ABSTRACT

This thesis consists of two parts. The first part deals with existerice and approximation techniques
for finding solutions of operator equations or fixed points of operators belonging to certain
classes of mappings. The classes of mappings studied include the K-posztz~~dee finzte operators,
the suppressive mappings and accretive-type rntippings. In particular, it is proved that for a real
Banach space X, the equation Au = f , f E X, where A is a Kpd operator with the same domain
as A’, has a unique solution. An iteration process is constructed ant1 shown to converge strongly
to the unique solution of this equation. Furtherniore, an asyrnptotzc version of Kpd operators is
introduced and studied and a convergence result is proved. Drawing from the ideas of Alber [I],
Alber and Guerre-Delabriere [2, 31, suppressive and accretive-type mappings are studied in more
general settings. In particular, it is proved that if I< is a closed convex nonexpansive retract of
a real uniformly smooth Banach space E, T : I< + E, a d-weakly contractive map such that a
fixed point x* E intK of T exists then a descent-like approxirnation sequence converges strongly
to z*. A related result deals with the approximation of a fixed point of T, when K is a subset of
an arbitrary real Banach space &, aiacl.R(T) := {.c E E : Tx = z} # 0. Moreover, asymptotically
d-weakly contractzve mappings are introtluc~ed and studied and convergence result,s are proved.
The second part of the thesis deals with matliematical modelliiig of irlfectious diseases. Models for
drug-resistant malaria parasites are presented both for single pol)ula.tions of hurnans and vectors
and also for multigroup populations. Eacll’ of the models results in a system of nonlinear ordinary
differential equations, which under suitable conditions leads to a locally stable equilibrium. The
ecological significance of these ecluilibriunl poirit s emerges as a by-product. For the compartmental
models, attention is devoted to the question of quailtitative agreement with published field
observations by the application of new nonlinear least squares techniques. A time dependent
immunity model is formulated arid used :is a baseline study to investigate parameter behaviour.
Furthermore, the multigroup models are studied in Rn. The ultimate intention is to extend to
infinite dimension, thereby providing a link between the analysis of these models and some well
known and developed HilbertIBanach space theory.

 

 

TABLE OF CONTENTS

 

1 GENERAL INTRODUCTION AND PRELIMINARIES
2 Existence, Uniqueness and Approximation of a Solution for a K-Positive Definite
Operator Equation 17
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 A Local Approximation Methods for the Solution of K-Positive Definite Operator
Equations 2 6
3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
,.,.. :..’.”‘ ,..a Pa ‘
4 Approximations of Fixed points of weakly contractive Non-self Maps in Banach
Spaces 3 2
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 Preliminaries . . . . . . . . . . . .,I . . . . … . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Iterative Methods for Fixed points of Asymptotically weakly contractive Maps 43
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6 Mathematical Modelling of Drug Resistant Malaria Parasites and Vector Populat
ions 5 6
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.2 Simple host-vector model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.3 Resistant parasites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7 Some Malaria Models Treating both Sensitive and Resistant Strains in Single
and Multigroup Populations 69
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7.2 Single population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.3 Multigroup population …………………………… 74
8 A Mathematical Model for Malaria Treating both Sensitive and Resistant
Strains in a Spatially Distributed Population 79
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8.2 Spatially Distributed Population Model . . . . . . . . . . . . . . . . . . . . . . . . 80
8.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

 

CHAPTER ONE

 

