Abstracts

The main problem in eradicating cancer cells in vivo using oncolytic viruses is that, to date, there is no full comprehension of the dynamics between oncolytic viruses and the tumor microenvironment. A mathematical model of oncolytic virus spread using a reaction-telegraph equation is presented. The model is based on an experiment where a spatial domain is populated with three components, namely, tumor cells, extra-cellularmatrix (ECM), and an oncolytic virus. Analysis of the model explores several cases in which the effect of oncolytic virus spread on a tumor and the role of the extra-cellular matrix in facilitating viral spread is evaluated. Analysis of the model includes proving existence of positive unique solutions for the temporal and spatiotemporal cases and determining steady state solutions and investigating their stability. Under a steady state assumption of tumor cells, exact solutions for the virus and ECM densities are determined. Numerical experiments are as well carried out to supplement the analitical results. Temporal model stability results reveal several possibilities which will be discussed in detail. Numerical simulations reveal that tumor cells would initially eat up the ECM but as the virus load increases, tumor cells are later lysed and eliminated and the ECM would grow to maximum capacity. Exact solutions depict that, with time, the ECM density either degenerates or exponentially grows while the virus load decays. Spatiotemporal simulations show that the tumor density grows towards the edges of the domain while the virus load decreases at the centre and the ECM density, which is initially uniform throughout the domain, becomes normally distributed and reduces with time.

Background: Brucellosis is a neglected zoonotic disease impacting global development and health. The elimination of brucellosis in livestock to prevent human brucellosis is embedded in complex socio-economic, epidemiological, public and animal health systems. Risk factors for bovine brucellosis are determined using multivariable logistic regression models. However, these models fail to separate the influence of distal determinants or the complex interrelations between multiple factors that result in brucellosis and the effects thereof at the human-cattle-farm interface.

Aim/Objectives: To model the complex interactions between proximate and distal determinants for bovine brucellosis and the effect of these on herd production variables and cattle handler symptoms of brucellosis.

Materials/Methods: Cattle farms participating in the provincial state veterinary services bovine brucellosis control program, between 2014 and 2016 were eligible for recruitment into the study. Farms were classified as either a case or control farms, where a cattle herd with 2 or more serological cattle reactors on RBT and confirmatory CFT > 60IU/ml was considered a case herd, and a control herd was one with a history of no serological reaction. Farms were also categorized into a zoonotic case farm, if one or more of the cattle handlers tested seropositive to the ELISAIgG test, or a zoonotic control farm, if no cattle handler tested seropositive on the ELISAIgG. A case control study design, using structured questionnaires and a combination of on-field and telephone interviews, was used on these farms to determine herd level risk factors for bovine brucellosis. A cross sectional sero-survey of cattle handlers on all case farms and a random selection of control farms was conducted from March to November 2016. Risk factors for four outcomes: farm study status, farm parcel status, zoonotic study status and herd symptoms status were systematically selected through univariate analysis for inclusion into a multilevel mixed-effects logistic regression model, with the group variable being farm parcel status. Variables with Odds Ratios p < 0.2 were retained. All variables selected through the univariate analysis were related to each other in a causal loop diagram, using Vensim PLE (2019), to generate a hypothetical causal web. A GSEM path model was then constructed to test this model using STATA 14.

Results: Human and herd management risk factors identified in this study, reveal that movement of cattle into a herd (p = 0.034), the presence of antelope on a farm (p = 0.010), and government sponsored farms (p = 0.004) are significantly associated to case farms. Herds were significantly clustered by farm parcel status (LR test vs. logistic model: chibar2(01) = 11.10; p >= chibar2 = 0.0004). Exposure of one or more cattle handler on a farm (ElisaIgG seropositive), was significantly associated to the presence of goats on the farm (p = 0.043). The GSEM pathway model of interaction between distal and proximate determinants is described (log likelihood = - 714.220).

Discussion: This complex causal web of brucellosis at the human-cattle-farm interface can be used to illuminate options for multi-faceted, multi-stakeholder, collaborative awareness and risk mitigation strategy.

Oncolytic virotherapy is an emerging cancer treatment modality that uses naturally
occurring or genetically engineered viruses to destroy cancerous cells. However, this ther-
apeutic approach faces many challenges including the immune system’s response to the
virus and/or infected cells, which might impede the success of therapy. Additionally, clin-
ical 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 be used to increase viral potency as long as the replication within normal cells is
well understood and controlled. In the present paper we address by means of mathematical modeling the following question: How can oncolytic virus infection of some normal cells in the vicinity of tumor cells enhance oncolytic virotherapy? We formulate a mathematical model describing the interactions between the oncolytic virus, the tumor cells, the normal cells, and the antitumoral 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. 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. 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. Moreover, our findings indicate that when infected tissues can be regenerated, oncolytic viral infection of normal cells could improve cancer treatment.

