transmission dynamics and spatial spread of vector borne diseases modelling, prediction and control

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TRANSMISSION DYNAMICS AND SPATIAL SPREAD OF VECTOR BORNE DISEASES: MODELLING, PREDICTION

AND CONTROL by

RONGSONG LIU

A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS

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Abstract

In this thesis, we study the transmission dynamics and spatial spread of vector borne diseases using mathematical models incorporating different factors, contributing to the com- plicated transmission dynamics and spatiotemporal spread patterns of vector borne diseases We focus on the demographic and disease ages of hosts, the culling structured by the age of the

vector, the spatial movement of the disease hosts, and the heterogeneity in host populations

We address the above issues and the relevant modeling and analysis techniques via a detailed case studying of the invasion and spread of West Nile virus in North America Models involved and developed include patchy models with both long-range and short-range dispersal, delay differential systems, non-local delayed reaction diffusion equations and impulsive differential

equations We develop necessary mathematical techniques, and carry out qualitative analysis and numerical simulations to describe the transmission dynamics and spatiotemporal patterns

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Acknowledgements

I am indebted to my supervisors Professor Jianhong Wu and Professor Huaiping Zhu for their valuable instruction and advice on my study of mathematics During my four years of

study, they provide me tremendous opportunities to attend summer schools, conferences and

workshops to open my eyes, to communicate with others, and to gain brand knowledge I would like to thank Professor Stephen A Gourley for his valuable instruction and col-

laboration in some parts of this work

I would like to thank Dr Peter Buck and Dr Jiangping Shuai at Public Health Agency of Canada for giving me the access to the surveillance data and further insightful discussions and collaboration I also would like to thank Dr Ping Yan at Public Health Agency of Canada for his suggestions in a few key issues related to the parameter assignment and identification I would thank the Department of Mathematics and Statistics of York University, Lab- oratory for Industrial and Applied Mathematics, and Laboratory of Mathematical Parallel

Systems for their supports I would also like to thank Mathematics for Information Technol-

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Contents

Abstract iv

Acknowledgements Vv

1 Introduction 1

2 Vector-borne diseases with structured population: nonlinear dynamics dur-

ing the maturation phase 6

2.1 Introduction 2.000 ee ee 6

22 Model derivation Q Q Q HQ kg gà vo 8 22.1 Positivity ofsolutions Q Q Q Q Q Q Q Q ee 15 2.2.2 Global convergence to disease free state .-.00 18 2.2.3 Local stability of disease-free equilibrium 23 2.2.4 Numerical simulations 0 00 eee eee eee 28

2.3 Discussions 2 Q Q Q Q Q Q n u nạ kg v N g k k kia 30 3 Eradicating vector borne diseases via age-structured culling 36

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3.2 Culling of immatures: model derivation 0 0000s

3.3 Applications to the control of WestNilevirus

3.3.1 Culling of immature mosquitoes 0000000048

3.3.2 Culling of mature mosquitoes 000 e ee nae 3.4 Simulations and discussions .0 00.00 ee eee eee ees Spatial spread of vector borne disease in patches: asymptotic dispersal 4.1 Introduction 2.2 0 0, V KV kh va 4.2 The model and disease free equilibrium .-2-004 4.3 Basic reproduction number and impact of directional dispersal

4.3.1 Identical patches and symmetric dispersal

43.2 Nonsymmetric dispersal of birds

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6.3.1 Birds ee 111 6.3.2 Mosquitoes Q Q Q Q ga 111

6.3.3 Bird ecology in the absence of West Nilevirus 113

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List of Tables 2.1 3.1 4.1 4.2

Meaning of parameters 0 0 ee Q Q Quy va 29

Parameter values used for the simulations Those that vary from simulation to simulation are shown in the figure captions Literature used [34, 7,58] 62 Definition for parameters in the model 000000 eae 76 Values of parameters in the simulations *Since most mosquitoes will stay

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List of Figures 2.1 2.2 2.3 3.1 3.2 3.3 3.4 4.1

The DFE is stable 2 1 Q Q LH LH HQ ng Q n v và La The DFE is unstable and the solution evolves to an endemic equilibrium

The DFE is unstable and the solution is oscillating, Parameter values are b; = 0.95, c; = 0.4, b= 10, At = 7 and other parameters have the values shown in Table 3.1 0 20.0 00 00200005 Parameter values are b; = 0.5, c; = 0.65, b = 10, At = 7 and other parameters have the values shown in Table 3.1

Parameter values are b; = 1, c; = 0.93, b= 10, At = 14 and other parameters

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4.2 4.3 4.4 4.5 5.1 6.1 6.2 6.3 6.4 6.5 6.6

The mosquitoes can fly to their nearest neighboring patches only, but birds can

reach as far as the mth neighboring patches In other words, partition of the

region into patches is based on the flying range of mosquitoes The pictures of birds and mosquitos are taken from http://westnilemaps.usgs.gov/ 74

Flow chart of the West Nile virus in a patch environment 93 The numbers of infected birds on each patch with respect to the time To

simulate the impacts of different birds’ diffusion patterns, h;, the diffusivity to left, is fixed at 0.005 while hg, the diffusivity to right, is varied 94

The numbers of infected birds on each patch with respect to the time when

birds skip some patches 2 ee va 95

Parameter values are 3; = 3.15 x 107°, 6 = 1.5 x 107>, Bn = 2.925 x 105, am = 0.75 x 107°, Dig = 13 km?/day [31, 39] (the diffusion rate of infected adult), D;; = 6 km?/day (the diffusion rate of infected juvenile), D,, = 0.1 km? /day (the diffusion rate of mosquito), and other parameters have the values shown in Table 3.1 For these values, the minimum speed Cmin, computed as described in the text, equals 2.623164094 km/day 104 Flow chart of the model (6.2.1) and (6.2.2) 2 00.0005 117 The densities of mosquitos in trap B2 2 0.000000 e 118

