A mathematical model of the population dynamics of disease ...

Journal of Biological Dynamics Vol. 5, No. 4, July 2011, 335–365

A mathematical model of the population dynamics of disease-transmitting vectors with spatial consideration Siewe Nourridinea , Miranda I. Teboh-Ewungkemb * and Gideon A. Ngwaa a Department

of Mathematics, University of Buea, P.O. Box 63, Buea, Cameroon; b Department of Mathematics, Lafayette College, Easton, PA 18042, USA (Received 29 September 2009; final version received 10 July 2010 )

A deterministic model with spatial consideration for a class of human disease-transmitting vectors is presented and analysed. The model takes the form of a nonlinear system of delayed ordinary differential equations in a compartmental framework. Using the model, existence conditions of a non-trivial steadystate vector population are obtained when more than one breeding site and human habitat site are available. Model analysis confirms the existence of a non-trivial steady state, uniquely determined by a threshold j parameter, R0 , whose value depends on the distribution and distance of breeding site j to human habitats. Results are based on the existence of a globally and asymptotically stable non-trivial steady-state human population. The explicit form of the Hopf bifurcation, initially reported by Ngwa [On the population dynamics of the malaria vector, Bull. Math. Biol. 68 (2006), pp. 2161–2189], is also obtained and used to show that the vector population oscillates with time. The modelling exercise points to the possibility of spatial–temporal patterns and oscillatory behaviour without external seasonal forcing. Keywords: vector-breeding sites and human habitat; Hopf bifurcation; flight range domain; population dynamics; compartmental framework 2000 AMS Subject Classification: 34A, 34C, 92B

1.

Introduction

Vector-borne diseases (e.g. malaria, dengue, fever, yellow fever, lyme disease, trypanosomiases, and leishmania), amongst all the human infectious diseases, continue to remain a public health concern and a severe burden on economies, causing high human mortality in the world. These diseases have not only posed problems to national economies, but have also caused poverty and low living standards, especially in countries in the tropical and subtropical regions of the worlds. For example, the vector-borne disease, malaria, caused by the plasmodium parasite and transmitted from one human to another by the female anopheles vector mosquito, continues to plague the world especially the developing nations. By the WHO World malaria report [22], the parasite, and hence malaria, caused an average of nearly 900,000 thousand deaths in 2006, of which 85% *Corresponding author. Email: [email protected] Author Emails: [email protected]; [email protected]

ISSN 1751-3758 print/ISSN 1751-3766 online © 2011 Taylor & Francis DOI: 10.1080/17513758.2010.508540 http://www.informaworld.com

336

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were of children under the age of five. Also, dengue fever, yellow fever, trypanosomiases, and leishmania are all highly prevalent tropical and subtropical diseases. Some vector-borne diseases, such as malaria, dengue, and yellow fever, that used to be common in some developed nations of the world have been successfully put under control [3]. However, these diseases are still a threat to developing nations and hence a potential threat to many regions of the world. Given recent trends in climate change [2], global warming [6,7], and increased movement between different nations, disease-transmitting vectors may be able to (re)-colonize and survive in zones not formerly possible. Therefore, understanding the population dynamics of vectors that transmit diseases is an important area of scientific inquiry. The main agents necessary for a successful transmission of a vector-borne disease are: the transmitting vector,1 the infectious agent,2 and the host.3 For the success of these diseases, the infectious agents adapt their life cycle so that part of it is harboured in the host and the other part in the vector, with the vector being the vehicle that transports the disease agent from one host to another. Hence, it is essential to study and understand the mechanism by which these transmitting vectors operate from a mathematical modelling perspective. Here, our focus will be on these vectors, with the main objective being to understand the dynamics of the transmitting vehicles – the vectors, by taking into consideration their behavioural aspects, their interaction with hosts, and their breeding patterns and habitats. Our special interest will be on insect-transmitting vectors though the ideas can easily be applied to any blood-sucking vector. The geographical and local distribution of different kinds of disease-transmitting vectors in different regions is determined by a complex set of factors – climatic and geographic factors, or may even be associated with hosts endangered by the vectors [15]. However, the basic transmission pattern for most vector-borne diseases are similar. The vectors interact with the host in some way, for the most part, in search of blood or other factors that can enhance its successful existence over different generations. During this interaction and search, the vector may either be successful or may fail and seek to try again, or may be killed. Upon a successful interaction, the vector may be infected by an infectious agent or on the other hand infect the host with an infectious agent. In our analysis, we will consider that a vector actively seeks the host [8] and hence seeks the interaction with a host. In addition, the vector lives, rests, and breeds in a habitat (vector-breeding site) and the host also lives and reproduces in its habitat with both habitats interacting. In trying to model the disease-transmitting vectors, we will focus on blood-feeding vectors with distinct stages of development, which include an egg laying, larval development, and adult eclosion vector stage, harboured in different breeding sites and habitats. In our modelling, these metamorphic stages of development will be accounted for. In addition, the model will take into consideration the human populations in their habitats that are endangered by the disease-transmitting vectors (not necessarily the disease) and the interactions between the two habitats. Mathematical models to study such vector dynamics and in particular mosquito dynamics have been rare. Most models have focused on the population dynamics of disease agents, e.g. in the case of malaria see [11,13,14,17,19], or on the dynamics of the disease agent within a host [9] or within a vector [18,20]. Ngwa [12] introduced a deterministic delayed differential equation model of the population dynamics of the malaria vector when only a single host habitat and single vector (mosquito) habitat or breeding site is concerned. His model incorporated vector deaths in stages prior to the adult vector eclosion stage. With his model, he showed that when a non-zero steady-state vector population density exists, it can be stable but can also be driven to instability via a Hopf bifurcation to periodic solutions. Here, we extend the deterministic model by Ngwa [12], to include more than one vectorbreeding site and also more than one host habitat. The goal will be to understand how variation in the number of human habitats or variation in the number of breeding sites enhances or affects the interaction between the vector habitat or breeding sites and the host habitat dynamics and how this ultimately affects the dynamics and existence of the vector populations. Secondly, the solutions

Journal of Biological Dynamics

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Figure 1. Schematic framework showing the inter-relationship between the different classes of vector populations and also the life cycle of vectors at breeding site.

derived from the mathematical model will be obtained and studied to ascertain that oscillatory dynamics known to exist within disease vector populations are captured. The compartmental framework on which we base our model is displayed in Figure 1. The rest of the paper is organized as follows: In Section 2, we define the notations used, describe the methodology used, outline the essential assumptions, and briefly derive the mathematical model. In Section 3, the analysis of the interactive model is presented. In addition, we analyse the existence and stability of the steady-state vector and host population densities. In Section 4, we simplify our model for the vector population densities by studying a simple case (one-human habitat–one-vector-breeding site). For this simple case, we establish the stability of the non-zero steady state with help of Hopf bifurcation methods. We round up with a conclusion and discussion in Section 5.

2.

Derivation of the model

The basic model divides the entire vector population into three compartmental classes representing physiological status. These classes are: the class of fed and reproducing vectors returning from human habitats to vector-breeding sites represented by the variable U ; the class of unfed and resting vectors present at vector-breeding sites represented by the variable V ; and the class of unfed vectors questing (or foraging) for food (blood meal) in human habitats represented by the variable W . Each class of vectors is again subdivided into subclasses representing spatial locations. If we assume that there are M human habitats x1 , . . . , xM , and N vector-breeding sites y1 . . . , yN , then the classes of vectors U , V , and W are subdivided into subclasses Ui , Vj , and Wi , i = 1, . . . , M, j = 1, . . . , N, respectively, where Ui ’s represent fed vectors returning from location xi , Wi ’s represent unfed vectors foraging for food at location xi , and Vj ’s are unfed and resting vectors resting at the vector-breeding site yj . With these considerations, at time t we have U (t) =

M i=1

Ui (t);

V (t) =

N

Vj (t);

W (t) =

j =1

M

Wi (t).

(1)

i=1

If Nv (t) is the total density of vectors at time t, then we have Nv (t) = U (t) + V (t) + W (t).