Maize Lethal Necrosis Disease (MLND) is a syner-
gism interaction between Maize Chlorotic Mottle Virus (MCMV)
and one of several viruses from the family Potyviridae which in-
clude Sugarcane Mosaic Virus (MCMV). In this talk, we
model the dynamic of SCMV and MCMV within a single growing
season of maize, where the transmission of the viruses is through
vectors. We study the qualitative analysis of the model including the
stability of the disease free equilibrium (DFE).
We use the extension of Lyapunov’s method to establish sufficient
condition for the global asymptotical stability of the DFE.

Malaria is ranked first among deadly mosquito-borne diseases in the sub-Saharan Africa. In this talk, we present a deterministic model for the transmission dynamics of malaria. The model, consisting of seven mutually exclusive compartment representing human and vector dynamics assumes a human-mosquito-human transmission. In the stability analysis, it is shown that the model has a locally asymptotically stable disease-free equilibrium (DFE) whenever the associated reproduction number is less than unity. Although there is no specific vaccine for malaria, extending the model by incorporating vaccination shows that, any future malaria vaccine will have positive impact in the community by reducing malaria burden.

We discuss microscopic and continuum cell-cell adhesion models and their derivation based on the underlying microscopic assumptions. We analyse the behavior of these models at the microscopic level based on the concept of H-stability of the interaction potential. We will derive these macroscopic limits via mean-field assumptions. We propose an improvement on these models leading to sharp fronts and intermingling invasion fronts between different cell type populations. The model is based on basic principles of localized repulsion and nonlocal attraction due to adhesion forces at the microscopic level. The new model is able to capture both qualitatively and quantitatively experiments by Katsunuma et al. (2016) [J. Cell Biol. 212(5), pp. 561--575]. We also review some of the applications of these models in other areas of tissue growth in developmental biology. We will analyse the mathematical properties of the resulting aggregation-diffusion and reaction-diffusion systems based on variational tools. We will discuss the numerical methods used for their simulation in the discussion sessions.

The lectures consist of two separate parts.

The first part is devoted to fragmentation-coagulation processes modelled through the mean-field model, the origins of which go to M. Smoluchowski. Depending on whether we treat matter as consisting of basic blocks, or as a continuum, the model takes the form of either an infinite system of nonlinear and nonlocal ordinary differential equations, or of an integro-differential equation. In the lecture we shall discuss some new applications of such equation and recent results regarding their solvability and long term behaviour.

In the second part we shall look at multiple scale systems; that is, systems driven by mechanisms acting on widely different time scales. An example of such a system is epidemiological system that include demographical processes. If we consider a disease such as flu, then the typical time scale for it a few days as compared to the time scale of human life that is 80 years; that is, almost 30000 day – the difference of over 3 orders of magnitude. Systems with coefficients differing by that much are stiff and thus inherently difficult to analyse in a robust way. Mathematically speaking, such problems belong to the class of singularly perturbed problems and there exist efficient theories that yield simplified models that are not stiff and yet preserve salient features of the original dynamics. In the talk we shall describe one approach to such systems and illustrate it on several applications.

In this talk, we present a deterministic model for the transmission dynamics of yellow
fever virus with delay in a bird-mosquito setting. The model is analysed for its qualitative properties. Analysis of the model show the existence of the phenomenon of backward bifurcation (where the stable disease-free equilibrium co-exists with stable endemic equilibrium when the associated reproduction number is less that unity). Also, condition for global asymptotic stability of the disease free
equilibrium of the model is computed. Numerical simulations to access the effect of delay (or the incubation period) are also presented.

We developed and analyzed a mathematical model for a within vector-host dynamics of the plasmodium falciparum parasite by taking into account the effect of human-antibodies during gametes fertilization and subsequent developmental stages of the parasite within the mosquito. Our model integrates the developmental stages of the parasite within the mosquito such as gametogenesis, fertilization and sporogenesis culminating in the formation of sporozoites. Quantitative and qualitative analyses for the within the vector-host model are performed. Our analysis for the within vector host dynamics showed that an increase in the efficiency of the antibodies in inhibiting fertilization results in lowering the density of sporozoites that are eventually produced. The average and total cumulative sums load of sporozoites produced during the within vector process and the entire within human-vector life cycle are quantified. We also showed that control of sporozoites within the mosquito is possible by boosting the efficiency of antibodies.

Physical aggregations such as fish schools, bird flocks, bacterial colonies, and insect swarms have characteristic structures governed by interactions between group members and the group's interaction with the environment. Nonlocal aggregation diffusion equations are used to describe the dynamics of a continuous population density. Diffusion, either local or nonlocal, allows for short-range repulsion and nonlocal aggregation allows for long-range attraction.
The interplay between attraction and repulsion allows for the emergence of a variety of stationary swarming patterns. With the use of numerical simulations, we identify the conditions on the attractive and repulsive forces that result in convergence in time towards steady states with characteristic morphologies. Additionally, the energy associated with the equation acts as a Lyapunov functional for the evolution of the equation. Hence, further properties of stable steady states are identified by considering conditions for energy minimizers.

