## Generalised matrix and compartmental population models

##### Abstract

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.

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**Citation**

Ph. D Thesis#####
**Sponsorhip**

University of Nairobi#####
**Publisher**

School of Mathematics, University of Nairobi

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**Description**

Master of Science