Applied Mathematics

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Age structured models of mathematical epidemiology.
(2013) Massoukou, Rodrigue Yves M'pika.; Banasiak, Jacek.
We consider a mathematical model which describes the dynamics for the spread of a directly transmitted disease in an isolated population with age structure, in an invariant habitat, where all individuals have a finite life-span, that is, the maximum age is finite, hence the mortality is unbounded. We assume that infected individuals do not recover permanently, meaning that these diseases do not convey immunity (these could be: common cold, influenza, gonorrhoea) and the infection can be transmitted horizontally as well as vertically from adult individuals to their newborns. The model consists of a nonlinear and nonlocal system of equations of hyperbolic type. Note that the above-mentioned model has been already analysed by many authors who assumed a constant total population. With this assumption they considered the ratios of the density and the stable age profile of the population, see [16, 31]. In this way they were able to eliminate the unbounded death rate from the model, making it easier to analyse by means of the semigroup techniques. In this work we do not make such an assumption except for the error estimates in the asymptotic analysis of a singularly perturbed problem where we assume that the net reproduction rate R ≤ 1. For certain particular age-dependent constitutive forms of the force of infection term, solvability of the above-mentioned age-structured epidemic model is proven. In the intercohort case, we use the semigroup theory to prove that the problem is well-posed in a suitable population state space of Lebesgue integrable vector valued functions and has a unique classical solution which is positive, global in time and has the property of continuous dependence on the initial data. Further, we prove, under additional regularity conditions (composed of specific assumptions and compatibility conditions at the origin), that the solution is smooth. In the intracohort case, we have to consider a suitable population state space of bounded vector valued functions on which the (unbounded) population operator cannot generate a strongly continuous semigroup which, therefore, is not suitable for semigroup techniques–any strongly continuous semigroup on the space of bounded vector valued functions is uniformly continuous, see [6, Theorem 3.6]. Since, for a finite life-span of the population, the space of bounded vector valued functions is a subspace densely and continuously embedded in the state space of Lebesgue integrable vector valued functions, thus we can restrict the analysis of the intercohort case to the above-mentioned space of bounded vector valued functions. We prove that this state space is invariant under the action of the strongly continuous semigroup generated by the (unbounded) population operator on the state space of Lebesgue integrable vector valued functions. Further, we prove the existence and uniqueness of a mild solution to the problem. In general, different time scales can be identified in age-structured epidemiological models. In fact, if the disease is not terminal, the process of getting sick and recovering is much faster than a typical demographical process. In this work, we consider the case where recovering is much faster than getting sick, giving birth and death. We consider a convenient approach that carries out a preliminary theoretical analysis of the model and, in particular, identifies time scales of it. Typically this allows separation of scales and aggregation of variables through asymptotic analysis based on the Chapman-Enskog procedure, to arrive at reduced models which preserve essential features of the original dynamics being at the same time easier to analyse.
Analysis and numerical solutions of fragmentation equation with transport.
(2012) Wetsi, Poka David.; Banasiak, Jacek.; Shindin, Sergey Konstantinovich.
Fragmentation equations occur naturally in many real world problems, see [ZM85, ZM86, HEL91, CEH91, HGEL96, SLLM00, Ban02, BL03, Ban04, BA06] and references therein. Mathematical study of these equations is mostly concentrated on building existence and uniqueness theories and on qualitative analysis of solutions (shattering), some effort has be done in finding solutions analytically. In this project, we deal with numerical analysis of fragmentation equation with transport. First, we provide some existence results in Banach and Hilbert settings, then we turn to numerical analysis. For this approximation and interpolation theory for generalized Laguerre functions is derived. Using these results we formulate Laguerre pseudospectral method and provide its stability and convergence analysis. The project is concluded with several numerical experiments.
Asymptotic analysis of singularly perturbed dynamical systems.
(2011) Goswami, Amartya.; Banasiak, Jacek.
According to the needs, real systems can be modeled at various level of resolution. It can be detailed interactions at the individual level (or at microscopic level) or a sample of the system (or at mesoscopic level) and also by averaging over mesoscopic (structural) states; that is, at the level of interactions between subsystems of the original system (or at macroscopic level). With the microscopic study one can get a detailed information of the interaction but at a cost of heavy computational work. Also sometimes such a detailed information is redundant. On the other hand, macroscopic analysis, computationally less involved and easy to verify by experiments. But the results obtained may be too crude for some applications. Thus, the mesoscopic level of analysis has been quite popular in recent years for studying real systems. Here we will focus on structured population models where we can observe various level of organization such as individual, a group of population, or a community. Due to fast movement of the individual compare of the other demographic processes (like death and birth), the problem is multiple-scale. There are various methods to handle multiple-scale problem. In this work we will follow asymptotic analysis ( or more precisely compressed Chapman–Enskog method) to approximate the microscopic model by the averaged one at a given level of accuracy. We also generalize our model by introducing reducible migration structure. Along with this, considering age dependency of the migration rates and the mortality rates, the thesis o ers improvement of the existing literature.
