Jacek Banasiak

University of Pretoria | UP · Department of Mathematics and Applied Mathematics

A deterministic nonlinear ordinary differential equation model for mosquito dynamics in which the mosquitoes can quest for blood either within a human population or within non-human/vertebrate populations is derived and studied. The model captures both the mosquito's aquatic and terrestrial forms and includes a mechanism to investigate the impact o...

A climate-based metapopulation malaria model is formulated by incorporating human travel between zones with varying climatic factors, effective and counterfeit drug treatments, and time-periodic parameters for the mosquito population to understand the effect of human travel on malaria transmission. We study the existence, uniqueness, and stability...

We consider a recently introduced model of mosquito dynamics that includes mating and progression through breeding, questing and egg-laying stages of mosquitoes using human and other vertebrate sources for blood meals. By exploiting a multiscale character of the model and recent results on their uniform-in-time asymptotics, we derive a simplified m...

In this paper, we develop and analyze a mathematical model for spreading malaria, including treatment with transmission‐blocking drugs (TBDs). The paper's main aim is to demonstrate the impact the chosen model for demographic growth has on the disease's transmission and the effect of its treatment with TBDs. We calculate the model's control reprodu...

In this paper, we develop and analyze a mathematical model for spreading malaria, including treatment with Transmission Blocking Drugs (TBDs). The paper’s main aim is to demonstrate the impact the chosen model for demographic growth has on the disease’s transmission and the effect of its treatment with TBDs. We calculate the model’s control reprodu...

A mathematical model is developed for describing malaria transmission in a population consisting of infants and adults and in which there are users of counterfeit antimalarial drugs. Three distinct control mechanisms, namely, effective malarial drugs for treatment and insecticide-treated bednets (ITNs) and indoor residual spraying (IRS) for prevent...

The paper concerns the well-posedness and long-term asymptotics of growth--fragmentation equation with unbounded fragmentation rates and McKendrick--von Foerster boundary conditions. We provide three different methods of proving that there is a strongly continuous semigroup solution to the problem and show that it is a compact perturbation of the c...

This is a brief report on the BIOMATH-2021 International Conference and School for Young Scientists held in Pretoria, South Africa.

In this paper, we provide a systematic way of finding explicit solutions for a class of continuous fragmentation equations with growth or decay in the state space and derive explicit solutions in the cases of constant and linear growth/decay coefficients.

We present a general approach to proving the existence of spectral gaps and asynchronous exponential growth for growth-fragmentation semigroups in moment spaces $L^{1}(\mathbb{R}_{+};\ x^{\alpha }dx)$ and $L^{1}(\mathbb{R} _{+};\ \left( 1+x\right) ^{\alpha }dx)$ for unbounded total fragmentation rates and continuous growth rates $r(.)$\ such that $...

We present a general approach to proving the existence of spectral gaps and asynchronous exponential growth for growth-fragmentation semigroups in moment spaces \begin{document}$ L^{1}( \mathbb{R} _{+};\ x^{\alpha }dx) $\end{document} and \begin{document}$ L^{1}( \mathbb{R} _{+};\ \left( 1+x\right) ^{\alpha }dx) $\end{document} for unbounded total...

In this paper we present an explicit formula for the semigroup governing the solution to hyperbolic systems on a metric graph, satisfying general linear Kirchhoff's type boundary conditions. Further, we use this representation to establish the long term behaviour of the solutions. The crucial role is played by the spectral decomposition of the boun...

The paper is concerned with a system of linear hyperbolic differential equations on a network coupled through general transmission conditions of Kirchhoff's-type at the nodes. We discuss the reduction of such a problem to a system of 1-dimensional hyperbolic problems for the associated Riemann invariants and provide a semigroup-theoretic proof of i...

In this paper, we provide a systematic way of finding explicit solutions for a class of continuous fragmentation equations with growth or decay in the state space and derive explicit solutions in the cases of constant and linear growth/decay coefficients.

In this paper we present an explicit formula for the semigroup governing the solution to hyperbolic systems on a metric graph, satisfying general linear Kirchhoff's type boundary conditions. Further, we use this representation to establish the long term behaviour of the solutions. The crucial role is played by the spectral decomposition of the boun...

In this paper, we provide a brief survey of mathematical modelling of malaria and how it is used to understand the transmission and progression of the disease and design strategies for its control to support public health interventions and decision-making. We discuss some of the past and present contributions of mathematical modelling of malaria, i...