Comparisons of the simulated and reported densities of mosquitos in trap B2 119 Simulation and surveillance density of dead crow in area of trap B2 120

Weekly density of dead crow in area of trap B2 120

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Chapter 1

Introduction

Vector borne diseases are infectious diseases that are carried by insects from one host to

another Examples include arboviral encephalitides (such as West Nile virus), malaria, yellow

fever, dengue fever, lyme disease and plague In many of these diseases it is the mosquito that carries the virus, but ticks and fleas can also be responsible The diseases can be spread to

humans, birds and other animals

Much has been done in terms of modeling and analysis of the transmission dynam- ics and spatial spread of vector borne diseases Dr Ross was awarded the second Nobel Prize in Medicine for his demonstration of the dynamics of the malaria transmission between

mosquitoes and humans He formulated a mathematical model that predicted that malaria outbreaks could be avoided if the mosquito population could be reduced below a critical threshold level, field trials supported his conclusions and led to sometimes brilliant successes

in malaria control [16] Using some simple deterministic models of recurrent epidemic be- haviors, Anderson and May [1] examined the relevance of heterogeneity in infectious disease

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and genetic heterogeneity in host population were considered Dietz [11] set up a simple model

for a parasitic disease such as malaria that led to a form of the logistic differential equation Two alternative types of control action, periodic drug administration to the host population and reduction of the contact rate by attacking the vector population, were considered Their effect on the long-term average proportion of affected hosts was investigated Volz [55] stud- ied a model for the growth and spread of a communicable disease carried by a vector The

problem was formulated as a partial delay differential equation with time periodic contact and recovery rate The epidemiological assumptions of the model were listed and then the

problem was formulated in a Banach space, and the existence and uniqueness of solutions was established Monotone techniques were used to prove that the disease either died out, or the proportion of infectious humans tended to a periodic solution which attracted all other solutions (except the zero one) Feng, etc [14] studied a system of differential equations modelling the population dynamics of dengue fever The model was constructed to examine both the epidemiological trends of the disease and conditions that permitted coexistence in competing strains They argued that the existence of competitive exclusion in the system was a product of the interplay between the host superinfection process and frequency-dependent (vector to host) contact rates Wonham et al [58] built a single-season model of West Nile

virus cross-infection between birds and mosquitoes, incorporating specific features unique to the ecology of West Nile They showed that mosquito control decreased, but bird control

increased, the chance of an outbreak Bowman et al [7] proposed a single-season ordinary differential equation model for the transmission dynamics of WNv in a mosquito-bird-human

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strategy (or strategies) could make the basic reproduction number Ro < 1

In this thesis, we study the transmission dynamics and spatial spread of vector borne

diseases by mathematical models which incorporate several key factors and contribute to the complicated and interesting transmission dynamics and spatiotemporal spread patterns of

vector borne diseases In particular, we focus on the demographic and disease ages of hosts, the age structured culling of vector, the spatial movement of the diseased hosts, and the heterogeneity in host populations Some of these factors have been intensively studied in isolation using various models, but some factors such as the heterogeneity in host populations (which may play a key role in the transmission cycle) and the correlation and interaction of

demographic and disease ages and spatial movement (which require extreme care in modeling and analysis and can lead to models with different levels of complexity) have not been studied yet, to our best knowledge

We address the above issues and the relevant modeling and analysis techniques via a case study of the invasion and spread of WNv in North America Models to be formulated and discussed include patchy models with both long-range and short-range dispersal, delay differential systems, non-local delayed reaction diffusion equations and impulsive differential

equations We obtain some qualitative results and carry out numerical simulations to describe the spatiotemporal patterns of disease spread whose transmission dynamics is modeled by the aforementioned nonlinear dynamical systems

In Chapter 2, we derive from a structured population model a system of delay differential equations with discrete and distribute delay describing the interaction of five sub-populations,

namely susceptible and infected adult and juvenile vectors and infected adult reservoirs, for

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eradication and sharp conditions for the local stability of the disease free equilibrium are

obtained using comparison techniques coupled with the spectral theory of monotone linear semiflows

There are no commercially available human vaccines for WNv In general, vector (mosquito) borne diseases are prevented in two major ways: personal protective measures and public

health measures to reduce the population of infected mosquitoes Personal measures include reducing time outdoors particularly in early evening hours, wearing long pants and long sleeved

shirts and applying mosquito repellent to exposed skin areas Public health measures often require spraying of insecticides to kill juvenile (larvae) and adult mosquitoes In Chapter 3, we qualitatively assess the effectiveness of larvicides and insecticide sprays in the control of vector borne diseases and the best timing of using adulticide and larvicide This again leads to structured population models, now taking the form of differential equations with impulses The use of impulsive differential equations as models of control seems to be a relatively unde-

veloped application area Based on the simulation results, we find out whether it is necessary to reduce susceptible mosquitoes substantially in order to eliminate infected mosquitoes by

culling depends on the disease induced death rate of infected birds If the disease induced death rate of birds is large enough, the disease will die out in both mosquito and bird pop- ulation even if the mosquito population is large While if the disease induced death rate of infected birds is small, the WNv will sustain in the mosquito and bird population except all mosquitoes die out One of our results is that culling does not necessarily help WNv control

