal Thermophilic Aerobic Digestion (ATAD) process. The process relies on promoting non-pathogenic thermophilic bacteria to digest organic matter and kill pathogens through metabolic heat generation.

18 May 2015

### ​​Eat, Pray, Love.

#### The life of a mosquito.

Claire Dufourd (Dpartment of Mathematics and Applied Mathematics, UP)

12 May 2015

### ​​On a seasonally responsive malaria model.

Kenneth Dukusa (Dpartment of Mathematics and Applied Mathematics, UP)

The abstract is available here.

05 May 2015

### ​​Mathematical Model or Statistical Model?

Open discussion.

A note related to the topic is available here.

(Click on the image to view the article)

21 April 2015

### ​​Characterising intraurban air pollution exposure using land use regression models

Bukola Olutola (School of Health Systems and Public Health, UP)

14 April 2015

24 March 2015

### ​​Mathematical modelling and analysis of Lipodystrophy syndrome in HIP patients Grid

#### (Project 10 from the 2nd joint UNISA-UP workshop)

Michael Chapwanya (Dpartment of Mathematics and Applied Mathematics, UP)

17 March 2015

### ​​The Diagnosis, Prediction and Prognosis of Exacerbated Asthma; the development of an Asthma Grid

Moses Kebalepile (School of Health Systems and Public Health, UP)

10 March 2015

### ​​Interspecific interactions between co-infecting ectoparasite communitiesof an ancient African mammal

Dr. Heike Lutermann (Department of Zoology and Entomology , UP)

24 February 2015

### ​​Application of NexGen health risk assessment framework to Gold nanoparticles exposure assessment

Tshepo P. Moto (School of Health Systems and Public Health, UP)

Abstract: Nanoparticles can be considered as particles in the range from 1 to 2 nanometres (nm) (clusters of atoms/molecules) to particles that are smaller than 100nm at least in one dimension. Nanoparticles are not a recent discovery though the use of them through nanotechnology applications appears to be an emerging field that has produced risk for the environment and human health. At present the synthesis and use of nanoparticles takes place in various applications in various fields including science, technology, medicine, diagnostics, etc. In spite of advances in technology the toxicology of nanoparticles is inadequately understood as there are no sufficient methods to test nanoparticles for health, safety and environmental impacts, especially in the lower than 50nm particles. The NexGen health risk assessment framework offers an accelerated method to determine risk as a result of exposure to chemical and biological substances.

17 February 2015

18 November 2014

11 November 2014

### ​​Convergent responses of ecosystem processes in Afromontane grassland

Cornel du Toit (NZG)

4 November 2014

### ​​The Hybrid Dilemma

Anri van Wyk (NZG)
 Abstract: Human-mediated hybridization poses a serious threat to the genetic integrity of threatened inter-fertile species or subspecies, such as the bontebok and blesbok as well as Blue and Black wildebeest in South Africa. The identification of pure and admixed populations is key to sound biodiversity conservation management and practices. Where identification based on morphometric measurements are complex, molecular analysis can provide a tool to identify hybrids or introgression in populations. These molecular methods however rely on estimated cut-off parameters, that when inaccurately set, could result in the misidentification of pure or hybrid animals. The effect of these misidentifications, however small, is currently unknown and has not been assessed for any animal population. We would therefore aim to model the effect of different cut-off parameters on these animal populations over the long term, in order to ascertain the effect of each on population integrity.

21 October 2014

### ​​A model for the hydration of a navy bean grain

Michael Chapwanya (Department of Mathematics and Applied Mathematics, UP)

Abstract: In this talk, we present a theoretical treatment of navy beans from a soft condensed matter perspective to describe the hydration kinetics of a single grain. The swelling of the grain is modelled as a moving boundary problem.  We present our current results.

14 October 2014

### ​​Digging deep for answers - contributions of density- and frequency-dependent factors on ectoparasite burden of a social mammal

Heike Lutermann (Department of Zoology and Entomology, UP)

07 October 2014

### ​​Visit at the NZG

16 September 2014

### ​​Mickens' SIR epidemic model with root dynamics

Jean Lubuma (Department of Mathematics and Applied Mathematics, UP)

9 September 2014

### ​​Switching from Exact Scheme to Nonstandard Finite Difference Scheme for Linear Delay Differential Equation​

Shitu Hassan Adamu (Department of Mathematics and Applied Mathematics, UP)​

2 September 2014

### ​​Influence of temperature on development of the Australian sheep blowfly

Chris Weldon (Department of Zoology and Entomology, UP)

26 August 2014

### ​​Investigating the behaviour of a predator-prey model with predator population saturation

Quay van der Hoff (Department of Mathematics and Applied Mathematics, UP)

12 August 2014

### A step toward Mathematical Models

Berge Tsanou (Department of Mathematics and Applied Mathematics, UP)