GENERAL INTRODUCTION AND
PRELIMINARIES
This thesis is divided into two parts. The contributions of the first part fall within the general
area of operator theory while the second part is concerned with applications to mathematical
modelling of communicable diseases, in particular, malaria models. For the first part, we shall,
‘ in particular, devote attention to existence and approximation methods for finding solutions of
operator equations or fixed points of certain nonlinear mappings defined in a subset of a Banach
space. The classes of mappings st~di&l’k’cl;ide: the K-positive definite operators, the suppressive
operators and certain accretive-type operators.
It is well known that many physically significant problems can be modelled in terms of an
I I
initial value problem of the form
where A is an accetive-type operator defined in an appropriate Banach space. It is clear that if
the solution of equation (1.0) is independent oft, then Au = 0 and the solutions of this equation
correspond to the equilibrium of the system (1.0). Consequently, considerable research
efforts have been devoted within the past half a century to finding techniques for the determination
of zeros of accretive-type operators (see, for e.g., [17, 18, 19, 22, 52, 54, 581). The study of
operator equat.ions is partly linked with fixed point theory for; u is a fixed point of the operator
A if and only if u is the solution of the operator equation ( I – A)u = 0, where I is the identity
operator. The classical importance and application of fixed point theory can be seen largely in
the theory of ordinary differential equations. The existence or construction of a solution to a
differential equation is often reduced to the existence or location of a fixed point for an operator
defined on a subset of a space of functions. In this thesis we shall.employ fixed point techniques
where appropriate.
Direct and iterative methods for finding solutions of operator equations or fixed points of an
operator defined in an appropriate Banach space have been studied by many authors. These studies
have given rise to the development of results and techniques which are now widely available
in the literature.
Petryshyn [54] considered the operator equation .4u = f , in a Hilbert space, when A is Kpositive
definite. Let HI be a dense subspace of a Hilbert space, H. An operator T with domain
‘ D(T) 2 HI is called continuously HI invertzble if the range of T, R(T), with T considered as
an operator restricted to HI is dense in H and T has a bounded inverse on R(T). Let H be a
complex and separable HilberC space%ndaA ‘be a linear unbounded operator defined on a dense
domain D(A) in H with the property that there exist a continuously D(A)-invertible closed
linear operator K with D(A) c D(K), and a constant c > 0 such that
(1.0) (Au, Ku) 12 c(. ~.. Ku~(~u ,E D(A),
then A is called I<-positive definite (Kpd) (see e.g., Petryshyn [54]). If K = I (the identity operator)
inequality (1.0) reduces to (Au, u) 2 cJu(a~n~d ,in this case, A is called positive definite.
If in addition c = 0, A is a positive operator (or accretive operator). Positive definite operators
have been studied by several authors (see e.g.[15, 17, 18, 19, 28, 38, 581). It is clear that the
class of K-pd operators contains among others, the class of positive definite operators, and also
contains the class of invertible operators (when K = A) as its subclasses. Furthermore, Petryshyn
[54] remarked that for a proper choice of K, the ordinary differential operators of odd order, the
weakly elliptic partial differential operators of odd order, are members of the class of K-pd operators.
Moreover, if the operators are bounded, the class of Kpd operators forms a subclass of
symmetrizable operators studied by Reid [58].
In [54], Petryshyri proved the following theorem.
Theorem P If A is a Kpd operator and D(A) = D(K), then there exists a constant a! > 0 such
that for all u E D(K),
ll.4ull < allKuII.
Furthermore, the operator A is closed, R(A) = H and the equation Au = f , f E H, has a
unique solution.
In the case that K is bounded and A is closed, F. E. Browder[l9] otained a result similar to
the second part of Theorem P.
In chapter two of this thesis, we extend the definition of a Kpd operator to real Banach spaces,
X . In particular, if X is a real – separable Ban,ach space with a strictly convex dual, we prove that ,, . a*,. +. ‘,>
the equation Au = f , f E X, where A is a Kpd operator with the same domain as K has a unique
solution. Furthermore, if X = Lp (or lp), p > 2, and is separable, we construct an iteration
process which converges strongly to this solution.
(1
In chapter three, we contiriue with the study of Kpd operators. We prove convergence of a
simplified iteration sequence to the solution of a Kpd operator in real uniformly smooth Banach
spaces. We extend the notion of a Kpd operator to F’rCchet differentiable operators, and in this
case, prove convergence of an iteration scheme to the unique solution of a Kpd operator equation