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the virus when the birds’ long-range dispersal dominates the nearest neighbor interaction and

diffusion of mosquitoes and birds By perturbation theory, we find the directional dispersal of birds slows down the spatial spread of WNv in North American

Besides the discrete patch model, the continuous spread of the diseases is also considered in

Chapter 5 Based on the age-structured model for host proposed in Chapter 2, we incorporate

the spatial movements by considering the analogue reaction-diffusion systems with non-local delayed terms A formal calculation of the minimal wave speed for the traveling waves is given

and compared with filed observed data

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Chapter 2

Vector-borne diseases with structured population: nonlinear dynamics during the maturation phase

2.1 Introduction

Vector borne diseases are infectious diseases that are carried by insects from one host to another Much has been done in terms of modeling and analysis of the transmission dynamics

and spatial spread of vector borne diseases, see Murray [36], Brauer and Castillo-Chavez [8],

Edelstein-Keshet [12], Kot [29], Jones and Sleeman [28], Wonham [58, 57], etc However, one important biological aspect of the hosts—the stage structure-seems to have received little attention although structured population models have been intensively studied (see Diekmann

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stage-structure with spatial dispersal has been receiving considerable attention in association

with the theoretical development of the so-called reaction-diffusion equations with non-local

delayed feedback (see the papers by Gourley, So and Wu [22] and Gourley and Wu [23] and

the references therein)

The physiological structure of hosts has a profound impact on the transmission dynamics of vector borne diseases In the case of West Nile virus the transmission cycle involves both mosquitoes and birds, the crow species being particularly important Nestling crows are crows

that have hatched but are helpless and stay in the nest, receiving more or less continuous care from the mother for up to two weeks and less continuous care thereafter Fledgling crows are old enough to have left the nest (they leave it after about five weeks) but they cannot fly very well After three or four months these fledglings will be old enough to obtain all of their food by themselves As these facts demonstrate, the maturation stages of adult birds, fledglings

and nestlings are all very different from a biological and an epidemiological perspective, and a realistic model needs to take these different stages into account For example, in comparison

with grown birds, the nestlings and fledglings have much higher disease induced death rate,

much poorer ability to avoid being bitten by mosquitoes, and much less spatial mobility

(35, 6, 44] In this Chapter we shall in fact assume there is only one pre-adult stage for the host population, which in the West Nile virus context could be thought of as the fledgling stage of crows

The aim of this Chapter is to formulate a model for the evolution of some vector borne

diseases, whose transmission dynamics and patterns are similar to those of West Nile virus

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infectious), coupled with a scalar delay differential equation for the adult vector population under the assumption that the total vector population is maintained at a constant level We then use the standard technique of integration along characteristics to reduce the model to a system of five coupled delay differential equations for the susceptible and infected juvenile and adult reservoir population and the adult infected vector The model derivation is carried out in detail in section 2.2, together with some detailed biological and epidemiological explanations of all terms involved

We consider the qualitative behaviors of the reduced ordinary delay differential system in subsections 2.1-2.4 We establish the positiveness and boundedness of the reduced system

and we emphasize the need to restrict the initial data to the subset which is biologically

and epidemiologically realistic We then establish a concrete criterion, expressed in terms of

the model parameters, for the disease eradication This is achieved using some comparison

techniques and differential inequalities We also obtain a necessary and sufficient condition for the disease free equilibrium to be locally asymptotically stable—this is done using an application of the spectral properties of a linear delay differential system due to Smith [46] The sharpness

of the disease eradication condition is then tested using the available data and parameters for West Nile virus, and our simulations show that sustained oscillation can occur should this

disease eradication condition be violated

2.2 Model derivation

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We will also refer to the reservoir as the host, and assume that the host population is

age-structured We start with a simple division of the host population as susceptible hosts

s(t,a) and infected hosts i(t,a) at time t and age a These host populations are assumed to evolve according to the McKendrick von-Foerster equations for an age-structured population:

H+ SS = —d,(a)s(t, a) ~ (a)s(t,a)mi(t), (2.2.1)

and

9 ði

ôi + ôa = —d,(a)i(t, a) + B(a)s(t, a)mi(t), (2.2.2)

where m,(t) is the number of infected adult mosquitoes satisfying another equation below,

and G(a) is the age-dependent transmission coefficient and it is assumed that conversion of

hosts from susceptible to infected occurs through interaction of susceptible hosts with infected

mosquitoes and, at this stage we assume that the rate of conversion is given by mass action We shall discuss the limitations of the model involving mass action, and shall indicate how our work can be extended to include a more standard incidence term that includes dividing

by the density of the host population The functions d,(a) and d;(a) are the age-dependent

death rates of susceptible and infected hosts

We shall further split the host population into juveniles and adults, defined respectively as those of age less than some number 7, and those of age greater than 7 We will work with

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The subscripts in these quantities refer to disease and juvenile/adult status; thus for example

the per capita death rates for susceptible juveniles and infected adults would be d,; and dia respectively The above choices enable us to formulate a closed system of delay differential

equations involving only the total numbers of hosts classified as adult susceptibles, adult in- fected, juvenile susceptibles and juvenile infected These total numbers are given respectively, using self explanatory notations, by

A,(t) = J “s(,ø)da, — A,(t) = / ~i(t,a)da, J, (t) = [ "s(t, a) da,

: (2.2.5)

Ji(t) = [ i(t, a) da

We assume no vertical transmission in the system (both from host and vector) On the

further assumption that the birth rate is a function of the total number of susceptible adult hosts, we have the following expressions for the birth rates s(t,0) and i(t, 0):

s(t,0) = b(A,(t)), i,0)=0, (2.2.6)

where 0(-) is the birth rate function for hosts (we shall later introduce B(-) as the birth rate function for mosquitoes) Equations (2.2.1) and (2.2.2) are solved subject to (2.2.6)