(2)

338


The human population, on the other hand, is divided into classes representing spatial locations. This means that if the density of the entire human population at time t is H (t), then H (t) is divided into classes Hi (t), i = 1, . . . , M, where each Hi (t) represents the density of humans present at location xi at time t, ∀i. We also have H (t) =

M

(3)

Hi (t).

i=1

For the analysis, it is assumed that vectors of type Vj are located at breeding sites yj , j = 1, . . . , N, in which they grow by undergoing four stages of metamorphosis (egg → larva → pupa → adult). They experience natural death at rate μVj . In the interactive model, we take into consideration only adult female vectors. Thus, all vectors present at the breeding site yj are either young emerging adult vectors or vectors that have just returned from the human habitat. Also, vectors located at yj , j = 1, . . . , N, can make visits to the human habitat xi , i = 1, . . . , mj , mj ≤ M, in search of a blood meal. When a vector leaves the breeding site and arrives at human habitat xi (to forage for blood meal), it immediately becomes a vector of type Wi , and the decision by such a vector to visit a particular human habitat, xi , is influenced by how far the human habitat is from the vector-breeding site, as well as the number of resource agents (humans) present at the human habitat. At habitat xi , vectors Wi interact with human according to standard mass action principle with a contact rate τi . This interaction is successful with probability p ∈ [0, 1] and a blood meal is taken, or else, it is unsuccessful with probability (1 − p) and the vector is assumed killed. Vectors of type Wi experience natural death with rate denoted by μWi . It is assumed that all vector-breeding sites are equally safe and that there is a constant alternative blood source for the vectors (may be from animals). However, we assume that the anthropophilic4 vectors, which form the basis of our mathematical study, have such a strong preference for human blood that they would fail to live in the absence of humans. Vectors which have successfully obtained a blood meal at human habitat xi join the class Ui of fed vectors returning to breeding site yj , j = 1, . . . , ni , ni ≤ N . We assume that the decision by such a vector to go to a particular breeding site depends on how far the breeding site is from the human habitat, and perhaps on the degree of safety and availability of other vectors at that site. Vectors of type Ui experience a natural death at rate μUi . In this paper, we do not take into consideration intervisitation of breeding sites by vectors. Next, humans reside at human habitat xi , i = 1, . . . , M, and for each i, the human population density at time t is labelled Hi (t). We assume that there is a net constant migration, Ci , of humans in habitat xi and that the humans experience a per capita natural death rate μHi . The term Ci incorporates births, immigration, and emigration. Humans from habitat xi migrate to other human habitat xk at rate wik , i, k = 1, . . . , M. In human habitat xi , the humans are visited by vectors and there is an interaction between the two species based on standard mass action contact. It is assumed at this stage that the humans do not suffer any deaths that can be directly associated with their contact and interaction with the vectors.5 However, every vector that interacts with a human and fails to obtain a blood meal is assumed killed. For a two-dimensional consideration, let d2 : R2 → R be the usual metric on R2 . In other words, ∀x, y ∈ R2 ,

with x = (x1 , x2 ), y = (y1 , y2 ), we have d2 (x, y) = x − y = (x1 − y1 )2 + (x2 − y2 )2 .

Then, d2 is the standard Euclidean distance between the points x and y in R2 . We then define the closed ball centred at x ∈ R2 of radius d, denoted Bd (x; d2 ) by Bd (x; d2 ) := {y ∈ R2 : d2 (x, y) ≤ d},

where d > 0.

(4)


339

This closed ball is a disc with radius d and centre x in Euclidean R2 . Define exp(−yj − xi 2 ), yj ∈ Bd (xi ; d2 ) Aij := 0, otherwise j = 1, . . . , N and

Bj i :=

exp(− xi − yj 2 ), xi ∈ Bd (yj ; d2 ) 0, otherwise i = 1, . . . , M.

(5)

(6)

Then, Aij and Bij are smallest when yj − xi is ‘far’ from zero and largest when yj − xi is ‘near’ zero. Aij and Bij may be regarded as flow rates from breeding site yj to human habitat xi and from human habitat xi to breeding site yj , respectively. It is evident that both forms for Aij and Bij suit our assumption that more vectors flow from breeding sites to nearest human habitats and vice versa. Additionally, in our definitions for Aij and Bij , the radius d will represent the maximum flight range of vectors. It can be shown [12] that the net rate of adult eclosion at breeding site j resulting from vectors coming from human habitat sites i, for i = 1, . . . , M, is χ˜ ij (t) = Aij λ(Ui (t − Te − Tl − Tg ))Ui (t − Te − Tl − Tg )e−μe Te −μl Tl −μg Tg ,

(7)

where Te , Tl , and Tg represent, respectively, the maturation times of the vector in previous life stages, λ : R+ → R+ is a birth rate function for vectors at breeding site yj , μe , μl , and μg are, respectively, the natural death rates for egg, larva, and pupa, the earlier life stages of the vector. Formula (7) represents the progeny of some fed vectors, Ui (t), which returned from human habitat xi to the breeding site yj and laid eggs Te + Tl + Tg units of time ago. Each of the exponential expressions e−μe Te , e−μl Tl , and e−μg Tg represent the probability of survival at a particular developmental stage (respectively, egg, larva, and pupa). Therefore, if we take into consideration the progeny of fed vectors from all human habitats that are within the flight range of vectors at the particular breeding site yj , we have the total sum χj (t) =

mj

χ˜ ij (t),

j = 1, . . . , N.

(8)

i=1

For simplicity, we assume that μe = μl = μg and set T = Te + Tl + Tg , with all parameters positive. 2.1. Dynamics of human population Let the density of human population at habitat xi be Hi (t). Let also Ci incorporate births, emigration, and immigration. Suppose that these resource agents, Hi , experience a constant natural death rate μHi and migrate to human habitat xj at rate wij (see the flow diagram in Figure 2 which describes this situation). Then, the rate of change of Hi (t) with respect to time t could be written as follows: M M dHi (t) = Ci − wij Hi (t) + wj i Hj (t) − μHi Hi (t), dt j =1,j =i j =1,j =i

where Ci , wij , and μHi are positive constants, ∀i, j = 1, . . . , M.

i = 1, . . . , M,

(9)

340

Figure 2.


Dynamic interplay or migration between humans at habitat xi and humans at habitat xj .

2.2. Dynamics of vector population with spatial characteristics Vectors at the breeding site (Vj ): From Equation (8) and the assumptions μe = μl = μg and T = Te + Tl + Tg , we have that χj (t) =

mj

Aij λ(Ui (t − T ))Ui (t − T )e−μe T ,

j = 1, . . . , N,

i=1

is the total progeny at breeding site yj of vectors Ui that returned from all human habitats xi , i = 1, . . . , M, that are within the flight range of vectors at breeding site yj , with T being the time lapse between egg-laying and adult vector eclosion. We assume that a fraction αv H H = , αv H + β v A K +H

where K =

βv A, αv

of vectors prefer human blood meal. Here, αv is the anthropophilic factor,6 while βv is the zoophilic factor7 of vectors. In addition, let us take Aij and Bij to be the respective flow rates of vectors from breeding site yj to some human habitat xi and that of vectors from human habitat xi to some breeding site yj as defined in Equations (5) and (6), respectively. Also, let these vectors at the breeding site yj die naturally at a constant rate μVj . Then using the flow diagram in Figure 3, showing the dynamic interplay between vectors at breeding site yj and human habitat xi , the rate of change of Vj (t) with respect to time t may be written as dVj (t) Hi = Aij λ(Ui (t − T ))Ui (t − T )e−μe T − Bj i Vj (t) dt K + Hi i=1 i=1 mj

+

mj

mj

Aij Ui (t) − μVj Vj (t),

j = 1, . . . , N,

i=1

Figure 3.

Schematic framework that models the flow of vectors Vj (t) from breeding site yj to human habitat xi .

