This paper presents mathematical models for tuberculosis and its dynamics under theimplementation of the direct observation therapy strategy (DOTS) in Nigeria. The models establishconditions for the eradication of tuberculosis in Nigeria based on the fraction of detectedinfectious individuals placed under DOTS for treatment. Both numerical and qualitative analysisof the models were carried out and the effect of the fraction of detected cases of active TB on thevarious epidemiological classes is investigated. The results showed that, provided that the fractionof detected infectious individuals exceeded a critical value, there exists a globally stable diseasefree equilibrium. However, if this critical detection level is not reached, the disease-free equilibriumwill be unstable even with the very high probability successful treatment under DOTS. Theresults showed that DOTS expansion in Nigeria must include significant increase in the detectionrate of infectious individuals; otherwise the effect in reducing the incidence in Nigeria will notbe achieved disregarding the tremendous efforts in any other direction, and the huge number ofundetected cases will make DOTS insignificant with respect to tuberculosis control.

A conceptual minimal model demonstrating spatially heterogeneous wave regimesinterpreted as pursuit-evasion in predator-prey system is constructed and investigated. Themodel is based on the earlier proposed hypothesis that taxis accelerations of prey and predators are proportional to the density gradient of another population playing a role of taxis stimulus. Considering acceleration rather than immediate velocity allows obtaining realistic solutions even while ignoring variations of total abundances of both modelled populations. Linear analysis of the model shows that stationary homogeneous regime becomes oscillatory unstable with respect to small heterogeneous perturbations if either taxis activities or total population abundances are high enough. The ability for active directed movement of both prey and predators is the necessary condition for spatial self-organization. Numerical simulations illustrate analytical results. The relation between the proposed model and conventional two-component systems with cross-diffusion is discussed.

Theoretical framework for linear stability of an anomalous sub-diffusive activator-inhibitor system is set. Generalized Turing instability conditions are found to depend on anomaly exponents of various species. In addition to monotonous instability, known from normal diffusion, in an anomalous system oscillatory modes emerge. For equal anomaly exponents for both species the type of unstable modes is determined by the ratio of the reactants' diffusion coefficients. When the ratio exceeds its normal critical value, the monotonous modes become stable, whereas oscillatory instability persists until the anomalous critical value is also exceeded. An exact formula for the anomalous critical value is obtained. It is shown that in the short wave limit the growth rate is a power law of the wave number. When the anomaly exponents differ, disturbance modes are governed by power laws of the distinct exponents. If the difference between the diffusion anomaly exponents is small, the splitting of the power law exponents is absent at the leading order and emerges only as a next-order effect. In the short wave limit the onset of instability is governed by the anomaly exponents, whereas the ratio of diffusion coefficients influences the complex growth rates. When the exponent of the inhibitor is greater than that of the activator, the system is always unstable due to the inhibitor enhanced diffusion relatively to the activator. If the exponent of the activator is greater, the system is always stable. Existence of oscillatory unstable modes is also possible for waves of moderate length.

We study the mathematical properties of a general model of cell division structuredwith several internal variables. We begin with a simpler and specific model with two variables, wesolve the eigenvalue problem with strong or weak assumptions, and deduce from it the long-timeconvergence. The main difficulty comes from natural degeneracy of birth terms that we overcomewith a regularization technique. We then extend the results to the case with several parameters andrecall the link between this simplified model and the one presented in [6]; an application to thenon-linear problem is also given, leading to robust subpolynomial growth of the total population.

The nonlocal Fisher equation has been proposed as a simple model exhibiting Turinginstability and the interpretation refers to adaptive evolution. By analogy with other formalismsused in adaptive dynamics, it is expected that concentration phenomena (like convergence to a sumof Dirac masses) will happen in the limit of small mutations. In the present work we study thisasymptotics by using a change of variables that leads to a constrained Hamilton-Jacobi equation.We prove the convergence analytically and illustrate it numerically. We also illustrate numericallyhow the constraint is related to the concentration points. We investigate numerically some featuresof these concentration points such as their weights and their numbers. We show analytically howthe constrained Hamilton-Jacobi gives the so-called canonical equation relating their motion withthe selection gradient. We illustrate this point numerically.