Let us now find a differential equation for A,(t) Differentiating the expression for A,(t) in (2.2.5), making use of (2.2.1), (2.2.3) and (2.2.4) and assuming (reasonably) that s(t, 00) = 0, we quickly find that

dA, dt

= 8(t,7) — dsaAs(t) — Barmi(t) A(t) (2.2.7) Next we need to find s(t,r) This will be achieved by solving (2.2.1) for 0 < a< 7 Set

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Then da ot mỊ t=t+a = —s¢(a) [d,(a) + 8(a)m;(€ + a)], dsg — ly Os so that

s(E+a,a) = s¢(a) = se(0)exp (— [(de(n) + 8(n)m(£ + n)) đn)

= (A,(6)) exp (— ["(de(n) + B(n)milé +)) dn) (2.2.8)

Putting ø =7 and £ = #— 7 and using (2.2.3), (2.2.4) gives

s(ty7) = B(A,(t =z)) exp (— [ (đu + 83ms(t— 7 +n)) dn)

Putting this into (2.2.7) gives, after a change of variables in the integral,

ue = b(A,(t — T)) exp (- L (ds; + Bymi(u)) du) — dsaA,(t) — Bami(t)A,(t) (2.2.9) t 7 In much the same way, we obtain the following equation for J,(t): dJ t — = b(A,())— b(A;(t— r))exp |— | (dạ; + Bymi(u)) du i CÍ, ) san —d,;J,(t) — G;m,(t)Js(t)

The differential equation for A,(t) turns out to be more complicated Differentiating the

expression for A;(t) in (2.2.5), assuming i(¢, co) = 0 and using (2.2.3) and (2.2.4) gives

dA;j{t) —

UA) (6,7) — dia Ault) + 8am()A.(9, (2.211)

and we need to find i(t,7), by solving (2.2.2) for 0 < a <7 Setting i¿(@) = ¡(£ + a,a) and differentiating with respect to a, we find from (2.2.2) that

dig(a)

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Integrating this from 0 to a, and recalling that i¢(0) = i(€,0) = 0 by (2.2.6), we find that

i(€ + a,a) = %(a) = 6; Ệ e~ 45(0- m(€ + )s(€ + rị, n) dn

Therefore,

„

i(t,7) = ù [ eH Om(t — 7 +) s(t — 7 +n,0) dn

= 6; [_ ¿-4t~Ôm(E)s(E,£++ — 9 dé (2.2.12)

In this integral the second argument of s(€,€ + 7 — t) goes from 0 to 7, and therefore an expression for s(€,€ + 7 — t) can be obtained from the earlier analysis From (2.2.8),

s(£,£++ — 9) = W(A,(t—7)) exp (- | lu + 8m0) iv) ,

Insertion of this expression into (2.2.12) yields an expression for i(t,7) that involves only the state variables in (2.2.5) and m,(t), and insertion of this expression for 7(t,7) into (2.2.11)

finally gives the differential equation for A;(t) to be dA,(t sl ) = —digA;(t) + Bam (t)A,(t) (2.2.13) ‘ —diy(t-€) ‘

+ Ø;b(A,(# — 7) [ mi(€)e~4-9) exp ( — [ (dg; + Bjm,(v)) du ) dé

Similarly, the differential equation for J;(t) can be shown to be dJ,(t HO) = ast) + Bymalt) lt

(2.2.14)

= paste r) f° meer exp (— [5 (das + dyno) av) a

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in (2.2.13) and (2.2.14) The first two terms in the right hand side of (2.2.13) are easy to interpret They are, respectively, death rate of infected adults and conversion of susceptible

adults to infected adults via contact with infected mosquitoes The last term in (2.2.13) tells us the rate at which infected immatures become infected adults having contracted the disease in childhood This term is the rate at which infected individuals pass through age 7 Now, an

individual that is of age 7 at time t will have been born at time t — 7 Recall, however, that

all individuals are born as susceptibles This is why the birth rate b(A,(t — 7)) is involved The individuals we are presently discussing have each acquired the infection at some stage during childhood, so assume a particular individual acquires it at a time € € (t —7,t) This particular individual remained susceptible from its birth at time t — 7 until time ¢, and the probability of this happening is

exp (- [ * (de + Bym;(v)) iv)

The probability that the individual will survive from becoming infected at time € until be-

coming an adult at time t is

e dij (t-8)

These two exponentials both feature in the last term in (2.2.13) The product 6;m,(€) is the per capita conversion rate of susceptible juveniles to infected juveniles at time €, and &

running from ¢t — 7 to t totals up the contributions from all possible times at which infected

individuals passing into adulthood might have acquired the infection

Finally, we need differential equations for the mosquitoes Let mr(t) be the total number of (adult) mosquitoes, divided into infected mosquitoes m,(t) and susceptible mosquitoes

mr(t) — m,(t) Death and reproductive activity for mosquitoes are assumed not to depend on

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assumed to obey

oral) =e" B(mp(t — 2)) — dmmr(t), (2.2.15)

where d; and d,, denote the death rates of larval and adult mosquitoes respectively and ø is

the length of the larval phase from egg to adult The function B(-) is the birth rate function for mosquitoes It is possible but unnecessary to write down a differential equation for larval mosquitoes Infected adult mosquitoes m;(t) are assumed to obey

dt —

—dymi(t) + Bn(mr(t) — mi(t))( Ji(t) + wA;(t)) (2.2.16)