(10)


341

Figure 4. Changes in the population density of questing vectors Wi with respect to time t, resulting from their interaction with humans at the human habitat xi .

where the sum is taken over all human habitat sites that communicate with the vector-breeding site, yj . Here, we assume that a human habitat site and a vector-breeding site communicate with each other if both sites are within the flight range, d, of the vector. Vectors visiting resource agents (questing vectors) (Wi ): Vectors from a particular breeding site, yj , are attracted to human habitat, xi (see the flow diagram in Figure 4), at the rate Bj i

Hi K + Hi

and return to breeding site yj with rate Aij , which is determined solely by proximity (we have chosen not to include whether the presence of vectors at a breeding site may aid returning vectors in choosing that breeding site because we are not aware of any strong scientific evidence that supports these claims). We consider that these questing vectors interact with humans in human habitats according to the principle of mass action, with contact rate τi∗ , and that they succeed with probability p ∈ [0, 1] and fail with probability (1 − p), in which case they are assumed killed. We further consider that the questing vectors experience a constant natural death rate μWi . Figure 4 displays a flow diagram that shows the changes occurring in the vectors after interacting with the humans. Then at the human habitat xi , i = 1, . . . , M, the rate of change of the population density of the questing vectors with respect to time is i Hi dWi (t) = Bj i Vj (t) − τi∗ Wi (t)Hi (t) − μWi Wi (t), dt K + H i j =1

n

i = 1, . . . , M,

(11)

where τi∗ , Bj i , μWi , j = 1, . . . , ni , i = 1, . . . , M, are positive constants and the sum is taken over all vector-breeding sites that communicate with human habitat site xi . Vectors returning from resource agent sites (Ui ): Let Ui be the vectors which succeed in their quest for blood at the human habitat xi . They can move to any of the breeding sites yj , j = 1, . . . , N, within their flight range, at rate Aij . Suppose also that the fed vectors experience a natural constant death rate μUi as they return to the breeding sites. Then using the flow diagram (Figure 5), the rate of change of Ui (t) with respect to time t can be written as ni dUi (t) = pτi∗ Wi (t)Hi (t) − Aij Ui (t) − μUi Ui (t), dt j =1

i = 1, . . . , M,

(12)

where τi∗ , Aij , μUi , i = 1, . . . , M, j = 1, . . . , N, are positive constants and the sum is taken over all breeding vector sites which communicate with the human habitat site xi .

342


Figure 5.

Dynamic interplay between fed vectors Ui at human habitat xi returning to breeding site yj .

Next, we make the following assumption. Assumption 2.1 Uk = Wk = Vl = 0 for all integers k and l not belonging to the sets {1, . . . , mj } and {1, . . . , ni }, respectively, i = 1, . . . , M, j = 1, . . . , N. Then without loss of generality, setting aij = Aij , bj i = Bj i

Hi , K + Hi

τi = τi∗ Hi ,

with i = 1, . . . , mj , j = 1, . . . , N,

(13)

and assembling Equations (9)–(12) together with Assumption 2.1, we obtain the following system of 3M + N equations in the 3M + N variables Hi , Ui , Wi , and Vj , i = 1, . . . , M, j = 1, . . . , N. M M dHi (t) = Ci − wij Hi (t) + wj i Hj (t) − μHi Hi (t), i = 1, . . . , M; dt j =1,j =i j =1,j =i M M dVj (t) −μe T aij λ(Ui (t − T ))Ui (t − T )e − μVj + bj i Vj (t) = dt i=1 i=1

+

M

aij Ui (t),

j = 1, . . . , N;

(14)

i=1

dWi (t) = bj i Vj (t) − (τi + μWi )Wi (t), i = 1, . . . , M; dt j =1 ⎞ ⎛ N dUi (t) = pτi Wi (t) − ⎝μUi + aij ⎠ Ui (t), i = 1, . . . , M. dt j =1 N

Notice that the equation for Hi (t) decouples from the rest of the system. To complete the formulation, it is expedient to give an explicit functional form to the function λ: R → R and also to supply the initial data for our model. We begin with the functional form.


343

Definition 2.1 A function λ : [0, ∞[→ R is a suitable birth rate function for the vectors of type U if it satisfies the following three assumptions: A1: λ(U ) > 0, ∀U ≥ 0; A2: λ(U ) is continuously differentiable with λ (U ) = (dλ(U )/dU ) < 0, ∀U ≥ 0; A3: There exists a positive number, called the vectorial basic reproduction number (denoted by R0 ), such that λ(0+ ) < λ(0+ ) = λ0 . λ(∞) < R0 Assumptions A1 and A2 ensure the existence of λ−1 (x) for x > 0, with λ(∞) < x < λ0 , while Assumption A3 ensures the existence of a threshold parameter, R0 , with the property that, when R0 > 1, a positive non-trivial equilibrium, given by x ∗ = λ−1 (x0 ),

with x0 ∈]λ(∞), λ0 [,

exists. This equilibrium does not exist if R0 ≤ 1. The choice of the functional form of λ(U ) is based on the fact that in ecology nonlinearity in the dynamics of the population of a single species can arise due to competition, usually for resources, between members of the population. When there is competition for a common resource between members of the same species, two types of competition can be identified, namely contest competition (competition for a resource that is partitioned unequally so that some competitors obtain all they need and others less than they need (i.e. there are winners and losers)) and scramble competition (competition for a resource that is inadequate for the needs of all, but is partitioned equally among contestants, so no competitor obtains the amount it needs) [5]. It seems reasonable to assume a contest type competition for mosquito dynamics based on the fact that there is abundance of humans for blood meal; however, some of the vectors are killed (losers) in their quest for a blood meal. In this paper, the following form of birth rate function, which satisfies Assumptions A1–A3, will be used: Ui λi (Ui ) = λi (0) 1 − , Ui Li , i = 1, . . . , mj , j = 1, . . . , N, (15) Li where λi (0) > 0 and Li > 0 are positive constants. To ensure that Assumption A1 is satisfied, Li is assumed very large and may be identified as the carrying capacity of environment created by habitat xi , while λi (0) may be regarded as the limiting rate at which vectors from human habitat xi would lay eggs if the population of the vectors should become very small. This functional form, commonly known as the Verhulst–Pearl logistic growth model, has been used in previous studies [12]. Other nonlinear birth rate functions can be used. For example: the Beverton–Holt function (λi (Ui ) = λi (0)/(1 + Ui /Li )) or the exponential or modified form of the Skellam function (λi (Ui ) = λi (0)e−Ui /Li ), or the Maynard–Smith–Slatkin function (λi (Ui ) = λi (0)/(1 + (Ui /Li )n ) for some positive n). See [1] for more on these functional forms. The main reason for using the Verhulst–Pearl function is because of its linearity. In fact, it is a general form for a first linear approximation to any nonlinear form of birth rate function satisfying Assumptions A1–A3. The mathematical assessment of the role of nonlinear birth in the population dynamics of disease-transmitting vectors is under investigation. Next we define initial data for t ∈ [−T , 0] as follows: (Hi (t), Vj (t), Wi (t), Ui (t)) = (hi (t), vj (t), wi (t), ui (t)), t ∈ [−T , 0], i = 1, . . . , M, j = 1, . . . , N,

(16)

where hi (t), vj (t), wi (t), and ui (t) are some continuously differentiable functions. Thus, the equations governing the rate of change of the total human and vector population densities, simply

344


obtained by adding up the relevant equations from above (as given by Equations (2) and (3)), are dH (t) Ci − μH i Hi , = dt i=1 i=1 M

M

(17)

N M M dNv (t) = aij λ(Ui (t − T ))Ui (t − T )e−μe T − (1 − p)τi Wi (t) dt j =1 i=1 i=1 ⎛ ⎞ N M M −⎝ μVj Vj (t) + μWi Wi (t) + μUi Ui (t)⎠ , j =1

i=1

(18)

i=1

with appropriate initial conditions, (H (t), Nv (t)) = (h(t), nv (t) = v(t) + w(t) + u(t)),

t ∈ [−T , 0].