We study the existence and some properties of travelling waves in partially degenerate reaction-diffusion systems. Such systems may for example describe intracellular calcium dynamics in the presence of immobile buffers. In order to prove the wave existence, we first consider the non degenerate case and then pass to the limit as some of the diffusion coefficient converge to zero. The passage to the limit is based on a priori estimates of solutions independent of the values of the diffusion coefficients. The wave uniqueness is also proved.

A mathematical model of the tumour growth along a blood vessel is proposed. Themodel employs the mixture theory approach to describe a tissue which consists of cells, extracellularmatrix and liquid. The growing tumour tissue is supposed to be surrounded by the hosttissue. Tumours where complete oxydation of glucose prevails are considered. Special attention ispaid to consistent description of oxygen consumption and growth processes based on the energybalance. A finite difference numerical method is proposed. The level set method is used to trackan interface between the tissues. The simulations show localization of the tumour within a limiteddistance from the vessels and constant expansion velocity along the vessels.

The emergence of multidrug resistance among gram-negative bacilli is complex. Numerousfactors need to be considered, including the biological fitness cost of resistance, fitnesscompensatorymutations and frequency and type of antibiotic exposure. A mathematical modelevaluating these complex relationships was developed in an individual colonized with strains ofpan-susceptible, single-, two- and multidrug-resistant (MDR) gram-negative bacilli (GN). The effectof bacterial fitness, compensatory mutations and the frequency of three-antimicrobial regimenexposure to predominance of multidrug-resistant strains were quantified. The model predicts thatinitially, in the absence of antibiotic exposure, the biologically fitter pan-susceptible strain predominatesover the resistant strains. Over time, the fitness of the MDR strains increases fasterwith repeated antimicrobial exposure, through compensatory-fitness mutations. Increasing thefrequency of exposure to the three-antimicrobial regimen or, increasing the initial fitness of theresistant strains, substantially decreases the time to MDR-GN predominance. The model impliesthat when MDR-GN strains evolve into strains that are fitter than susceptible strains, a reductionin antimicrobial exposure may not result in a decrease of MDR-GN, since the absence of selectiveantimicrobial pressure would no longer favor susceptible strains. The model also implies thatantimicrobial cycling may promote the emergence of MDR-GN.

In this work we study the inflammatory process resulting in the development of atherosclerosis. We develop a one- and two-dimensional models based on reaction-diffusion systems to describe the set up of a chronic inflammatory response in the intima of an artery vessel wall. The concentration of the oxidized low density lipoproteins (ox-LDL) in the intima is the critical parameter of the model. Low ox-LDL concentrations do not lead to a chronic inflammatory reaction. Intermediate ox-LDL concentrations correspond to a bistable system and can lead to a travelling wave propagation corresponding to a chronic inflammatory reaction. In this case the disease development depends on the initial condition. If the concentration of monocytes in the intima is sufficiently high, which can be caused by an inflammation related to other factors, then the development of atherosclerosis can start. Otherwise, the system returns to the stable disease free equilibrium. High ox-LDL concentrations correspond to a monostable system and even a small perturbation of the non inflammatory case leads to a travelling wave propagation which corresponds to a chronic inflammatory response.

Atherosclerosis always develops in plaques, and the reasons are not clear. We test the hypothesis that plaque morphology results from a self-perpetuating propagating process driven by macrophages (Mphs). A computer model of atherogenesis was written in which the computer screen represents a surface view of a flattened area of an arterial wall on which greatly accelerated atherogenesis is depicted. Rate of Mph recruitment from blood monocytes is set as a steeply rising function of the number of Mphs locally present. Smooth muscle accumulation depends on Mph number, Mphs have a probability of death/loss, and lipid accumulation results directly from the death of Mphs. The program runs in reiterative cycles. From an initially normal wall, fatty streak-like foci of Mphs form at random sites, which may progress or regress. Some develop into progressive focal lesions resembling advanced plaques, which are Mph-rich and have a central fibrous cap-like central region of smooth muscle cells. Lipid accumulates centrally in them. To investigate a fetal origin of atherosclerosis, the simulation was initially loaded with Mphs: lesion development was greatly enhanced. These results strongly resemble atherosclerosis in vivo, and support the Mph-dependent hypothesis of spreading plaque growth.