in arbitrary Banach space. This result is also derived directly from the inverse function theorem.
Let E be a real normed linear space with dual E*. We denote by J the normalized duality mapping
from E to 2E* defined by
Jx = {f E E* : (x, f) = = /l f 112),
where (., .) denotes the generalized duality pairing. It is well known that if E* is strictly convex
then J is single-valued and if E* is uniformly convex thcn J is unifornlly continuous on bounded
subsets of E (see e.g., [6]). We shall denote the single-valued duality mapping by j.
Let K be a subset of a real Banach space E. A map T : K -+ K is called a strict contraction
if t,here exists k E [0, 1) such that llTx – Tyll 5 kllx – yll, and it is called nonexpansive if, for
arbitrary x, y E K, /ITx – Tyl/ < 1/17: – yl 1. The map T is called pseudocontractive if, for each
x, y E K, there exists j(x – y) E J(x – y) such that
Let I< be a nonempty convex subset of a real normed linear space E. For strict contraction
mappings, nonexpansive and pseudocontractive self-mappings T of K with a fixed point in K,
three well known iterative methods,+thewlebmted Picard method, the Mann iteration method (see,
for example [48]) and the Ishikawa iteration method (see, for example [41]) have been successfully
employed to approximate such fixed points. If, however the domain of T, D(T), is a proper subset
of E (and this is the case in several applications), and T maps K into El these iteration methods
may not be well defined. Under this situation, for Hilbert spaces, this problem has been overcome
by the introduction of the metric projection (see, for e.g. [20, 261). The advantage of this is
that if K is a nonempty closed convex subset of a Hilbert space H and PK : H -+ K is the
metric projection of H onto K, then PK is nonexpansive. This fact which is central in most of the
proofs actually characterizes Hilbert spaces and consequently, it is not available in general Banach
spaces. In this connection, Alber [I] recently introduced a generalized projection operator IIK in
an arbitrary Banach space E which is an analogue of the metric projection in Hilbert spaces.
We recall the definition of the metric projection operator PK : E -+ K, K C E. The operator
PK : E -+ K is called a metric projection operator if it assigns to each x E E its nearest point
2 E K i.e, fi is the solution for the minimization problem
Now let E be a real normed linear space and K C E. Consider the operator IIK : E -+ K defined
by
IIlcz = 2; fi : V(x, 2) = inf V(x, 0,
CEK
where
Observe that in a Hilbert space H, (1.1)r educes to V(x,() = llx – (I[$,x E H, < E K .
Existence and uniqueness of’ the operator IIK follow from the properties of the functional
V(x,() and the strict monotonicity of the mapping J (see for e.g. [4]).I n a Hilbert space, ITK
becomes PK .
Using the Lyapunov functional defined in (1.1), Alber and Guerre-Delabriere [2] introduced
the following classes of non-self mappings.
., ,? . 4 “1, 0. ‘>a ‘
Let K be a nonempty subset of a Banach space E. A map T : K -+ E is called strongly
suppressive on I< if there exists 0 < q < 1 such that for all x, y E K,
T is called weakly suppressive of class Ca(t) if there exists a continuous and nondecreasing function
Q,(t)d efined on !J?+ such that Q, is positive on !J?+{o), Q,(O) = 0, lim Q,(t)= +m and Vx,y E K ,
t++m
The map T is called n,onextensive if
It is easy to see that each weakly suppressive mapping is nonextensive, and each strongly suppressive
opcrator is weakly suppresive with @it) = (1 – q)t. In Hilbert spaces, nonextensive
I operators are nonexpansive and vice versa and strongly suppressive operators coincide with strict 1
contractions.
Assuming the existence of fixed point’s for the above classes of operators, Alber and Guerre-
I Delabriere [4] proved convergence theorems with the help of generalized projection maps. They
1 also introduced the following classes of operators. 1
A mapping T with domain D(T) and range R(T) in E is called weakly contractive (see e.g.,
I [2, 3, 41 ) if there exists a contiiiuous and nondecreasing function : [0, m) := %+ + Xf such
I that @ is positive on W+ {0), Q(0) = 0, lirn,, @it) = co and for x, y E D(T) there exists 1
j(x – y) E J(x – y) such that
(1.5) IITx -Tyll I llx – yll – @(I12 – 911).
It is called d-weakly contractive if for all x, y E D(T) there exist j(x – y) E J(x – y ) and @(t) as
I above such that
‘i’
The d-weakly contractive operators include several important classes of nonlinear operators. 1
In particular, they include the weakly contractive operators.
11 . .
In [4], Alber and Guerre-Delabriere proved, interalia, the following theorem.
Theorem AG Let T : G + H be a d-weakly contractive map, G a closed convex bounded subset
of a Hilbert space H and suppose that a fixed point x* E int(G) of T exists. Then the sequence
{x,) defined by
XI E G; zn+l := PG(xIL- ailI(xn- Tx,)), R = 1,2,. ..,
6
where PG is the metric projection onto the set G, {a,) is a sequence of positive numbers such
