Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol167.Issue1.Sept2000:

In¯uence of vertical and mechanical transmission on the

dynamics of dengue disease

Lourdes Esteva

a,*

, Cristobal Vargas

b

a

Departamento de Matematicas, Facultad de Ciencias, UNAM, Mexico, D.F. 04510, Mexico b

Departamento de Matematicas, CINVESTAV-IPN, A.P. 14-740, Mexico, D.F. 07000, Mexico

Received 1 February 1999; received in revised form 21 July 1999; accepted 23 July 1999

Abstract

We formulate a non-linear system of di€erential equations that models the dynamics of transmission of dengue fever. We consider vertical and mechanical transmission in the vector population, and study the e€ects that they have on the dynamics of the disease. A qualitative analysis as well as some numerical examples are given for the model. Ó _{2000 Elsevier Science Inc. All rights reserved.}

Keywords:Dengue disease; Vertical transmission; Interrupted feeding; Endemic equilibrium; Threshold number

1. Introduction

The dynamics of dengue fever are in¯uenced by many factors involving humans, the mosquito vector and the virus, as well as the environment which directly or indirectly a€ects all three populations involved and the interrelations among them. The altitude, climate and humidity are important variables that directly a€ect transmission dynamics of dengue viruses by acting as determinant factors on the mosquito population [15,17,27]. Dengue viruses are only endemic in tropical areas of the world where climate and weather allow continuous breeding populations of mosquitoes. In subtropical and temperate regions of the world, periodic epidemics of dengue may occur, but the viruses are not endemic and must be introduced before transmission is initiated [9]. Mosquitoes can transmit arbovirus without the participation of the human host. In the tran-sovarialor verticaltransmission [9] the virus is transmitted from the infected mother to the eggs. There have been laboratory and ®eld evidences that transovarial transmission exists to certain

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*_{Corresponding author. Tel.: +52-56 224 858; fax: +52-56 224 859.}

E-mail addresses:lesteva@servidor.unam.mx (L. Esteva), cvargas@math.cinvestav.mx (C. Vargas).

0025-5564/00/$ - see front matter Ó _{2000 Elsevier Science Inc. All rights reserved.}

degree in some species of Aedes mosquitoes [1,13,14,23±25]. However, the role of transovarial transmission in the maintenance of cycle of dengue viruses is not clearly determinated [14,26,28].

A. aegyptiis the most important urban vector of dengue viruses. Vertical transmission in this vector has been observed at a relative low rate [25]; however new studies begin to show that this could not be necessarily true [1,14]. In contrast, A. albopictus, considered less important for the transmission of dengue virus to man, has a degree of transovarial transmission sucient to assist in the maintenance of dengue virus in nature [3,19,26]. A. albopictus has been incriminated as responsible for a large amount of sporadic dengue transmission in rural parts of Asia [9,20,28]. This reinforces the hypothesis suggested by several authors [4,7±9,25] that viruses in semirural, rural and forested areas could be maintained by more ecient vector species thanA. aegyptiby combining transovarial transmission with periodic outbreaks in human or monkey populations. It has been suggested that an important factor that makesA. aegyptisuch an ecient vector of dengue fever is its habit of taking partial blood meals [9,22]. It is not uncommon that a singleA. aegypti bites several persons in the same room or house before becoming satiated. This is probably a survival mechanism, since the slightest movement of the person being bitten will distract the mosquito and interrupt the blood feeding.

After feeding on a person whose blood contains the virus, A. aegypti can transmit dengue immediately in a mechanical way, by a change of host when its blood meal is interrupted, or after an incubation period of 8±10 days, during which time the virus multiplies in its salivary glands [28].

In this paper we continue the study of the dynamics of dengue disease started in [5,6]. Here we analyze the role of vertical transmission and mechanical transmission due to interrupted feeding in the dynamics of dengue disease. To this end we formulate a model for the dynamics of dengue disease taking into account both aspects. The purpose of the model is to treat in a qualitative and quantitative manner the main features of the process.

2. The model

The model is represented by the following transfer diagram:

whereV1 is the class of susceptible mosquitoes,Vmthe class of mosquitoes that have acquired the

mosquitoes, consisting of individuals which are infected but not yet infectious, V2 the class of

infectious mosquitoes,Sthe class of susceptible humans,Ithe class of infectious humans,Ris the class of removed humans.