(19)

We observe that in the present formulation, the equation governing the human population, though influencing the size of the vector population, is decoupled from the system in the sense that those equations can be analysed separately. It is also clear from Equations (13) and (14) and the above formulation that if Hi (t) → 0, ∀i, then Nv (t) → 0 as t → ∞, since then τi → 0 ⇒ Ui (t) → 0. We take up the analysis of the interactive model in the next section.

3. Analysis of the interactive model In this section, we examine the spatially explicit model and show that there exists a steady-state spatial distribution of vector populations over the entire region of study (N > 0, M > 0 arbitrary) and derive conditions under which such population densities could exist. Before this, we briefly analyse the dynamics of the human population. 3.1. Existence of a steady-state human population density Here, we examine the existence and stability of steady-state solutions for the density of human populations. In this section, we shall assume that the death rate of humans, μHi , are identical for all habitats. In addition, we assume that the migration of humans between any two human habitats, wij , is also identical so that μHi = μ and wij = w, where w and μ are constants. This means that the human habitats are identical. To examine and analyse the stability of the steady state of the human population, we state and prove a lemma and theorem that will be essential. Lemma 3.1 Let An be an n × n real symmetric matrix depending on positive parameters x, y. If in addition, for each n ∈ N, the matrix An has only two distinct elements such that −(y + (n − 1)x) i = j, (An )ij = x i = j, then for n ≥ 2, An has only two distinct eigenvalues. Furthermore, all the eigenvalues of the matrix are real and negative and satisfy |λIn − An | = (λ + y + nx)n−1 (λ + y), where In is the identity matrix of size n × n.


Proof

345

Let n ≥ 2 be an integer. We easily verify that |λIn − An | = (λ + y + nx)n−1 |En |,

where En is the n × n matrix ⎛ λ + y + (n − 1)x ⎜ −1 ⎜ ⎜ −1 ⎜ En = ⎜ −1 ⎜ ⎜ .. ⎝ . −1

−x 1 0 0 .. .

−x 0 1 0 .. .

−x 0 0 1 .. .

0

0

0

⎞ · · · −x ··· 0 ⎟ ⎟ ··· 0 ⎟ ⎟ . ··· 0 ⎟ ⎟ .. ⎟ .. . . ⎠ ··· 1

Then, direct computation |En | by cofactor expansion shows that at the kth stage of the expansion, for k ≤ n, for any n ≥ 2, we have the relation |En |k = |En |k−1 − x,

k ≥ 2,

⇒ |En |n = |En |2 − (n − 2)x, Now,

λ + y + (n − 1)x |En |2 = −1

n ≥ 2.

−x = λ + y + (n − 2)x. 1

Hence, |λIn − An | = (λ + y + nx)n−1 (λ + y). This therefore gives us λ = −y

or

λ = −y − nx.

We now use Lemma 3.1 to prove the following theorem. Theorem 3.2 The linear system of equations M M dHi (t) = Ci − wij Hi (t) + wj i Hj (t) − μHi Hi (t), dt j =1,j =i j =1,j =i

i = 1, . . . , M,

(20)

with wij = w, μHi = μ (w and μ are constants), for any values of Ci , has a unique non-zero steady-state distribution of humans over the M habitat sites whose value, Hi∗ , for each i, is given by w Ci + = Cj , μ + Mw μ(μ + Mw) j =1 M

Hi∗

i = 1, . . . , M.

(21)

Moreover, the postulated steady state is globally and asymptotically stable. Proof We examine Equation (20) when the time derivatives are set to zero. Let Hi∗ , i = 1, . . . , M, be the steady-state solution. Then on setting wij = w and μHi = μ, i, j = 1, . . . , M,

346


as assumed, we obtain that Hi∗ satisfies the system (μ + (M −

1)w)Hi∗

M

= Ci + w

Hj∗ ,

i = 1, . . . , M.

(22)

j =1,j =i

If we let H∗ =

M

Hj∗ ,

(23)

i = 1, . . . , M,

(24)

j =1

and then substitute in Equation (22), we obtain Hi∗ =

Ci + wH ∗ , Mw + μ

which when we substitute into Equation (23), and then solve for H ∗ , results to 1 Cj . μ j =1 M

H∗ =

(25)

We next substitute Equation (25) into Equation (24) to obtain Equation (21). To establish the stability of this steady state, let Hi = Hi∗ + hi ,

|hi | 1, i = 1, . . . , M.

(26)

If we substitute Equation (26) into Equation (20) with wij = w, μHi = μ, for all i, j , we obtain the analogous linear system M dhi = −(μ + (M − 1)w)hi + whj . dt j =1,j =i

(27)

If we seek solutions of the form hi (t) ∝ eξ t , where ξ is an eigenvalue that measures the temporal growth of the solution at time t, this leads to the solvability condition |ξ IM − AMM | = 0,

(28)

where IM is the identity matrix of size M and AMM is the M × M matrix with every off-diagonal elements being w and all diagonal elements being −μ − (M − 1)w. That is, (AMM )ij =

−μ − (M − 1)w, i = j, w, i = j .

(29)

Hence from Lemma 3.1, we obtain that |ξ IM − AMM | = (ξ + μ)(ξ + μ + Mw)M−1 .

(30)

Thus, the eigenvalues are all real and negative, and the steady-state solution is globally and asymptotically stable for all values of the parameters.


347

Remark 3.1 The extension of the results of Theorem 3.2 to cover the general cases, where wij , μHi are different, is solved by observing that Equation (20) can be written in matrix form as ⎛

⎞ dH1 (t) ⎜ dt ⎟ ⎞ ⎛ ⎞ ⎛ ⎜ ⎟ C1 H1 ⎜ dH2 (t) ⎟ ⎜ ⎜H ⎟ ⎜C ⎟ ⎟ ⎜ dt ⎟ ⎜ 2⎟ ⎜ 2⎟ ⎜ ⎟ = (AMM )ij ∗ ⎜ . ⎟ + ⎜ . ⎟ , ⎜ ⎜ . ⎟ ⎜ . ⎟ ⎟ .. ⎜ ⎟ ⎝ . ⎠ ⎝ . ⎠ ⎜ ⎟ . ⎜ ⎟ HM CM ⎝ dH (t) ⎠ M dt

(31)

where the matrix (AMM )ij is defined as (AMM )ij =

−μHi −

M

j =1,j =i

wij ,

i = j, i = j,

wj i ,

(32)

from which the steady states can be easily obtained and the system analysed based on the properties of (AMM )ij . However, for our analysis, we will consider the simpler case where the human habitats are assumed identical since we are using human habitat here to refer to a location in space where there are humans from whom mosquitoes can quest for blood. The real structure and property of that spatial location does not really make a difference, since the differentiating feature is the presence of humans which has been captured by the model. 3.2.

Existence of a steady-state vector population density

We examine in this section the spatially explicit model for the distributions of vectors. We show that there exists a spatial distribution of vector population densities over the entire region of study and derive conditions under which such steady population densities could exist. Our focus is now on the following three equations (taken from system (14)): ⎞ ⎛ N dUi (t) = pτi Wi (t) − ⎝μUi + aij ⎠ Ui (t), dt j =1

i = 1, . . . , M,

dVj (t) aij λi (Ui (t − T ))Ui (t − T )e−μe T = dt i=1

(33)

M

− μVj +

M i=1

bj i Vj (t) +

M

aij Ui (t),

(34)

j = 1, . . . , N,

i=1

dWi (t) bj i Vj (t) − (τi + μWi )Wi (t), = dt j =1 N

i = 1, . . . , M.

(35)

The steady states are obtained by setting the time derivatives in Equations (33)–(35) to zero and solving for the 2M + N quantities Vj∗ , j = 1, . . . , N, Wi∗ , i = 1, . . . , M, and Ui∗ , i = 1, . . . , N.

348


We then establish that Wi∗ =

1 μW i + τ i

N

bj i Vj∗ ,

i = 1, . . . , M.