that Cy a, = co and lim,,+a a, = 0 converges strongly to s*. Moreover, there exist a constant
C > 0 and a bounded sequence {x,,) c {x,), 1=1,2, … such that
Furthermore,
For a Banach space El the modulus of smoothness of E is the function p~ : [0, co) -+ [0, co)
defined by
E is said to be uniformly smooth if lim = 0. Typical examples of such spaces are the
7+0+
Lebesgue Lp, the sequence lpl and the Sobolev Wpm spaces, 1 < p < oo.
It is known (see, e.g. [38, 39, 631) that pE(0) = 0. Moreover, hE(r) = r-lpE(r) is continuous,
nondecreasing and hE (0) = 0. IN
In chapter four of this thesis, we extend Theorem AG in various directions. In particular,
we prove that Theorem AG remains true in real uniformly smooth Banach spaces and without
the boundedness condition imposed on the domain. Furthermore, we prove a related convergent
theorem in our general setting when the fixed point x* of T exists but is not necessarily in the interior
of G. Finally we prove a convergent theorem for approximating a fixed point of a uniformly
continuous d-weakly contractive and bounded self-map T of G with F(T) # 8, in arbitrary real
Banach spaces.
In 1972, Goebel and Kirk [39] introduced a class of mappings generalizing the class of nonexpansive
operators. Let K be a nonempty subset of a normed space E. A mapping T : K + K is
called asymptotically nonexpansive if there exists a sequence {Ic,,), Ic, 2 1, such that limn,, Ic, =
1, and I ITnx – TnyJI 5 Icnl lx – yll for each x, y in K and for each integer n 2 1.
Later in 1993, Bruck et a1 introduced and studied another class of asymptotic nonexpansive maps.
A mapping T : K + K is called asympiotically nonexpansive i n the intermediate sense (see e.g.,
Bruck et a1 [22]) provided T is uniformly continuous and
lim sup sup (IIT7’x – Tnyll – IIx – yII) 5 0.
n,tm i :,,ycK 1
Asymptotic pseudocontractive operators have also been introduced and studied, first by Schu (see
e.g., [59]) and then by a host of other authors, as a generalization of asymptotic nonexpansive
maps. T : K + K is called asymptotic all?^ pseudocontractive if there exists a sequence {Ic,),
Ic, 2 1, lim Ic, = 1 such that
for each x, y E K.
. ,?. 4 “7. .*. , ,> ‘
It is easy to see that asymptotically pseudocontractive maps include the asymptotic nonexpansive
ones. These classes of maps have been studied by various authors.
Motivated by .Goebel and Kirk [39], Bruck eta1 [22] and Schu [59] , we introduce and study,
I1
in chapter five, the class of asym,ptotically d-weakly contractive maps.
A map T with domain D(T) and range R(T) in E will be called asymptotically d-weakly contractive
if there exists a continuous and nondecreasing function @ : [0, m) := R+ + R+ such
that @ is positive on R+{O), g(0) = 0, limt,,@(t) = m and for x, y E D(T), there exists
j(x – y) E J(x – y) such that
where {k,) is a real sequence such that k, 2 1, lim,,,k, = 1 and P is a retraction from R(T)
to D(T).
Clearly, if T is a self-map, then condition (*) reduces to the following one:
With the help of the d-weakly contractive maps, we prove a version of Theorem AG in real
uniformly smooth Banach spaces. A corollary of our theorem extends Theorem AG from Hilbert
spaces to real uniformly smooth Banach spaces. Furthermore, the boundednes requirement on the
domain in Theorem AG is dispensed with in our more general setting. A related result deals with
the approximation of fixed points of uniformly continuous asymptotically d-weakly contractive
self-maps with nonempty fixed point sets, in arbitrary real Banach spaces.
Part two
Malaria is a mosquito-borne infection caused by protozoa of the genus plasmodium. Four
species of the parasite, namely: P. falciparuin, P. vivax, P. ovale, and P. malariae infect humans.
Malaria remains the most important of the tropical diseases, being widespread throughout the
tropics, but also occurring in many temperate regions.
The parasites are transmitted by the bite of infected female mosquitoes of the genus Anopheles.
Mosquitoes become infected by feeding on the blood of infected people, and the parasites
then undergo another phase of reproduction in the infected mosquito. Clinical symptoms such as
fever, pains, and sweats may develop a few days after an infected mosquito bite.
In many parts of Africa, where malaria has long been highly endemic, people are infected so
frequently that they develop a degree of acquired immunity, and may become asymptomatic carriers
of the infection [36]. Treatment and control have become more difficult in recent years with
the spread of drug resistant strains of malaria parasites [14, 36, 621. Drugs such as chloroquine,
nivaquine, quinine, and fansidar are used for treatment. More recent and more powerful drugs
‘include mefloquine, and halofant,rine.
It is estimated that 267 milli6~’@%pk $re presently infected, with 107 million clinical cases
annually; the number of countries affected is put at 103 [62].
The biology of the four species of plasmodium,is generally similar and consists of two discrete