Thus, the rate at which mosquitoes become infected is given by mass action as the product of susceptible mosquitoes mr(t) — m,(t) and infected birds which may be either juvenile or

adult The presence of the factor a@ is to account for the possibility that juvenile and adult birds might not be equally vulnerable to being bitten Again, we defer the discussion of a more standard incidence term to the final section

Certain assumptions will be made concerning the birth function B(-) for the mosquitoes

These assumptions, which are ecologically reasonable, are geared towards ensuring that the

total number mr(t) of mosquitoes stabilises and does not tend to zero (otherwise the disease is automatically eradicated and the model is not interesting) These assumptions are

B(0) = 0, B(-) is strictly monotonically increasing, there ex-

ists m* > 0 such that e~®?B(m) > dam when m < m%, and (2.2.17) e~47 B(m) < dam when m > m*

The quantity m% > 0 in (2.2.17) is an equilibrium of (2.2.15), and mr(t) — mp as t > co provided mr(0) > 0 and m7(6) # 0 on 8 € [—c,0} (see Kuang [30]) Accordingly, equa-

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thereby lowering the order of the system to be studied, which we now note consists of equa- tions (2.2.9), (2.2.10), (2.2.13), (2.2.14) together with

dm; (t)

dt

= —dmmi(t) + 8„(m‡ — m;(f))(J;) + œA;(Ð) (2.2.18) which is the asymptotically autonomous limiting form of (2.2.16) Note that this system does not explicitly involve the delay o, but this delay is still involved via the quantity m+

2.2.1 Positivity of solutions

We will prove that the system consisting of equations (2.2.9), (2.2.10), (2.2.13), (2.2.14) and (2.2.18) have a positivity preserving property It is easy to appreciate that this sys- tem cannot have a positivity preserving property for completely arbitrary non-negative initial data (a glance at the terms in the right hand side of (2.2.14) makes this clear) However,

positivity preservation does hold when some components of the initial data satisfy certain

relations ‘These relations are easily seen to be the only ones that make sense ecologically and

therefore are easily admitted We therefore now append to the abovementioned system the

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where 4?(Ø) and m?(9) are prescribed continuous functions of the variable 6 € [—7,0] and

A°(0) is also a given value Note that J,(0) and J;(0) have to be calculated from the initial

data for A, and m, This is ecologically reasonable, after all J,(0) is the number of juvenile susceptibles at time ¢ = 0 The integral on the right in the expression for J,(0) is simply

accounting for all these juvenile susceptibles at t = 0 Each one was born at some time

€ € [—7, 0], hence the presence of the birth rate b(A°(£)), and each has to have survived and remained susceptible until time 0, hence the exponential term which represents the probability

of this actually happening The interpretation of the expression for J;(0) is similar but more complicated Of the infected juveniles J;(0) at time 0, each one was born at some time

€ € [—71,0] as a susceptible, and each of these newborns at time € then became infected at

some subsequent time 7; € [£, 0|

We will now prove the following positivity preservation result

Proposition 2.2.1 Let (2.2.17) hold Then each component of the solution of the system consisting of equations (2.2.9), (2.2.10), (2.2.13), (2.2.14) and (2.2.18) fort > 0, subject to the initial conditions (2.2.19), remains non-negative for allt > 0 Also, m,(t) < m* for all t>0 If, furthermore, the function b is bounded, then each component of the above solution is also bounded for allt > 0

Proof First we will show that m;(t) < m7 for all t > 0 Suppose the contrary, then there

must exist a time t; such that m,(t;) = m} and dm;,(t,)/dt > 0 Evaluating (2.2.18) at time t; immediately gives a contradiction

Next we prove non-negativity of A,(t), for t € (0,7] in the first instance On this interval,

dA,(t)

dt

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By comparison, A,(t) is bounded below by the solution of the corresponding differential equa-

tion obtained by replacing > by =, and this differential equation contains a factor of A,(t) in

its right hand side Since A,(0) > 0, it follows that A,(t) > 0 for all ý € (0,7] This argument can be continued using the method of steps, and we conclude that A,(t) > 0 for all t > 0

Non-negativity of J,(t) will be shown next This can be seen by noting that the solution

of (2.2.10), subject to the initial value for J,(0) given in (2.2.19), is

Jatt) = [ˆ B(A,(©) ep (~ [Ha + Bjms(u)] au) a (2.2.20)

which is non-negative because A, is non-negative

We still have to prove non-negativity of A,(t), J;(t) and m,(t) It will be helpful to note

that the solution of (2.2.14), subject to the initial value for J;(0) given in (2.2.19), is

H(t) = f° HALE) { [/Bjmalnpertotmer fete dn} ae, (2.2.21)

which is non-negative if m,;(t) is non-negative Therefore, it suffices to prove non-negativity

of A;(t) and m,(t) These two functions can be viewed as the solution (A,(t), mi{t)) of the

system of differential equations consisting of (2.2.13) and

dt -

(fh Ale) { ff imi(nyertotme Lele iro an} ae + As(t))

for t > 0, with initial data taken from (2.2.19), but with A,(t) thought of simply as some

—dmm;(t) + Bm (mp —™; (t))

(2.2.22)

prescribed non-negative function Recalling that m,(t) < m> we now note that, even though

this system does not satisfy a quasimonotonicity condition, Theorem 2.1 on p81 of Smith [46]

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The boundedness of A,(t) is simple since by (2.2.9),