(36)

j =1

Also, substituting Equation (36) into Equation (33) when the right-hand side is equated to zero gives N 1 pτ i ∗ Ui = bj i Vj∗ , i = 1, . . . , M. (37) μWi + τi j =1 μU i + N j =1 aij Equations (36) and (37) show that we can calculate the steady-state values, Ui∗ and Wi∗ , once the Vj∗ ’s are known. We see that when Vj∗ = 0, ∀j , we have Wi∗ = 0 and Ui∗ = 0, ∀i. If Vj∗ = 0, for some j , then the dynamic interaction between the human and vector populations establishes a non-zero steady-state population of vectors of all categories in all the human and vector sites that are within the flight range of the vectors from the particular breeding site yj . For each j , referring to Equation (34) and using the postulated form (15) for λi , the Vj∗ ’s should satisfy the N equations M i=1

U∗ λi (0) 1 − i Li

e

−μe T

+ 1 aij Ui∗ − Pj Vj∗ = 0,

j = 1, . . . , N,

(38)

where Pj = μVj +

M

bj i ,

j = 1, . . . , N,

(39)

i=1

are positive constants. Now, using the form of Ui∗ given in Equation (37), we set 1 pτ i −μe T , i = 1, . . . , M, + 1) κi (T ) = (λi (0)e μW i + τ i μU i + N j =1 aij 2 2 λi (0)e−μe T pτi 1 ξi (T ) = , i = 1, . . . , M, N Li μW i + τ i μUi + j =1 aij Sj k (T ) =

M

κi (T )aij bki ,

j, k = 1, . . . , N,

(40)

(41)

(42)

i=1

Qj kl (T ) =

M

ξi (T )aij bki bli ,

j, k = 1, . . . , N,

(43)

i=1

so that Equation (38) for the steady state VJ∗ then takes the form N k=1

Sj k (T )Vk∗ − Pj Vj∗ −

N N

Qj kl (T )Vk∗ Vl∗ = 0,

j = 1, . . . , N.

(44)

k=1 l=1

Qj kl (T ) and Sj k (T ) as defined above are all positive constants. We also note that Sj k (T ) = Skj (T ) and Qj kl (T ) = Qj lk (T ). This relation simply captures the symmetry in the problem under consideration which arises due to the fact that when a human habitat site xi is within a vector breeding site yj , then the two sites communicate. In Equation (44), all parameters are positive


349

and the relation must hold for all values of j . We observe that Vj∗ = 0, ∀j , is a solution. That is, the trivial steady state is the same for sites which are within a communicable distance. But when Vj∗ = 0, for some j , the result is a system of N nonlinear equations for which the description of the solutions is not apparent in view of all arbitrary values of the parameters. However, from a mathematical (and physical) perspective, all we wish to establish is whether there exists a non-negative solution, the steady-state solution, for the system (44). Definition 3.1 Let Sv and Sh be the sets of all vector-breeding and human habitats sites, respectively, and S := Sv ∪ Sh . Let x ∈ S, d > 0 the maximum flight range of vectors from location x. Define Rx := {y ∈ S|d2 (x, y) ≤ d, x ∈ Sh , y ∈ Sv }. Then, we call Rx the flight range domain of vectors from location x, where d2 is the metric defined in Section 2, with domain Sv × Sh . We similarly define the flight range domain for Ry . Definition 3.2 Let xi ∈ Sv , i = 1, . . . , r, r ≤ N, Rxi ’s the flight range domains of vectors at xi ’s, i = 1, . . . , r, respectively. We say that the Rxi ’s are disjoint if Rxi ∩ Rxj = ∅,

i = j, ∀i, j = 1, . . . , r.

Definition 3.3 Let xi ∈ Sv , i = 1, . . . , r, r ≤ N, Rxi ’s the flight range domains of vectors at xi ’s, i = 1, . . . , r, respectively. We say that the Rxi ’s are one-point-intersected if there exists a unique z ∈ Sh s.t.

r

Rxi = {z}.

i=1

In this case, d2 (z, xi ) = d2 (z, xj ), ∀i, j = 1, . . . , r. Definition 3.4

(i) Let A ⊆ S. We define the distance from a site y to the set A as d2 (A, y) := min{d2 (x, y)| x ∈ A}.

(ii) Let xi ∈ Sv , i = 1, . . . , r, r ≤ N, Rxi ’s the flight range domains of vectors at xi ’s, i = 1, . . . , r, respectively. We say that the Rxi ’s full-intersect if there exist at least two sites z1 , z2 ∈ Sh s.t. z1 , z2 ∈

r

Rxi .

i=1

We set Rf :=

r i=1

Rxi .

Definition 3.5 Let xi ∈ Sh and yj ∈ Sv , i = 1, . . . , r1 , j = 1, . . . , r2 , r1 ≤ M, r2 ≤ N. Suppose ∃ x ∈ Sh and y ∈ Sv s.t. xi ∈ Ry and yj ∈ Rx , ∀ i, j. Define Uh :=

r 1 i=1

Rxi

∩ Ry ,

350


⎛ Uv := ⎝

r2

⎞ R yj ⎠ ∩ R x ,

j =1

Ch := {x ∈ Uh |d2 (x, z1 ) = d2 (x, z2 ), ∀ z1 , z2 ∈ ∂Uh }, Cv := {y ∈ Uv |d2 (y, z1 ) = d2 (y, z2 ), ∀ z1 , z2 ∈ ∂Uv }, where ∂A is the boundary of A, for any set A (Notice that Ch = {xh } and Cv = {yv }, where xh ∈ Uh and xv ∈ Uv , are singleton sets). We call Uh the unit of the human habitats xi , i = 1, . . . , r1 , and Uv the unit of the vector-breeding sites yj , j = 1, . . . , r2 . We call xh the centre of Uh and yv the centre of Uv . Remark 3.2 We similarly define the concepts of disjoint, one-point-intersection and fullintersection flight range domains when xi ’s, i = 1, . . . , r, r ≤ M, are taken in Sh . We state and prove the following proposition using the case study approach that is later generalized. Proposition 3.1 The system (44) has exactly a two-solution set in which Vj∗ ≥ 0, ∀ j ∈ {1, 2, . . . , N}. The trivial solution, Vj∗ = 0, ∀ j ∈ {1, 2, . . . , N}, which always exist, and a non-trivial solution, with Vj∗ > 0, ∀ j ∈ {1, 2, . . . , N}, whose existence is uniquely determined j

by a threshold parameter R0 in the sense that j

(a) if R0 ≤ 1 then the system has no positive non-zero real solutions; j (b) if R0 > 1 then the system has exactly one real positive non-zero solution. Proof In all the proofs, we suppose that Vj∗ > 0, ∀ j , since the solution Vj∗ = 0, ∀ j, is trivial. Without loss of generality, we rewrite Equation (44) replacing Vj∗ by Vj , ∀ j : N k=1

Sj k (T )Vk − Pj Vj −

N N

Qj kl (T )Vk Vl = 0,

j = 1, . . . , N.

(45)

k=1 l=1

Case 1: N=1 (a) We first consider the situation where human habitats are not found within flight range domain of vectors at breeding site y (Figure 6).

Figure 6.

Diagram showing various human habitats not belonging to flight range domain Ry .


351

In this case, we deduce from (5), (6), and (13) that ai1 = 0 = b1i ,

i = 1, . . . , M.

This implies that S11 = 0, Q111 = 0,

and

P1 = μV1

so that Equation (45) is simply μV1 V1 = 0.

(46)

The solution of Equation (46) is the trivial steady state V1 = 0 which is what we expected, since an assumption made during the derivation of our model was that the vectors do not survive in the absence of human population. Therefore, if vectors cannot interact with humans at human habitats, then their population will die out. Assumption 3.1 Within their flight range domain, vectors located at breeding site y always have a unit of human habitats, Uy , centred at xc , where they find blood meal to live and prosper. (b) We now suppose that human habitats belong to the flight range domain of vectors located in breeding site y (Figure 7).

Figure 7.

Diagram showing various human habitats belonging to flight range domain Ry .

Figure 8.

Diagram showing centre of unit Uy as a representative of the whole unit.