phases-sexual and asexual . The parasite &rates to the liver where it remains latent for several
days while replicating. The latent period is followed by penetration of red blood cells and asexual
replication within them. Asexual parasites in the blood, after surviving some developmental
period, give rise to sexual stages called gametocytes. Gametocytes can remain in the blood for
more than two years [36].
The emergence of drug-resistant strains of malaria parasite has become a significant health
problem. Recent pronouncernents by the World Health Organisation indicate the availability of
strains that are resistant to virtually all known drugs.
Among the four species of plasmodium, P. falciparum causes the most serious illness and it is
the most widespread in the tropics. This paper therefore focuses on the dynamics of P. falciparum
malaria, although the analysis is similar for all forms of malaria infections.
An early fundamental model in the art of mathematical modelling of malaria, due to Ross-
Macdonald describes the basic features of the interaction between infected humans (y) and infected
mosquito vectors (9). The model is defined as follows:
where a = bpg, p, r., p are some constants defined as follows (see for e.g. [12]):
N is the size of the human population;
M is the size of the femtle mosquito population;
. ,. , . *1. ,t’ ,
$$ is the number of female mosquitoes per human host;
/3 is the rate of biting on man by a single mosquito (number of bites per unit time);
b is the proportion of infected bites on ,man that produce an infection;
r is the per capita rate of recovery for humans (: is the average duration of infection in the
human host) ;
p is the per capita mortality rate for mosquitoes (iis t he average lifetime of a mosquito).
In this simple model, the total population of both humans and vectors is assumed fixed, so that the
dynamical variables (y and q) are the proportion infected in each population. The first equation
describes changes in the proportion of humans infected. New infections are acquired at a rate
that depends on the following factors:
(i) the number of mosquito bites per person per unit time (pg)
(ii) the probability that the biting mosquito is infected (q)
(iii) the probability that a bitten human is uninfected (1 – y)
(iv) the probability that an uninfected person thus bitten will actually become infected (b).
Infections are lost by infected people returning to the uninfected class, at a characteristic recovery
rate r. Similarly, the second equation describes changes in the proportion of mosquitoes infected.
Population changes are determined by the following factors:
(i) the number of bites per mosquito per unit time (p)
(ii) the probability that the biting mosquito is uninfected (1 – q)
(iii) the probability that the bitten human is infected (y).
The loss term (pq) arises from the death of infected mosquitoes. The loss terms for the infected
humans and infected vectors both involve death and recovery. For human hosts the recovery rate
is typically faster than the death*r.~t&,”whgrefaosr vectors the opposite is the case. The origin is
a local asymptotic equilibrium for this model if pr > [email protected] Thus infection dies out if the product
of the death rates (p and r) for the two populations is large in the sense that pr > [email protected] Thus the
above formulation is a sensible approximation.
I
However, this model is highly simplified. The model simply assumes that an infected individual
either recovers to join the susceptible group N(l – y) or dies. It fails to distinguish between
the various infected categories of human and vector hosts. Thus it cannot describe accurately the
recent trend in malaria infection.
This basic model has been studied, modified and generalized in different directions by various
authors (see for e.g., [12, 13, 14, 471). Aron and May [12] extended this model by introducing
another population group x- the latent infected humans (infected but not yet infectious). They
conjectured that if the incubation interval in the mosquito has duration T the second equation in
the basic model could be replaced by the two dynamical systems:
where the circumflex denotes evaluation at time T in the past: % = y(t – T); etc. Here, the two
categories of mosquito (uninfected and infected-and-infectious) are now replaced by three categories:
a proportion, 1 – q – z, that are uninfected; a proportion q that are infected and infectious;
and a third, new proportion z that are latent (infected but not yet infectious).
Bailey [13] considered two interacting populations-human hosts and mosquito vectors with
each group consisting of three subgroups, viz susceptibles, infectives, and isolated (recovered and
immune). These are designated by x, y, and z respectively for the human populations and x’,
y’, and z’ respectively for the vector populations. It follows that the number of new infections
