Generalised matrix and compartmental population models. - Mwambi, H.G. (University of Nairobi, 1997)
The present thesis is concerned with the development of mathematical models for structured population species. The structuring or classification may be due to age, stage of development or a combination of both in a more general perspective. The class of matrix population models are examples of such models and have been the subject of theoretical and practical study for many years. In this work attention is focussed on vector population species which are carriers of disease agents for animals. This has therefore necessitated the investigation of a class of models which deal with the interaction of vector population species and the host population. In particular the study combines both discrete and continuous population models in order to achieve its goal. Multi-dimensional coupled differential equations have proved handy in this respect. Chapter I gives a general introduction to the work. In section 1.1 an introductory description of mathematical population models is given. In section 1.2 an overview of preliminary concepts and notations are introduced. In this section a brief description of matrix population models and continuous time models is also given. A brief review of relevant literature is presented in section 1.3. In this section, literature review specific to a stage structured population species, the brown ear tick is also given. Sections 1.4 and 1.5 deal with the statement of the problem together with the specific objectives' of the study. In section 1.6 the importance of this study is briefly mentioned. in the last section of the chapter the methodology ofhow data was acquired and analysed is given. This section is important because the study involved a practical application to validate the models. The data was for the three host brown el;&,r .tick the causal vector for East Coast fever. Chapter II reviews basic models for age structured populations. After a brief introduction we present the Lotka's Integral equation in section 2.2 where age and time are treated as continuous variables. The solution to this equation is reviewed in section 2.3 first by elementary mathematical methods in sub-section 2.3.1 and by Laplace transforms in sub-section 2.3-.2. In particular it is shown that the solution has a real root which determines the direction of increase of a population. The asymptotic behaviour .ofthe solution is given in sub-section 2.3.3. In section 2.4 we review the partial differential equation describing the evolution of the population density n(x, t) which is known as the McKendrick-von Foester equation. This model is a hyperbolic initial boundary value problem. Section 2.5 deals with the discretized age and time matrix model which requires a thorough understanding of the life table survivorship function, presented in sub- section 2.5.1. The actual formulation of the matrix model is given in sub- section 2.5.2. It is in this section where we demonstrate the connection of the matrix model and the McKendrick von Foerster model. One of the core problem in application of matrix population models is in the estimation of the matrix inputs. The derivation of the inputs is discussed in section 2.6 for two types of populations namely the birth flowpopulations and birth pulse populations. These are presented in sub-section 2.6.1 and 2.6.2 respectively. Chapter III deals with the time homogeneous matrix model and its properties. After an introduction in section 3.1 the model is presented in section 3.2 for an age structured population. The chapter brings in the idea of the complete population projection matrix which includes both pre- and post- reproductive individuals. It isshownthat after a long enough time it is the pre-reproductive part of the population which determines the projection matrix of interest. Section 3.3 outlines a list ofproperties of the population projection matrix. The theory of directed graphs wasused quite extensively to achieve 'this. The Perron-Frobenius theorem for both primitive and irreducible matrices is generally stated since it is important in the study of the limiting properties of the population projections. Sub- section 3.3.1 thus talks about the stable population theory showing the asymptotic behaviour ofthe population structure~ It is shown that the limiting population structure is independent of the initial population structure. This property is egordic in nature. Upto section 3.3 the population is structured according to age but the aim of the study is to generalise the classification. Thus in section 3.4 we present a generalized matrix model where classification is according to hath stage of development and according to age within the stage. This model is more general and can be used to study the dynamics of many population species such as insects, arthropods, plants and many more. Estimation of to matrix inputs for such a model is discussed in section 3.5. In sub-section 3.5.1 we consider estimation from transition frequency data while in sub-section 3.5.2 we consider estimation from stage duration data. Finally in sub-section 3.5.3 we consider estimation from experimental cumulative distributions. The connection between the transition probabilities in the classical Leslie model and those from experimental cumulative distributions is given in section 3.6. In chapter IV we present a mathematical model for the brown ear tick which is a three host tick and is a vector for the East Coast fever(ECF). It is a stage structured population. In section 4.1 we present several modeling approaches including terminology and definitions. In section 4.2 we present a continuous time compartmental model, cyclic in nature. The model is related to that by Metz and Diekmann(1986) for physiologically structured populations since individuals have to age within a stage with reference to chronological time before transitting to the next stage. The characteristic polynomial for the system is derived in this section and the dependence of the spectral bound on various population parameters is discussed through the implicit function theorem. We also derive the general persistent stage structure in this section. Section 4.3 gives a discussion on vector-host interaction where an additional equation describing the dynamics of the host population is added into thesystem of the 11 coupled differential equations mentioned above. Conditions for population increase or decline and co-existence of the pop- ~, ulation species are discussed. The reproduction number for the tick population as a function of host density is also discussed. Section 4.4 gives a discussion on the stability analysis of the model. Section 4.5 is on the phenomenon of competition of ticks on host which acts 'as-a regulatory mechanism for the population species not to increase without bound. In sub-section 4.5.1 we give a discussion of generalcyclic triangular systems with respect to density dependence on mortality and transition rates. An alternative method of deriving the reproduction number is also presented. In sub-section 4.5.2 we present a discussion on positive invariance paying attention to the qualitative behaviour of the s-ystem, distinguishing where the system is dissipative. In order to establish dissipativeness we find a bounded set that attracts all orbits and which is positive invariant. Sub-section 4.5.3 is on the connection between spectral radius and spectral bound; while we finish this' chapter with a simulation experiment of the model based on the brown ear tick data. In chapter V we consider the spatial distribution of the tick vector populattion. Section 5.1 is an introduction to this topic. Section 5.2 discusses on host distribution ofvector parasites then we derive the model in sub-section 5.2.1. The nullhypothesis that the on host distribution of parasites is in general asymmetric and follows the negative binomial distribution is discussed in detail. Sub-section 5.2.2 discusses the parameter estimation in the model by the MLE method, assumingthe parameters are functions of several host-specific attributes. In section 5.3density dependence and host heterogeneity on susceptibility to parasites is discussed. The effect of this on the stability of the parasite-host model is discussed insub- section 5.3.1. Section 5.4 is about the effect of on host parasite load on the reproduction ratio of the parasite population. The general model is discussed in sub-sections 5.4.1 and 5.4.2. Section 5.5 suggests a possible area of future study aimingtowards a general stochastic dynamical model particularly with respect to vectorparasite populations such as ticks. Inchapter VI wedemonstrate an application of the already developed theories tothebrown ear tick(R. appendiculatus) based on Zimbabwe data. The application isbased on a time dependent multiple matrix model incorporating seasonality and heterogeneity in the vegetation. Section 6.1 gives a general introduction while the , model is given in section 6.2. In section 6.3 we deal with the problem of matrix parametrization estimating all the required matrices in the system stating all the assumptions made: Section 6.4 is on sensitivity analysis of the model parameters and conclusions. In chaptef Vll some comments regarding the significance of the results arrived at in this thesis are made. Some areas which we think need further investigation are also pointed out.