352


Taking into consideration Assumption 3.1 and using Figure 8, we shall use the centre xc of the unit Uy to represent all the human habitats within the flight range of vectors located at y (we therefore think as if we have a system with one breeding site and one human habitat found within the flight range of vectors located at the breeding site), set V1 = V ,

a11 = a,

and

b11 = b,

and substitute these into Equation (45) to have the equation (S − P )V = QV 2 ,

(47)

from which we derive the non-zero steady-state solution V =

S−P . Q

(48)

In order that V > 0, we should have S − P > 0. We therefore derive an expression for R10 : R10 =

S . P

(49)

Case 2: N=2 In this case, Equation (45) becomes Q111 V12 − S11 V1 + (Q112 + Q121 )V1 V2 − S12 V2 + Q122 V22 = −P1 V1 Q211 V12 − S21 V1 + (Q212 + Q221 )V1 V2 − S22 V2 + Q222 V22 = −P2 V2 .

(50)

Let y1 and y2 be the two breeding sites in the study. Let the centres of the units of each of the two breeding sites be, respectively, x1 and x2 .

Figure 9.

Disjoint flight range domains with the respective centres representing all the belonging human habitats.


353

(a) Disjoint flight range domains (Figure 9): In this case, Ry1 ∩ Ry2 = ∅, therefore vectors in Ry1 do not visit sites in Ry2 , and vice versa. Then by Equation (13), we obtain aij = 0

and bj i = 0,

∀ i = j.

(51)

Substituting Equation (51) into Equations (42) and (43), we get Sj k = 0

and Qj kl = 0,

∀ j = k, j = l, k = l.

(52)

Substituting Equation (52) into Equation (50) gives Q111 V12 − S11 V1 = −P1 V1 ,

(53)

Q222 V22 − S22 V2 = −P2 V2 , which has as solution set when Vj > 0, ∀j , the singleton V1 =

S11 − P1 S22 − P2 , V2 = Q111 Q222

.

(54)

We observe that for j = 1, 2, Vj > 0 ⇔ Sjj > Pj . This implies the following definition j

R0 :=

Sjj , Pj

j = 1, 2.

(55)

(b) One-point-intersection flight range domains (Figure 10): We consider here vectors present in human habitats x, and intersection of the flight range domains Ry1 and Ry2 . At this site x, vectors decide (according to our assumptions), after a successful interaction with humans, to return either to the breeding site y1 , or to the breeding site y2 , with equal chance. In Equation (13) where we define the rates at which vectors move from human habitat xi to vector-breeding site yj , and from breeding site yj to human habitat xi , respectively, we see that aij and bj i depend essentially on the distance from xi to yj . With

Figure 10. habitats.

One-point-intersected flight range domains with the respective centres representing all the belonging human

354


this, we obtain that at habitat x aij = a > 0

and bij = b > 0,

∀ i, j.

(56)

Substituting Equation (56) into Equations (39), (42), and (43), we have Sj k = S > 0,

Pj = P > 0,

Qj kl = Q > 0, ∀ j, k, l.

(57)

This means that at habitat x, all parameters are constant and positive (this is easily verified). Substitute Equation (57) into Equation (45) to obtain the following system of equations: QV12 − SV1 + 2QV1 V2 − SV2 + QV22 = −P V1 , QV12 − SV1 + 2QV1 V2 − SV2 + QV22 = −P V2 .

(58)

Equate the two equations in Equation (58) to obtain V1 = V2 . Substitute this equality in any one of the equations in Equation (58) to have the following solution set 2S − P 2S − P , V2 = . (59) V1 = 4Q 4Q For j = 1, 2, we see that Vj > 0 ⇔ 2S > P . Therefore, we choose j

R0 :=

Figure 11. habitats.

2S , P

j = 1, 2.

(60)

Full-intersected flight range domains with the respective centres representing all the belonging human


355

(c) Full-intersection flight range domains (Figure 11): Let y1 , y2 ∈ Sv , and Ry1 and Ry2 be the flight range domains of vectors present at y1 and y2 , respectively. Let us define the following: Rf := Ry1 ∩ Ry2 ; d1 := d2 (y1 , Rf ) (= d2 (y2 , Rf )); dRf := (d − d1 ); θ := (y 1 y2 x); d1 + (dRf /2) 2d1 + dRf = ; cos θ 2 cos θ exp(−d22 (xi , yj )), yj ∈ Bd˜ (xi ; d2 ); a ij := 0, otherwise; exp(−d22 (yj , xi )), xi ∈ Bd˜ (yj ; d2 ); b ij := 0, otherwise. d˜ :=

(61)

We make the following remark: Remark 3.3 x ∈ Rf must belong to the line (D), which is equidistant to both y1 and y2 . For, if x does not belong to (D), then vectors present in x will choose to return to the nearest breeding site, that is y1 or y2 . Thus, cos θ cannot be equal to zero. By considering the terms defined in Equation (61), we proceed as in the case of one-pointflight range domains above, with d˜ in the place of d, a˜ ij and b˜j i in places of aij and bj i , respectively. We therefore obtain similar results, that is V1 = V˜1 =

2S˜ − P˜ = V˜2 = V2 , ˜ 4Q

(62)

and ˜ j ˜ j = 2S > 0, R0 = R 0 ˜ P

for j = 1, 2,

(63)

˜ P˜ , and Q ˜ are obtained by replacing aij and bj i by a˜ ij and b˜j i , respectively, in where S, Equations (39), (42), and (43). Case 3: N=3 In this case, Equation (45) becomes Q111 V12 − S11 V1 + Q122 V22 − S12 V2 + Q133 V32 − S13 V3 + (Q121 + Q112 )V1 V2 + (Q131 + Q113 )V1 V3 + (Q132 + Q123 )V2 V3 + P1 V1 = 0 Q211 V12 − S21 V1 + Q222 V22 − S22 V2 + Q233 V32 − S23 V3 + (Q221 + Q212 )V1 V2 + (Q231 + Q213 )V1 V3 + (Q232 + Q223 )V2 V3 + P2 V2 = 0 Q131 V12 − S31 V1 + Q322 V22 − S32 V2 + Q333 V32 − S33 V3 + (Q321 + Q312 )V1 V2 + (Q331 + Q313 )V1 V3 + (Q332 + Q323 )V2 V3 + P3 V3 = 0.

(64)

356


(a) Disjoint flight range domains: For similar reasons as in previous cases, aij = 0,

bj i = 0, ∀ i = j.

(65)

This implies that Sj k = 0,

Qj kl = 0,

∀ j = k, j = l, k = l.

(66)

Equation (64) then becomes (using Equation (66)) Q111 V12 − S11 V1 + P1 V1 = 0, Q222 V22 − S22 V2 + P2 V2 = 0,

(67)

Q333 V32 − S33 V3 + P3 V3 = 0. The non-zero solution of Equation (67) is the singleton set S11 − P1 S22 − P2 S33 − P3 . , V2 = , V3 = V1 = Q111 Q222 Q333

(68)

Again, for j = 1, 2, 3, Vj > 0 ⇔ Sjj > Pj ,

∀ j = 1, 2, 3.

Therefore, we set j

R0 :=

Sjj , Pj

∀ j = 1, 2, 3.

(69)

(b) One-point-intersection flight range domains (Figure 12): aij = a > 0,

bj i = b > 0, ∀ i, j ;

(70)

this implies that Pj = P > 0, Sj k = S > 0, Qj kl = Q > 0,

Figure 12.

(71) ∀ j, k, l.

One-point-intersected flight range domains with three vector-breeding sites.


357

Using Equation (71) into Equation (64), we obtain QV12 − SV1 + QV22 − SV2 + QV32 − SV3 + 2QV1 V2 + 2QV1 V3 + 2QV2 V3 + P V1 = 0, QV12 − SV1 + QV22 − SV2 + QV32 − SV3 + 2QV1 V2 + 2QV1 V3 + 2QV2 V3 + P V2 = 0, QV12 − SV1 + QV22 − SV2 + QV32 − SV3 + 2QV1 V2 + 2QV1 V3 + 2QV2 V3 + P V3 = 0. (72) Equate all the equations in Equation (72) to have V1 = V2 = V3 . Substituting the last equality in any one of the equations in Equation (72), we have the following non-zero solution set 3S − P 3S − P 3S − P V1 = , V2 = , V3 = . (73) 9Q 9Q 9Q From this, for j = 1, 2, 3, Vj > 0 ⇔ 3S > P ,

∀ j.