In many fields of science there is the chicken or the egg dispute—whether applications drive theory, or the theory makes applications possible. Actually, in mathematics, there is another option, when certain concepts existed both in applications and in pure theory, happily oblivious of each other. An example of such concepts are order and positivit...

Hyperbolic systems on networks often can be written as systems of first order equations on an interval, coupled by transmission conditions at the endpoints, also called port-Hamiltonians. However, general results for the latter have been difficult to interpret in the network language. The aim of this paper is to derive conditions under which a port...

Recently, promising clinical advances have been made in the development of antimalarial drugs that block the parasite transmission and also cures the disease and has prophylactic effects, called transmission-blocking drugs (TBDs). The aim of this paper is to develop and analyze a population level compartmental model of human-mosquito interactions t...

The paper is concerned with a system of linear hyperbolic differential equations on a network coupled through general transmission conditions of Kirchhoff's type at the nodes. We discuss the reduction of such a problem to a system of 1-dimensional hyperbolic problems for the associated Riemann invariants and provide a semigroup theoretic proof of i...

Hyperbolic systems on networks often can be written as systems of first order equations on an interval, coupled by transmission conditions at the endpoints, also called port-Hamiltonians. However, general results for the latter have been difficult to interpret in the network language. The aim of this paper is to derive conditions under which a port...

In this paper we prove the global in time solvability of the continuous growth--fragmentation--coagulation equation with unbounded coagulation kernels, in spaces of functions having finite moments of sufficiently high order. The main tool is the recently established result on moment regularization of the linear growth--fragmentation semigroup that...

The paper draws attention to the asymptomatic and mildly symptomatic cases of COVID-19, which, according to some reports, may constitute a large fraction of the infected individuals. These cases are often unreported and are not captured in the total number of confirmed cases communicated daily. On the one hand, this group may play a significant rol...

In this note we provide a new proof of the Tikhonov theorem for the infinite time interval and discuss some of its applications.

The existence and occurrence, especially by a backward bifurcation, of endemic equilibria is of utmost importance in determining the spread and persistence of a disease. In many epidemiological models, the equation for the endemic equilibria is quadratic, with the coefficients determined by the parameters of the model. Despite its apparent simplici...

In this paper, we show that for a large class of singularly perturbed problems, the classical Chapman‐Enskog asymptotic procedure leads, in a shorter way, to the same asymptotic expansion as the renormalization group (RG) approach. We also prove that the Chapman‐Enskog expansion gives the expected error estimates uniformly on [0,∞).

In this note we explore the concept of the logarithmic norm of a matrix and illustrate its applicability by using it to find conditions under which the convergence of solutions of regularly perturbed systems of ordinary differential equations is uniform globally in time.

This book features selected and peer-reviewed lectures presented at the 3rd Semigroups of Operators: Theory and Applications Conference, held in Kazimierz Dolny, Poland, in October 2018 to mark the 85th birthday of Jan Kisyński. Held every five years, the conference offers a forum for mathematicians using semigroup theory to discover what is happen...

In this note we provide a new proof of the Tikhonov theorem for the infinite time interval and discuss some of its applications.

A mathematical model for malaria in humans was developed to explore the effect of treatment on the transmission and control of malaria. The model incorporates effective antimalarial and substandard drugs as treatment for infectious humans. The reproduction number \(R_{0}\) is evaluated, and shown to increase due to the presence of partially recover...

In this paper, we consider discrete growth–decay–fragmentation equations that describe the size distribution of clusters that can undergo splitting, growth and decay. The clusters can be for instance animal groups that can split but can also grow, or decrease in size due to birth or death of individuals in the group, or chemical particles where the...

In this paper we prove the existence of global classical solutions to continuous coagulation-fragmentation equations with unbounded coefficients under the sole assumption that the coagulation rate is dominated by a power of the fragmentation rate, thus improving upon a number of recent results by not requiring any polynomial growth bound for either...

We consider structured population models in which the population is subdivided into states according to certain feature of the individuals. We consider various rules allowing individuals to move between the states; it may be physical migration between geographical patches, or the change of the genotype by mutations during mitosis. We shall see that...