John Hargrove and Glyn Vale
Centre of Excellence in Epidemiological Modelling and Analysis (SACEMA),
University of Stellenbosch, Stellenbosch, South Africa.

A recent publication suggests that tsetse flies (Glossina spp, vectors of trypanosomiasis) exhibit strongly negative density-dependent dispersal rates, that reducing tsetse fly densities may unleash dispersal and recolonization from neighbouring sites and that long-term benefits of control campaigns may thereby be jeopardized. Our study demonstrates, by contrast, that even if dispersal rates did indeed negative density-dependence, this would not make vector control more difficult. The theory is tested by modelling the results of a tsetse control programme in Zimbabwe. The data are well fitted using a model where tsetse are assumed to move according to a random movement (diffusion) process where the rate of diffusion is independent of population density. No evidence is found for negative density-dependent dispersal rates. It is suggested that the apparent effect is serendipitous, arising as an artefact of inappropriate analysis of inadequate genetic data. In the analysis, data have been pooled from different species of tsetse, which have different natural rates of dispersal. Moreover, the study takes no account of differences of season or of habitat. In order to provide convincing evidence on the nature if tsetse dispersal, measures are required of the dispersal rates of a single species of tsetse, at a single location and under approximately constant meteorological and habitat conditions.

Many countries especially in Africa have their national priority zoonoses in order to focus their limited resources to develop prevention and control strategies among other objectives for these top zoonotic diseases. Often, the less prioritised zoonoses are hardly reported due to several reasons and thus receive low support from national governments. However, these low ranked diseases need to be monitored because many have the potential to cause devastating health consequences in man and animals as well as negative socio-economic implications. In the face of lack of adequate data and limited resources allocated to these zoonoses, a paradigm shift in emphasising the traditional surveillance approaches for disease management is due. Concurrent incorporation of routine and traditional surveillance methods with mathematical, statistical or spatial models to predict and/or simulate probable occurrence and spread of the less prioritised zoonoses will give room to a more accurate epidemiological situation of such diseases and maximize data gathering. More so, employing the use of such models while planning and executing disease surveillance will be helpful in filling the gaps of weak surveillance capacities. Modellers, policy makers, epidemiologists and clinical health team members therefore need to collaborate actively as enshrined in the one health concept to promote the adequate understanding of the dynamics and efficient management of these less supported diseases.

Many systems in Life Sciences have been modeled by Reaction-Diffusion Equations (RDE). However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release) such that an appropriate formalism is necessary, using, for instance, Impulsive Reaction-Diffusion Equations (IRDE). While the study of traveling waves for monotone RDE has been done in several works, very little has been done in the case of (monotone) IRDE. Based on recursion equations theory, we aim to present in this talk a generic framework that handles two main issues of IRDE. First, it allows the characterization of spreading speeds in monotone systems of IRDE. Second, it deals with the existence of traveling waves for (nonlinear) monotone systems of IRDE. We apply our methodology to a system of IRDE that models tree-grass interactions in fire-prone savanna, extending previous results. Numerical simulations are also provided, including, numerical approximations of the spreading speeds.

We discuss the basic theory of Banach lattices and the operators acting between two Banach lattices (positive operators). The main aim is to discuss the properties of positive operators that have some applications in finding solutions to differential and integral equations. Since the theory of fixed point is an important tool to study various boundary value problems of ordinary differential equations, difference equations and dynamic equations on time scales, we will consider the applications of positive operators for finding positive solution of differential equations.

The present study examines the steady incompressible flow of a nanofluid over a rotating disk in the presence of binary chemical reaction, Brownian diffusion and activation energy. The model equations for flow are treated with the spectral quasilinearization method to ascertain the influence of various parameters on fluid properties of interest that include the velocity, temperature, concentration, heat and mass transfer rates and skin friction. Convergence of the numerical solutions was validated and monitored using the residual error analysis. The results have immense applications in scientific, industrial, biomedical and technological fields.

Mathematical models have seen a significant uptake in influencing policies in recent decades globally, especially in the control of infectious diseases. In this talk, we focus on how mathematical models should adapt to changing disease and policy landscapes with regards to interventions and new scientific discoveries. In particular, we focus on the recent "treatment" discoveries for Ebola and changing policies in the management of HIV/AIDS globally. The results have implications on model designing and even a reconsideration of already published research outputs. The intention is to generate robust discussions on how models should adapt to changes in policies and new scientific discoveries.