Salt) < bsup — dgqAg(t) — Øam(t)A;(t),

where Öaup = SUP4>9 (A) < 00 The boundedness of 4;(£) follows from (2.2.13) and the

boundedness of m,(t) by using (2.2.18) The boundedness of J,(t) and J,(f£) follows from (2.2.20) and (2.2.21) directly This completes the proof

2.2.2 Global convergence to disease free state

In this section we shall prove a theorem giving sufficient conditions for the system to evolve to

the disease free state (i.e conditions that ensure A;, J; and m; go to zero as t — 00) Since the differential equations (2.2.10) and (2.2.14) can be solved to give (2.2.20) and (2.2.21) respec- tively, it is sufficient to study the system consisting of equations (2.2.9), (2.2.13) and (2.2.22), with initial data taken from (2.2.19) These equations form a closed system for A,(t), A;(t)

and m,(t) Our aim will be to establish using these three equations a differential inequality

for the variable m,(t) only, and to use this to find conditions which ensure that m;(t) — 0 as t + oo Note that if m,(¢) — 0 then, from (2.2.21) it follows immediately that J;(t) — 0

and, furthermore, (2.2.13) then becomes an asymptotically autonomous ODE from which it is easily seen that A;(t) tends to zero

We will make certain assumptions concerning the birth rate function b(-) for hosts These

assumptions are:

b(0) = 0, B(A) > 0 when A > 0, beup := SUP4>9 D(A) < 00,

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These assumptions are not the same as those for the birth rate function B(-) for mosquitoes (assumptions (2.2.17)); note in particular that we do not require b(-) to be monotone

The reader will realise that the quantity A‘ in (2.2.23) is, in fact, a non-zero equilibrium

value for A,(t) in the case when the disease is absent Assumptions (2.2.23) are geared

towards ensuring that the population A,(t) of adult susceptible hosts does not go to zero even without the disease, otherwise the model is not interesting This is important because if e-447b(A) < d,„A for all A > 0 (which means that, in the absence of the disease, adult recruitment of susceptible hosts is insufficient to offset natural death of adult susceptible

hosts) then it is natural to expect that A,(t) — 0 even without the disease, and this can be

mathematically shown to be the case, using equation (2.2.9)

We will prove the following theorem Assumption (2.2.17) is needed to ensure the existence

of mp We shall need the functions a; and ap defined by

ay (€) = dmdig + Am diy + didi;

mm bsu B; * bsu e~4si7 Se 0mm‡a8,(T —— +€ (2.2.24) “gu (1— €-T(đu—đn~áu) , — s7 | _— _._ * , ề di; _ dm _ dj BmmpaB; sup and

dia m Tsu j ụ —a/7

ao(€) = Am died; — Sabie ~ di; BmMpaBo (ae + 9

s sa (2.2.25)

—dsi l— e7 7 (dij dn dey) *

— die ‘sjT dung hn d BmMpaB;bsup- 4 m 83

Theorem 2.2.2 Let (2.2.17) and (2.2.23) hold, and let A,(t), A;(t) and m,(t) satisfy (2.2.9),

(2.2.13) and (2.2.22), with initial data taken from (2.2.19) Assume further that

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where the functions a1, ao are defined by (2.2.24) and (2.2.25) Then (A;(£),m;(£)) — (0,0)

ast — oo

Remark It is not hard to check that (3.3.44) can be satisfied for some parameter values

It is satisfied, for example, when the contact rates G., 6; and Bm are sufficiently small, or when the mosquito capacity m7 is sufficiently small These are situations in which we intuitively

expect the theorem to hold As such, an obvious control measure to achieve disease eradication is to reduce the mosquito capacity Reducing G,, is an alternative approach

Proof of Theorem 2.2.2 For the reasons explained above, we may concentrate on

showing that m,(t) — 0 as t — oo From positivity of solutions, we find from (2.2.9) that dt < b(A;(— @)) Nuôi — d,„A,€) < Beupe” “957 — dsgAg (t) Hence b, —dgjT lim sup A,(t) < up t-—00 đạo By hypothesis (2.2.26) and by a continuity argument we may choose > 0 sufficiently small that

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Solving this differential inequality, and ignoring a transient term involving A;(0), we find that

Deupe 2% 7

Aj(t) < Ba ( 7 + 3 fred mv) dw

7 c (2.2.29)

+ ba, | e Ast~9 LỄ mị(Ệ)e 0-9 cựp (- J) (das + Byrmsto)) iv) dé deb

We shall use this estimate for A;(t) to obtain a differential inequality for m,(t) as follows From (2.2.22), and using positivity of m,;(t) and the bound on 0(-), dm; (t) dt t t ` (neff Bym;(n)e~ 23-9 e~ Je [dag +8jmmi(0)] ean dé + aAi(t)) so that, from (2.2.29), < —dymj(t) + Bmmp dt t ‡ TU ;ztn¿ + ưn‡bạp | Í 8pmá(n)e- 827902” felon Sin dg < —dmm (t) Deupe a7 + 8mm>+o8, ” + 2 / 's~4e(~9)m, (0) dụ t ap €

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To make progress we need to estimate some of these integrals If we change the order of integration in the first double integral of (2.2.30), we reach the following estimate: t rt [ fam me 146-947 8⁄-8)dn đệ = E.k 6m )e~#¿ữ~)e~4s;( =9) q£ đn 3, t—T qm t < Tỉ —4dj;(t—n) < Z mane dn t < 6 [ miln)er* aq (2.2.31) ds; 0 assuming t > T From equation (2.2.18) and Proposition 2.2.1 we have dm, (t) >— ;(t) rm dmm,(t) Integrating from € to gives m(Ệ) < m()c*®-9, E<y Using this and (2.2.31) we obtain ort = < —d„m;(£) + Hư | m,(n)e~ ts dn + Bama, (* THẾ “ tả fe “halt Yen (p) dep (2.2.32) sa 1 — e~7T(#¿ —dm—de;) * —d,;T —dig (—) + + 8„m‡o;b¿„pe4s Es [ € mi(wW) dp