A modelling framework that describes the dynamics of populations of the female Anopheles sp mosquitoes is used to develop and analyse a deterministic ordinary differential equation model for dynamics and transmission of malaria amongst humans and varying mosquito populations. The framework includes a characterization of the gonotrophic cycle of the...

In this paper we consider a delayed exchange of stability for solutions of a singularly perturbed nonautonomous equation in the case when a backward bifurcation of its quasi-steady (critical) manifolds occurs. This result is applied to provide a precise description of canard solutions to singularly perturbed predator–prey models of Rosenzweig–MacAr...

Fragmentation--coagulation processes, in which aggregates can break up or get together, often occur together with decay processes in which the components can be removed from the aggregates by a chemical reaction, evaporation, dissolution, or death. In this paper we consider the discrete decay--fragmentation equation and prove the existence and uniq...

In this paper we study the discrete coagulation--fragmentation models with growth, decay and sedimentation. We demonstrate the existence and uniqueness of classical global solutions provided the linear processes are sufficiently strong. This paper extends several previous results both by considering a more general model and and also signnificantly...

This is a brief exposition of dynamic systems approaches that form the basis for linear implicit evolution equations with some indication of interesting applications. Examples in infinite-dimensional dissipative systems and stochastic processes illustrate the fundamental notions underlying the use of double families of evolution equations intertwin...

In this note we consider the population the model of which, derived on the basis of ethnographical accounts, includes a projection matrix with both positive and negative entries. Interpreting the eventually negative trajectories as representing the collapse of the population, we use some classical tools from convex analysis to determine a cone cont...

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. Though nature was studied by the ancient civilizations of Mesopotamia, Egypt, the Indian subcontinent and China, the origins of modern bio...

In this article we consider asymptotic properties of network flow models with fast transport along the edges and explore their connection with an operator version of the Euler formula for the exponential function. This connection, combined with the theory of the regular convergence of semigroups, allows for proving that for fast transport along the...

In the paper we present a survey of recent results obtained by the author that concern discrete fragmentation–coagulation models with growth. Models like that are particularly important in mathematical biology and ecology where they describe the aggregation of living organisms. The main results discussed in the paper are the existence of classical...

Background and Significance of the topic: Complexity of many advanced models practically precludes its robust analysis. Fortunately, in many cases the models involve multiple time or size scales and thus yield themselves to asymptotic analysis that allows for significant simplifications of them without losing essential features of their dynamics. M...

Fragmentation-coagulation processes, in which aggregates can break up or get together, often occur together with decay processes in which the components can be removed from the aggregates by a chemical reaction, evaporation, dissolution, or death. In this paper, we consider the discrete decay-fragmentation equation and prove the existence and uniqu...

In recent papers it was shown that, under certain conditions, the -semigroup describing a flow on a network (metric graph) that contains terminal strong (ergodic) components converges to the direct sum of periodic semigroups generated by the flows on these components. In this note we shall provide an explicit description of these limit semigroups i...

The spread of a particular trait in a cell population often is modelled by an appropriate system of ordinary differential equations describing how the sizes of subpopulations of the cells with the same genome change in time. On the other hand, it is recognized that cells have their own vital dynamics and mutations, leading to changes in their genom...

In this paper we provide an elementary proof of the existence of canard solutions for a class of singularly perturbed predator-prey planar systems in which there occurs a transcritical bifurcation of quasi steady states. The proof uses a one-dimensional theory of canard solutions developed by V. F. Butuzov, N. N. Nefedov and K. R. Schneider, and an...

In this paper we provide an elementary proof of the existence of canard solutions for a class of singularly perturbed predator-prey planar systems in which there occurs a transcritical bifurcation of quasi steady states. The proof uses a one-dimensional theory of canard solutions developed by V. F. Butuzov, N. N. Nefedov and K. R. Schneider, and an...

In this paper we consider a general macro-model describing a metapopulation consisting of several interacting with each other subpopulations connected through a network, with the rules of interactions given by a system of ordinary differential equations. For such a model we construct two different micro-models in which each subpopulation has its ow...

In this paper we consider systems of transport and diffusion problems on one-dimensional domains coupled through transmission type boundary conditions at the endpoints and determine what types of such problems can be identified with respective problems on metric graphs. For the transport problem the answer is provided by a reformulation of a graph...

The standard version of the epidemiological model with continuous age structure (Iannelli, Mathematical Theory of Age-Structured Population Dynamics, 1995) consists of the linear McKendrick model for the evolution of a disease-free population, coupled with one of the classical (SIS, SIR, etc.) models for the spread of the disease. A natural functio...