Therefore, we define 3S , j = 1, 2, 3. P (c) Full-intersection flight range domains (Figure 13): We begin the following definition of a full point. j

R0 :=

(74)

Definition 3.6 Let yj ∈ Sv , j = 3, . . . , N, be breeding sites. Let ij be the lines equidistant from yi and yj , ∀i, j . Then there are sn of these lines, with sn = sn−1 + (n − 1), n ≥ 4; (75) (sn ) : s3 = 3. Denote these lines by D1 , D2 , . . . , Dsn , n ≥ 3, and define {F } :=

sn

Dk ,

n ≥ 3.

k=1

We call F the full point, or simply the F-point, of the yj ’s.

Figure 13.

Full-intersected flight range domains with three vector-breeding sites.

(76)

358


We consider in the current situation only human habitat x present at the F-point of y1 , y2 , and y3 , for it is the only human habitat within flight range of vectors that is equidistant from all breeding sites. As in Equation (61), we define the following: d˜ := d2 (y1 , x); exp(−d22 (xi , yj )), a ij := 0, exp(−d22 (yj , xi )), b ij := 0,

yj ∈ Bd˜ (xi ; d2 ); otherwise;

(77)

xi ∈ Bd˜ (yj ; d2 ); otherwise

and follow the same procedure as for the one-point-intersection case, with d˜ in the place of d, a˜ ij and b˜j i in places of aij and bj i , respectively. We therefore obtained similar results. That is 3S˜ − P˜ (78) V1 = V2 = V3 = V˜1 = V˜2 = V˜3 = ˜ 9Q and ˜ := R0 := R 0 j

j

3S˜ > 0, P˜

for j = 1, 2, 3,

(79)

˜ P˜ , and Q ˜ are obtained by replacing aij and bj i by a˜ ij and b˜j i , respectively, in where S, Equations (39), (42), and (43). General case: N≥2 (a) Disjoint flight range domains: As in previous cases, aij = 0,

bj i = 0, ∀ i = j.

(80)

Qj kl = 0, ∀ j = k, j = l, k = l.

(81)

This implies that Sj k = 0,

Using Equation (81) into Equation (45), we obtain the system of N equations Qjjj Vj2 − Sjj Vj + Pj Vj = 0,

j = 1, . . . , N,

which has as non-zero solution set the singleton Sjj − Pj N . Vj = Qjjj j =1

(82)

(83)

This result tells us that within their respective flight range domains, vectors live and prosper whenever there is available nutrient from humans in human habitats. Observe that in the present situation, the number N of vector-breeding sites does not affect the value of the respective Vj , ∀ j . From Equation (83), Vj > 0 ⇔ Sjj > Pj ,

∀ j.

Therefore, we define j

R0 :=

Sjj , Pj

j = 1, . . . , N.

(84)


359

(b) One-point-intersection flight range domains: As in previous cases, at the F-point, aij = a > 0, bj i = b > 0,

∀ i, j.

(85)

This implies that Sj k = S > 0,

Qj kl = Q > 0,

Pj = P > 0, ∀ j, k, l.

(86)

Using Equation (86) into Equation (45), we obtain the system of N equations Q

N k=1

Vk2

+ 2Q

N N −1

Vk Vl − S

k=1 l>k

N

Vk + P Vj = 0,

j = 1, . . . , N.

(87)

k=1

Equating all equations in system (87), the following relation is clear: Vj = V1 ,

∀j = 1, . . . , N.

We use this equality into any one of the equations of the system and obtain the equation N 2 QV12 − N SV1 + P V1 = 0.

(88)

We therefore solve this equation for the unknown V1 and have the solution V1 = Hence

NS − P . N 2Q

NS − P Vj = N 2Q

N (89) j =1

is the non-zero solution set of Equation (87). Vj > 0 ⇔ N S > P ,

∀ j,

so that we define NS , ∀j. P (c) Full-intersection flight range domains: We use the definitions in Equation (77) of case N and similar reasoning. The results in this case therefore are similar to those of case N That is, ⎧ N ⎫ ⎨ ⎬ ˜ − P˜ N S Vj = V˜j = . ˜ ⎩ ⎭ N 2Q j

R0 :=

(90) = 3, = 3.

(91)

j =1

Since this situation, with the definitions in Equation (77), is similar to the one-pointintersection flight range domains case, the conclusion is also similar to that of the quoted case. Then, we define ˜ j ˜ j := N S , ∀j = 1, . . . , N, R0 = R (92) 0 P˜ ˜ P˜ , and Q ˜ are defined as in case N = 3. where S,

360


We now state the following corollary which results from Proposition 3.1. Corollary 3.1 Define

N j

R0 :=

k=1

Sj k

Pj

j = 1, . . . , N.

,

(93) j

Then, Equation (44) has exactly one real non-negative solution when R0 > 1, ∀j, and no positive j real solution when R0 ≤ 1, for some j .

4.

Example: the one-vector-breeding site–one-human habitat model with no delay

Using M = 1, N = 1, T = 0, U1 = U, V1 = V , W1 = W, a11 = a, b11 = b, τ11 = τ , and L1 = L (as defined in Equation (15)) in the model (14), we obtain the simplified system which is studied at any time t1 dU = pτ W − (a + μ)U ; dt1 dV = aλ(U )U + aU − (μ + b)V ; dt1 dW = bV − (μ + τ )W. dt1

(94)

Then, using the expression of λ in Equation (15), we obtain V∗ =

(a + μ)(τ + μ) ∗ U , bpτ ∗

with U = 0

a+μ ∗ U , pτ L (a + μ)(b + μ)(τ + μ) ∗ λ0 − or U = −1 . λ0 abpτ W∗ =

(95)

For notational convenience, set C = (a + μ)(b + μ)(τ + μ),

M = abpτ,

N = C − M = μ3 + (b + a + τ )μ2 + (ab + aτ + bτ )μ + (1 − p)abτ, (96)

M , where λ0 = λ(0), N Q = 1 + γ + ρ, R = γ + ρ + ργ .

R0 = λ0

Also consider the change of variables: u=

U , U0

v=

V , V0

w=

W , W0

t∗ =

t1 , T0

(97)

where a+μ (a + μ)(τ + μ) U0 , V0 = U0 , pτ pτ b 1 1 U0 = U ∗ = L 1 − , and T0 = , R0 a+μ

W0 =

(98)


361

du = w − u = f 1 (u, v, w), dt dv = αu(1 − u) − ρ(v − u) = f 2 (u, v, w), dt dw = γ [v − w] = f 3 (u, v, w). dt

(99)

to have

It is now easy to see that system (99) has the steady states (u∗ , v ∗ , w∗ ) = (0, 0, 0)

and

(u∗ , v ∗ , w∗ ) = (1, 1, 1).