By the theory of monotone systems [46], m,;(t) < M;(t) where M,(t) is the solution of the differential equation obtained from (2.2.32) by replacing < by =, subject to the same initial

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the transform variable and M;(p) denote the Laplace transform of M;,(t), we find after some algebra that

M;(p) A(p) = mi(0)(p + dia)(p + dụ) (2.2.33)

where

A(p) = p? + (din + dia + diz)p? + a1(€)p + ao(€) (2.2.34)

with a;(e) and ao(e) given by (2.2.24) and (2.2.25) Recall that the small number e > 0 has

been chosen such that (2.2.27) holds This fact, together with the Routh Hurwitz criteria,

imply that all the roots of the cubic equation A(p) = 0 satisfy Rp < 0, and so the same is

true of all singularities of M;(p) By the inversion formula for Laplace transforms, M;(t) — 0 as t > oo Since 0 < m,(t) < M,(t), mi(t) — 0 as t — oo By (2.2.13), A;(t) 0 as t > oo

The proof of Theorem 2.2.2 is complete

2.2.3 Local stability of disease-free equilibrium

If (2.2.23) holds then the model (2.2.9), (2.2.10), (2.2.13), (2.2.14) and (2.2.18) has a disease- free equilibrium (DFE), obtained by substituting J; = 0,A; = 0 and m; = 0 into the right

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further insight, and we shall present a condition (namely, condition (2.2.38) below) which is both necessary and sufficient for Ep to be linearly stable Though we do not establish disease

eradication globally under this particular condition, it is clearly the weakest possible condition for disease eradication

We first require the following simple preliminary result which provides a condition for the linear stability of the disease free equilibrium Ep to perturbations in which the disease remains

absent

Lemma 2.2.3 Let (2.2.23) hold Then (A%, J7), given by (2.2.86), is a locally asymptotically stable equilibrium of the subsystem dJ,(t) _ b(A,(t)) — b(A,( — r))e” #27 — đ„;J;(Ê), i (2.2.37)

sO = W(As(t — 1))e*” — dae Aa),

if deg > |b'(A®) e747

Proof Obviously, (A*, Js) is an equilibrium of system (2.2.37) The linearisation of (2.2.37)

at this equilibrium has solutions of the form exp(At) whenever 2 satisfies

—r — ds; bI(At)(1 — e7 O+4s5)7) 0 —À — đạ¿ + b(A‡)e~Ô*+d»¿)r

Therefore, (A*, J*) is a locally stable solution of (2.2.37) if and only if all the roots A of —d = deg +b'(At)e—O+4es)" = 0 have negative real part It is straightforward to show that this

is the case if dsq > |b/(A*)|e~437 The proof is complete

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Theorem 2.2.4 Let (2.2.17) and the hypotheses of Lemma 2.2.3 hold and assume addition- ally that *\3, — e-ds;T — dsr dm > Bmp | SO |? eit (1 —e~4s ) dij ~ ds; ds; di; a ag (1 — en ies") (2.2.38)

+7 a4: +8;b(A)e” %4 toe] \

Then the disease free equilibrium Ep given by (2.2.35) is linearly asymptotically stable as a

solution of the full model (2.2.9), (2.2.10), (2.2.13), (2.2.14), (2.2.18)

Remark The hypotheses of Theorem 2.2.4 are the weakest possible hypotheses that can

guarantee the stated result Recall from earlier remarks that if (3.3.38) or (3.3.39) is violated

then the mosquito or host population is doomed irrespective of the disease If the two sides of (2.2.38) are equal then zero is an eigenvalue of the characteristic equation of the linearisation about Ey (equation (2.2.40) below), signalling the bifurcation of an endemic equilibrium As will be shown numerically at the end of this section, a Hopf bifurcation of periodic solutions may further bifurcated from this endemic equilibrium It remains to be a challenging problem to determine if these hypotheses of Theorem 2.2.4 are sufficient guarantee the global activity

of Eo

Proof We aim for a linear equation in m; only Making use of the expression (2.2.21) for

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Solutions of the form m;(t) = e** exist whenever satisfies b(A5) 8; Ề — e—đajT _ (1 — — A+dn = „mi | À + dj; _ ds; dsj ae i (2.2.40) —^_ |qA*+ 8;b(A*)e-dar(LC eT) + quan J4; + 8/804 A+dg-dy |Jƒ

The structure of the linear equation (2.2.39) is such that the linear stability of its zero solution can be determined by considering only the real roots of the characteristic equation (2.2.40)

This follows from Theorem 5.1 on p92 of Smith [46] and Theorem 3.2 of Wu [59] Our aim is therefore to show that, under condition (2.2.38), equation (2.2.40) does not have any non- negative real roots From simple graphical arguments, we see that it is sufficient to show that

the right hand side of (2.2.40) is monotonically decreasing as a function of A € R for \ > 0 Let F(A) denote the right hand side of (2.2.40), excluding the G,,m# factor It is sufficient

to show that F’(X) < 0 for all A > 0 Now

FO) = PONE iptayr) — H+ ayy)