occurring in the human population in time interval 6t is /3xy16t, where /3 is the infection rate.
Since the converse arrangement -is”YeQdired’to hold for vectors, namely that susceptible vectors
are infected by human infectives, the number of new infections occurring in susceptible vectors
in time interval 6t is therefore given by /3’x1y6t. If in addition the overall removal rates for the
two populations are assumed to 11e y and y’, respectively, then the numbers of removals occurring
in time 6t are yybt and y’y’6t for humans and .vectors. respectively. The system of differential
equations for the dynamic process involved is given as
X = -PxY’, X’ = -plxly + y’y’,
Y = Pxy1-YY7 Ij’ = /3’x’y – y’y1,
2 = YY 7
where p, p’, y, y’ are some constants. Due to its relatively short life-cycle, isolation by immunity
is negligible in the mosquito vectors, and hence the only isolation process is by death which is
assumed to occur equally in all groups. Hence i’ r 0 (see [14], pp. 61-68).
Let the initial states at t = 0 be (x,, yo, 0) and (xi, yb, 0). It is clear from the above equations
that for a proper epidemic outbreak to occur, y, > 0 and yo’ > 0. That is
Let N and N’ be the populations of humans and vectors respectively, so that N = x + y and
N’ = x’ + y’. It follows that if both y, and yL are small, then x, z N and xb z N’. Thus
Eliminating y, we have
Thus for an epidemic build-up to occur we must have the condition
NN’ > -YY’ PP’
This model, like the first, also describes the basic interaction between the infected human host
population and the mosquito vec.t,o;,~,~pul,ationb ut with an additional group in each population.
,?
Proposition PS [55] Consider the system
with f (., .) and g(.) continuous throughout a compact subset E of R~D.e fine P(E) as the projection
of E onto the x2 axis. Assume
I. E is positively invariant for the system,
and
2. x2 = i2 is (m equilibrium point that is globally asymptotically stable on a subset A of P(E) .
Then every trajectory startzng in -4′ = {(TIx, 2) : xz E A) tends asymptotically to a point of the
form (21, :t2), where li-l is an, eqzdibrium of XI = f (zl, &).
Consider the system
where wij, N;, gi are constants and wij > 0, Ni > 0: gi > 0, yi 2 0. Let y be the vector whose
components are yi, i=l, … ,n; A the matrix of linear terms, and Q(y) the vector of quadratic
terms in (1.7). Then the vector form of the system is
Lajmanovich and Yorke (see [44]) proved that solutions to system (1.8) are globally asymptotically
stable with limiting value dktermined by the stability modulus s(A) of the matrix A. This is
defined as the maximum real part of the eigenvalues of A, i.e.,
s(A) = max{Re X : X an, eigenvalue of A).
Precisely, the following theorem w.,a s I? g~ped,,byL.a jmanovich and Yorke.
THEOREM LY The solutions to system (1.8) approach the origin if s(A) 5 0 and approach a
unique positive equilibrium. 6 if s(A) > 0, provided yi(0) > 0 for some i. Furthermore in this case
0 < &(0) < IVi for’ each i=l, 2, … ,n. ,, . ..
Throughout this part, we shall adopt the following notations where applicable.
N denotes the total human population;
x denotes susceptibles (the number of people that are uninfected);
y denotes infectives (the number of people that are severely infected);
15
v denotes infected vectors (the population of vectors that can transmit the disease).
Where y and E’ occur simultaneously, we designate y as the number of individuals infected
with parasites that are sensitive to drugs and Y as the number of individuals that are infected
with the resistant parasites with or without the sensitive parasites. Also where applicable, v and
V denote respectively the population of vectors carrying sensitive strains and resistant strains,
with or without the sensitive strains of the parasite.
In chapter six, models describing recent trend in malaria infection are formulated. The first
model deals with single strain infection while the second incorporates resistant strains. Each of
the models results in a system of nonlinear ordinary dzfferential equations. A version of Proposition
PS is proved in R4. Our major theorem, Theorem 3 gives conditions for the existence of
different endemic states. Since major health efforts should be geared towards the elimination
of resistant parasites (and hence resistant infection) our findings and conclusions enable us to
forecast the trend in resistant infection.
In chapter seven, we formulate models treating both sensitive and resistant infections in single
and multigroup populations. We consider a case of a homogenous interacting population groups.
Theorem LY plays a major role i.n. , .t .h .e analvsis of our models. Results which forecast the trend .L .V ,
of resistant infection are established.
Finally in chapter eight we consider the more general multigroup population in Rn. Local
asymptotic equilibrium are obtained in R?,. A. general extension theorem is proved. The main
tool in this direction is the Poincar4-Bendisson theorem

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