Ngwa [12] established that system (99) undergoes a Hopf bifurcation in part in a parametric space where α = αc =

QR γ

with Q and R as defined in Equation (96). We use a step-by-step procedure, as explained in [4], to completely determine the nature of the periodic solutions arising from the Hopf bifurcation. Tedious but straightforward computations yield the approximation solutions

2πt A2 4π t 2 u(t) = 1 + cos D1 + √ + cos T˜ T˜ 3 R 2A1 A2 4πt + + 2 C − √ + 0( 3 ), √ − D2 sin T˜ 3 R R √ 2π t ρ − αc 2π t v(t) = 1 + 2 + R sin (ρ − Q) ρ cos (ρ + R)(ρ − Q) T˜ T˜ √ 1 4π t + 2 √ (A2 ρ − A1 R)(ρ − Q) + D1 (ρ 2 + R) cos T˜ 3 R √ 2 4π t 2 2 + √ (ρ − Q)(A1 ρ + A2 R) − D2 (ρ + R) sin T˜ 3 R √ 1 + 2 √ (ρ − Q)(−A2 ρ + A1 R) + C(ρ 2 + R) + 0( 3 ), R √ 2π t 2πt w(t) = 1 + cos − R sin T˜ T˜ A2 4π t A1 + 2 √ + + D1 (1 − Q) cos 3 T˜ 3 R 2A1 4π t 2A2 2 − D2 (1 − Q) sin + √ − 3 T˜ 3 R A2 + 2 (1 − Q)C − √ − A1 + 0( 3 ), R

(100)

(101)

(102)

362


where A1 :=

QR + μ∗ γ , 2(Q2 + R)

Q2 R + μ ∗ γ Q A2 := √ , 2 R(Q2 + R) B1 := −

(QR + μ∗ γ )2 [(Q2 + 4R)2 + 3Q2 ] , Q(Q2 + R)2 (Q2 + 4R)2

μ∗ 2 γ 2 (2R − Q) + Q[R 2 (2RQ − Q3 + 4) − 2Q2 μ∗ γ ] , √ 2 R(Q2 + R)2 (Q2 + 4R)2 QR + μ∗ γ C := − , 2Q(Q2 + R)

B2 :=

Q2 R + μ ∗ γ Q , (Q2 + R)(Q2 + 4R) √ √ QR R + μ∗ γ R D2 := , (Q2 + R)(Q2 + 4R) D1 := −

and μ∗ = α − α c ,

2 =

μ∗ + 0(μ∗ )2 ( provided μ˜ = 0), μ˜

with

√ (Q2 + 4R)(12A21 + 10A22 − 3 RB1 ) μ˜ = . √ 6γ R For the parameter values a = 1, b = 0.8, μ = 0.042, and τ = 4, we obtain the following values for the period, the amplitudes, and the characteristic exponent, respectively: T˜ 2.25 unit time , umax 0.045,

vmax 0.18,

wmax 0.14 unit distance ,

β˜ −5.12 × 10−3 .

Figure 14. Graph representing the behaviour of the periodic solutions u(t), v(t), and w(t) in the system of coordinates (u, v, w) with respect to time (tmin = 1000, tmax = 1005).


363

Figure 14 shows the graph of the periodic solutions u(t), v(t), w(t) plotted in the system of coordinates (u, v, w) with respect to time t, where t ∈ [tmin = 1000, tmax = 1005]. Observe that these solutions are positive and bounded around the steady state x∗ = (1, 1, 1).

5.

Conclusion and discussion

A mathematical model, originally introduced by Ngwa [12] to study the population dynamics of the malaria vector, was extended to include spatial component and dispersal in which vectors, from different breeding sites, can visit host habitats within their flight range domain and interact with them. During the interaction, the vectors can be successful, fail and seek to try again, or fail and be killed. In our modelling, the vectors were assumed to be located at N breeding sites spatially distributed among M human habitats. Vector deaths prior to the adult vector stage and a delay factor T were taken into consideration in the modelling. Analysis of the model equations under a given set of defined assumptions and using a Verhulst–Pearl birth function for the vectors led to the following results: (1) There is always a unique non-zero steady-state distribution of humans over the number of human habitats, which is globally and asymptotically stable for all reasonable parameter values. (2) For the vector population, a trivial steady-state solution always exists. In addition, there is a non-trivial steady-state vector distribution whose existence is uniquely determined by a j j threshold parameter R0 . When the threshold parameter R0 > 1, there exists exactly one real j positive non-trivial steady state solution, and if R0 ≤ 1, the only steady state is the trivial steady state. The non-trivial steady states were obtained for different distributions of the vector-breeding sites. (3) Furthermore, for the one-human habitat, one-vector-breeding site scenario with no time delay, Ngwa [12] partially established the presence of a Hopf bifurcation in a particular parameter space. Here, we use a step-by-step procedure, as explained in [4], to completely determine the nature of the periodic solutions arising from the Hopf bifurcation. The modelling techniques used can be applied to analyse the dynamics of many different vectors. In addition, the results obtained can be very useful in designing effective control strategies aimed at fighting and eradicating vector-borne diseases. Two major ways in which disease vectors can be controlled are chemical or biological. Biological control of vectors can be realized by the introduction of a predator (i.e. another organism which feeds on the vectors) at the vector breeding site. For example, in the case of swamp insect vectors, control can be implemented by the introduction of a species of fish which feeds on the larvae of the vectors. Usually, the effectiveness of this kind of control is long term, since the predator needs some time to establish itself in the environment. On the other hand, chemical control can be realized through direct application of a chemical and toxic substances in the vectors themselves or in their breeding sites, in order to interrupt their reproduction cycle. For example, pesticides like DDT and toxic bed nets are used to control mosquitoes and midges (see [16] for more on human disease vectors and their control measures). Therefore, understanding vector dynamics is crucial in the understanding how both biological and chemical control can impact vector-borne disease control. For example, continuous application of control measures (use of insecticides, bed nets, specialized candles, and mosquito traps [10]) when the vector population is at its minimum amplitude may result in a decrease in the growth of the vector population and possible extinction in extreme cases. On the other hand, the application of control measures when the population is at its peak may not be an efficient strategy.

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The model presented is the first to the best of our knowledge, and just the first step in completely understanding the population dynamics of vectors when there is more than one breeding site interacting with the host habitat. This is very important especially when applied to the Anopheles mosquito, the vector that transmits the malaria-causing agent – plasmodium. In order to control malaria, multiple interventions are needed [18,19] which involve a combination of: vector control [12]; reduction of contact between humans and vector [17]; control of the infectious agent, Plasmodium, both within the human host [19] and the vector [18]; and education (of the people that are most affected) [17] and control of the vector-breeding sites. Hence, our paper seeks to further the understanding of the vector population dynamics which could assist in the planning and implementation of new malaria intervention and vector control strategies. Many extensions exist, some of which are currently under investigation, and include the following: (1) Analysing the stability of the steady states obtained. (2) Using a shorter method of establishing the existence of steady states for any number of vectorbreeding sites and human habitats, and obtaining an explicit form of the steady states and analysing their stability. (3) Incorporating the fact that many of the vector-breeding sites are dynamic. In most endemic regions such as Cameroon, breeding sites are created due to muddy ponds and washed out due to heavy rains or dried up due to high heat. Hence, most of the breeding sites very close to human habitats are temporal and created at some rate and destroyed at some rate [21]. Therefore, incorporating the lifespan of breeding sites and the dynamics of their formation and destruction into our model will prove very crucial in understanding mosquito dynamics and will be important in control measures at eradicating vector-borne diseases that are transmitted by vectors that live in breeding sites and rely on host habitats for their main sustenance and survival. (4) Incorporate predation at the vector-breeding sites and see what the impact to the mosquito dynamics will be. (5) Include disease dynamics in the model in which questing vectors that feed successfully can in the act transmit an infectious agent that can cause a disease or be infected with an infectious agent that can again be transmitted at a later feeding.

Acknowledgements The first author (S. Nourridine) and the third author (G.A. Ngwa) wish to acknowledge support of a grant from the Faculty of Science, University of Buea, Cameroon, under the Faculty of Science Research Grant Scheme 2008/2009. The second author (M.I. Teboh-Ewungkem) wishes to acknowledge support of the NSF Grant Award No. OISE-0855380 that made it possible for all three authors to meet and to work at the University of Buea. All three authors will like to thank the reviewers for their time in reading this paper.

Notes 1. 2. 3. 4.

These are usually blood-sucking athropods. It could be a virus, protozoa, bacteria, fungi, or helminth. By host we mean the organism infected by the infectious agent and which interacts with the vector. These are vectors with a preference for human blood. If the vectors have a preference to animal blood, they are said to be zoophilic. 5. It is understood that once the infectious disease is involved and accounted for in the model, e.g. when either the vector or human is infected, then this assumption must be relaxed. 6. Human’s blood preference factor of the vectors (H=human density). 7. Animal’s blood preference factor of the vectors (A=animal density).


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