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and

e*(2?+2r+2)<2, +>0,

e*#(z2+2z+2) >2, z+<0

It is sufficient to show that #1(A) < 0 and #2(À) < 0 for all À > 0, with the #;(À) defined

by (2.2.41) It is very easily seen, using (2.2.42), that FZ(A) < 0 for all \ > 0 (in fact for all

À > —da) To show that #1(À) < 0, introduce € = À + đị; — đ;; and the function ø(£) defned

by

9() = z((d„;7) — ƒ((§ + đa;)7)),

then it is more than sufficient to show that ø'(£) < 0 for all £ €1 But

dé) = alt (E+ dy)r) — ƒ(du)]— 2ƒ (€ + dạ)

= z [ƒ'(( + d„)r) — ƒ'((€ + d„;)z)]

= (0—1)z?ƒ“(e)

for some numbers 6 € (0,1) and c € R which arise from applications of the mean value theo- rem Since f”(c) > 0 by (2.2.42) it follows that g’(€) < 0 as desired Thus, equation (2.2.40) does not have any non-negative real roots

With m,(t) — 0 it follows from (2.2.21) and (2.2.13) that J;(t) + 0 and 4;(£) — 0 Then

the hypotheses of Lemma 2.2.3, which are embodied within those of Theorem 2.2.4, imply

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2.2.4 Numerical simulations

Let us introduce the new variable W, defined by

Wì (t) = Ệ mác s9 exp (- [as + Bjm,(v)) iv) dé,

so that we can rewrite the model (2.2.9), (2.2.10), (2.2.13), (2.2.14) and (2.2.18) in the form:

_- = b(A,(0)) — b(A,( — r))e-#9re~ Ñ—v 8m94 — 4 J (8) — 8y/m,(8)2,(0),

“Ad = b(A¿(£— r))e”#%"e" lễ 8/9489 ủ (t) — Bymi(t) A(t),

a) = —dizJ;(t) + Bymi(t) Je(t) — Bjb(As(t — 7)) Wilt),

#20 = —digA;(t) + Bam(t)Ag(t) + 8;b(A;( — r))W:(),

Tạ, = —dmm() + (mr(t) — m:(f))8m(50) + œA¡;@));

mm Œ) _ wy) (dej — dig + Bymn(t — 7) + mi (t)em4si em Len OB c~durmm (§ — r), (2.2.43) The DFE of model (2.2.43) is the equilibrium in which

(Js, As, Ji, Ai, Mi, Wi) = (27, A}, 0, 0, 0, 0)

In the simulations reported below, we take the birth function of mosquitoes and that of birds

as

B(mr) = bamre""", b(A,) = by Age thê, (2.2.44) respectively These forms for the birth function have been used, for example, in the well- studied Nicholson’s blowflies equation [24]

Various parameter values are given in Table 2.1, taken from [34, 7, 58, 35] with reference to West Nile virus We took the initial conditions to be

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Table 2.1: Meaning of parameters

para | meaning of the parameter value

bp maximum per capita daily birds production rate 0.5

1/a, | size of birds population at which 1000

the number of new born birds is maximized

bm maximum per capita daily mosquito egg production rate 5

1/am, | size of mosquito population at which egg laying is maximized 10000 ds; mortality rate of uninfected juveniles (per day) 0.005 di; mortality rate of infected juveniles (per day) 0.05

dsa mortality rate of uninfected adults (per day) 0.0025 dia mortality rate of infected adults (per day) 0.015

dm mortality rate of mosquito (per day) 0.05

B; contact rate between uninfected juvenile and infected mosquito | variable Ba contact rate between uninfected adult and infected mosquito variable Bm contact rate between uninfected mosquito and infected juvenile | variable aBm | contact rate between uninfected mosquito and infected juvenile | variable T duration of more vulnerable period of bird (day) 160

ơ maturation time of mosquito (day) 10

dị mortality rate of larva mosquito (per day) 0.1

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for t € [_—7,0] and Á;(0) = 2 This, together with the matching condition (2.2.19), gives

J,(0) = 5470 and J;(0) = 0

In Figure 2.1, parameter values are 8; = 4.7015 x 107-6, 8, = 2.3705 x 1078, 6, = 1.1853 x 10-°, a@m = 4.3657 x 10-” and other parameters have the values shown in Table 2.1

In this case d,, is larger than the right hand side of (2.2.38) which equals 0.0382 One can

check the condition (2.2.38) is satisfied and the infected populations go to zero However, as we increase the contact rates, i.e., parameter values are @; = 6.7021 x 10-6, 8, = 3.3792 x 1078, Bm = 1.6896 x 10”, œđ„ = 6.2234 x 10” and other parameters have the values shown in Table 3.1 In this case d,, is larger than the right hand side of (2.2.38) which equals 0.0777 In this case, the condition (2.2.38) fails and the disease sustains in the bird and mosquito

population as shown in Figure 2.2 If we continue to increase the contact rates: parameter

values are 3; = 1.8 x 1075, 8, = 9.0756 x 10-°, Bn = 4.5378 x 107°, am = 1.6714 x 107° and

other parameters have the values shown in Table 3.1 In this case d,, is less than the right hand side of (2.2.38) which equals 0.5605 We eventually find oscillatory behaviors as shown in Figure 2.3 suggesting the possibility of a Hopf bifurcation to periodic solutions

2.3 Discussions

Throughout this Chapter simple mass action terms have been used In some virus infections, possibly including mosquito borne diseases, one might argue for the inclusion of a term which

represents the fact that a female mosquito takes a fixed number of blood meals per unit time (Anderson and May [2]) Such a modification involves dividing by bird density and has recently been utilized by Lewis et al [31] in some simpler models for WNv In this Chapter such a