One of the aims of systems biology is to build multiple layered and multiple scale models of living systems which can efficiently describe phenomena occurring at various level of resolution. Such models should consist of layers of various microsystems interconnected by a network of pathways, to form a macrosystem in a consistent way; that is, the o...

Models describing transport and diffusion processes occurring along the edges of a graph and interlinked by its vertices have been recently receiving a considerable attention. In this paper we generalize such models and consider a network of transport or diffusion operators defined on one dimensional domains and connected through boundary condition...

With the unifying theme of abstract evolutionary equations, both linear and nonlinear, in a complex environment, the book presents a multidisciplinary blend of topics, spanning the fields of theoretical and applied functional analysis, partial differential equations, probability theory and numerical analysis applied to various models coming from th...

Perturbations of Positive Semigroups with Applications is a self-contained introduction to semigroup theory with emphasis on positive semigroups on Banach lattices and perturbation techniques. The first part of the book, which should be regarded as an extended reference section, presents a survey of the results from functional analysis, the theory...

Mathematical Modelling in One Dimension demonstrates the universality of mathematical techniques through a wide variety of applications. Learn how the same mathematical idea governs loan repayments, drug accumulation in tissues or growth of a population, or how the same argument can be used to find the trajectory of a dog pursuing a hare, the traje...

In our terminology, a kinetic type equation describes an evolution of a population of objects, depending on attributes from a certain set Ω, subject to a given set of conservation laws. Equations of this type also are referred to as Master Equations.

Age structure of a population often plays a significant role in the spreading of a disease among its members. For instance, childhood diseases mostly affect the juvenile part of the population, while sexually transmitted diseases spread mostly among the adults. Thus, it is important to build epidemiological models which incorporate the demography o...

Multiple time scales are common in population models with age and space structure, where they are a reflection of often different rates of demographic and migratory processes. This makes the models singularly perturbed and allows for their aggregation which, while significantly reducing their complexity, does not alter their essential dynamic prope...

1 Small parameter methods - basic ideas.- 2 Introduction to the Chapman-Enskog method - linear models with migrations.- 3 Tikhonov-Vasilyeva theory.- 4 The Tikhonov theorem in some models of mathematical biosciences.- 5 Asymptotic expansion method in a singularly perturbed McKendrick problem.- 6 Diffusion limit of the telegraph equation.- 7 Kinetic...

We develop a Laguerre-type pseudo-spectral scheme for solving transport–fragmentation equations. The method converges rapidly for certain type of fragmentation problems and works under mild restrictions on the growth rate of the coefficients of the equation. Rigorous stability and convergence analyses are provided. Numerical simulations illustrate...

In this paper we extend some of the previous results for a system of transport equations on a closed network. We consider the Cauchy problem for a flow on a reducible network; that is, a network represented by a diagraph which is not strongly connected. In particular, such a network can contain sources and sinks. We prove well-posedness of the prob...

The paper is concerned with asymptotic analysis of a singularly perturbed system of McKendrick equations of population with age and geographical structure. It is assumed that the migration between geographical patches occurs on a much faster time scale than the demographic processes and is described by a reducible Kolmogorov matrix. We apply a nove...

In this chapter we analyse a model which describes motion of individuals who may switch the direction of motion according to the prevalent direction of other individuals in their neighbourhood. The small parameter in this model is related to the mean time between the changes of the direction of motion. The main result of the chapter is that if this...

In this chapter we generalize the examples from Chap. 2 by allowing for a continuous age structure of the population. This leads to the McKendrick model which is a system of partial differential equations with nonlocal boundary conditions. Here, the Tikhonov theorem cannot be applied and we use the asymptotic expansion introduced in Chap. 2. The pr...

In this chapter we provide a gentle introduction of the Chapman–Enskog-type asymptotic expansion and of the basic techniques of proving its convergence. To make the presentation not too technical, it is illustrated on systems of linear ordinary differential equations. The chapter begins with a survey of necessary results from linear algebra and the...

We introduce basic ideas of asymptotic analysis and present a number of models which describe complex processes, the components of which occur at significantly different rates. Such models in a natural way contain a small parameter which is the ratio of the slow and the fast rates, thus lending themselves to asymptotic analysis. We discuss, among o...