Categorical systems biology : an appreciation of categorical arguments in cellular modelling.
(2012) Songa, Maurine Atieno.; Banasiak, Jacek.; Amery, Gareth.
With big science projects like the human genome project, [2], and preliminary attempts to seriously study brain activity, e.g. [9], mathematical biology has come of age, employing formalisms and tools from most branches of mathematics. Recent results, [51] and [53], have extended the relational (or categorical) approach of Rosen [44], to demonstrate that (in a very general class of systems) cellular self-organization/self-replication is implicit in metabolism and repair/stability. This is a powerful philosophical statement and removes the need of teleological argument. However, the result carries a technical limitation to Cartesian closed categories, which excludes many mathematical languages. We review the relevant literature on metabolic-repair pathways, category theory and systems theory, before performing a critique of this work. We find that the restriction to Cartesian closed categories is purely for simplicity, and describe how equivalent arguments may be built for monoidal closed categories. Moreover, any symmetric monoidal category may be "embedded" in a closed one. We discuss how these constructions/techniques provide the formal structure to treat self-organization/self-replication in most contemporary mathematical (modelling) languages. These results signicantly soften the impact on current modelling paradigms while extending the philosophical implications.
Matrix models of population theory.
(2013) Abdalla, Suliman Jamiel Mohamed.; Banasiak, Jacek.
Non-negative matrices arise naturally in population models. In this thesis, we first study Perron- Frobenius theory of non-negative irreducible matrices. We use this theory to investigate the asymptotic behaviour of discrete time linear autonomous models. Then we discuss an application for this in age structured population. Furthermore, we study Liapunov stability of a general non-linear autonomous model. We consider a general nonlinear autonomous model that arises in structured population. We assume that the associated nonlinear matrix of this model is non-increasing at all density levels. Then, we show the existence of global extinction. In addition, we show the stability condition of the extinction equilibrium of the this model in the Liapunov sense.
On singularly perturbed problems and exchange of stabilities.
(2015) Kimba-Phongi, Eddy.; Banasiak, Jacek.
Singular perturbation theory has been used for about a century to describe models displaying different timescales, that arise in applied sciences; particularly, models displaying two timescales, namely slow time and fast time. Different techniques have been developed over time in order to analyze the limit behaviour and the stabilities of their solutions when the small parameter tends to zero. The nature of the limit equation obtained when the small parameter tends to zero plays a major role in understanding the behaviour of the solution of singularly perturbed problems. In this thesis, we analyze the behaviour of the solution of singularly perturbed problems in the following cases. First, when the limit equation displays the Allee effect. Next, when the limit equation is structurally stable or non-structurally stable and the standard Tikhonov theorem is applicable and finally, when the quasi-steady states of the degenerate equation intersect causing an exchange of stabilities. Furthermore, we perform numerical simulations in each case to support the analytic results.
Study of singularly perturbed models and its applications in ecology and epidemiology.
(2017) Seuneu Tchamga, Milaine Sergine.; Banasiak, Jacek.
In recent years the demand for a more accurate description of real life processes and advances in experimental techniques have resulted in construction of very complex mathematical models, consisting of tens, hundreds, if not thousands, of highly coupled di erential equations. The sheer size and complexity of such models often preclude any robust, theoretical or numerical, analysis of them. Fortunately, often such models describe phenomena occurring on vastly di erent time or size scales. We focused on complex processes with two time/size scales described by systems of ordinary di erential equations. In such a case, there is a small parameter that multiplies one or more derivatives. Using the Tikhonov Theorem, we have been able to understand the asymptotic behaviour of the solution to some complex epidemiological models. Furthermore, we present analysis based on the Butuzov theorem, which, for the purpose of the discussed models, was generalized to two dimensional non-autonomous problems. We applied the developed theory on an ecological model with interactions given by the mass action law.
Transport on network structures.
(2013) Namayanja, Proscovia.; Banasiak, Jacek.
This thesis is dedicated to the study of flows on a network. In the first part of the work, we describe notation and give the necessary results from graph theory and operator theory that will be used in the rest of the thesis. Next, we consider the flow of particles between vertices along an edge, which occurs instantaneously, and this flow is described by a system of first order ordinary differential equations. For this system, we extend the results of Perthame [48] to arbitrary nonnegative off-diagonal matrices (ML matrices). In particular, we show that the results that were obtained in [48] for positive off diagonal matrices hold for irreducible ML matrices. For reducible matrices, the results in [48], presented in the same form are only satisfied in certain invariant subspaces and do not hold for the whole matrix space in general. Next, we consider a system of transport equations on a network with Kirchoff-type conditions which allow for amplification and/or absorption at the boundary, and extend the results obtained in [33] to connected graphs with no sinks. We prove that the abstract Cauchy problem associated with the flow problem generates a strongly continuous semigroup provided the network has no sinks. We also prove that the acyclic part of the graph will be depleted in finite time, explicitly given by the length of the longest path in the acyclic part.

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Browsing Applied Mathematics by Author "Banasiak, Jacek."