The assumptions and parameters in the model are as follows:

(i) The human population sizeN, and the mosquito population sizeVare constant, with con-stant mortality ratesl and m, respectively.

(ii) Newborn mosquitoes from either the susceptible or mechanical transmission class are sus-ceptible. We also assume that the progeny of those in the latent class are all susceptible, since the parasite density within an individual host in this class may not have reached a level at which o€springs are likely to be much a€ected. Of the progeny of infectious individuals, we suppose that a fraction p is susceptible and qˆ1ÿp is infectious. In this paper we will assume that

p>0.

(iii) 1=rmis the average period of time that a mosquito stays in the class Vm before it becomes

latent (this period can be very short). During this period, mosquitoes can transmit the virus me-chanically.

(iv) 1=r is the latent period in mosquitoes (about 8±10 days). (v) 1=cis the infectious period in humans (about 4±7 days).

(vi)k1andk2denote the contact rates between susceptible humans and infected mosquitoes

be-longing to the classesVm andV2, respectively. In the model,k1 and k2 have the form

k1ˆbb1

V

N‡m; k2 ˆbb2 V N‡m;

whereb is the biting rate of mosquitoes (average number of bites per mosquito per day); mis the number of alternative hosts available as blood sources; andb1 andb2 are the transmission

probabilities (the probability that an infectious bite produces a new case) from mosquitoes belonging to the classes Vm and V2, to humans.

(vii)dis the contact rate between susceptible mosquitoes and infectious humans, and it is given by

dˆba N

N‡m;

whereais the transmission probability from humans to mosquitoes.

Since both populations are constant, it is equivalent to formulate the model in terms of the proportionsV1;Vm;Vl;V2;S;I;Rof individuals in each class. Also, sinceV1ˆ1ÿVmÿVlÿV2 and

Rˆ1ÿSÿI, it is enough to consider the following system of equations:

S0ˆl…1ÿS† ÿ …k1Vm‡k2V2†S;

I0ˆ …k1Vm‡k2V2†Sÿ …c‡l†I;

V_m0 ˆd_{I…1ÿVmÿVlÿV2† ÿ …rm‡m†Vm; …2:1†}

V_l0ˆrmVmÿ …r‡m†Vl;

in the region

Xˆ f…S;I;Vm;Vl;V2† 2R5‡ :06S‡I61; 06Vm‡Vl‡V261g:

3. Analysis of the model

It can be seen that the vector ®eldF, of system (2.1), on the boundary ofXdoes not point to the exterior of this region. Therefore, the solutions of (2.1) remain in X for t>0, and thus the problem is well posed.

Next, we will ®nd the equilibrium points of (2.1). To this end we expressS;Vm;VlandV2in terms

ofI.

Adding the equilibrium equations forSand I of system (2.1) we obtain

Sˆ1ÿc‡l

l I: …3:1†

From the equations forVl and V2:

Vl ˆ

rm

r‡mVm; …3:2†

V2 ˆ

rmr

pm…r‡m†Vm: …3:3†

Adding the last three equations of (2.1):

Vmˆ

pm…r‡m†dI

‰pm…r‡m† ‡rm…pm‡r†ŠdI ‡pm…r‡m†…rm‡m†

: …3:4†

We have the disease-free equilibrium E0 ˆ …1;0;0;0;0†, which always exists in X. In order to

®nd the non-trivial equilibria in X, we assume I 6ˆ0. Substituting S;Vm and V2 in the second

equation of (2.1), we obtain after some manipulations the following solution

I ˆT…R0ÿ1†; …3:5†

where

R0 ˆ

d_F

…c‡l†…rm‡m†

; …3:6†

F ˆ k1

‡ rrmk2 pm…r‡m†

…3:7†

and

T ˆ lpm…r‡m†…rm‡m†

d‰pm…r‡m†…F ‡l† ‡lrm…pm‡r†Š

: …3:8†

It is an easy matter to verify that the equilibrium E1ˆ …S;I;V_m;V_l;V2† whose coordinates

satisfy Eqs. (3.1)±(3.5) belongs to int…X† if and only if R0>1. Therefore we have the following

Lemma 3.1.LetR0 defined by(3.6).IfR061,the only equilibrium point of the system(2.1)is the

disease-free equilibrium E0. If R0>1,there also exists a unique endemic equilibrium E1 in int…X†

whose coordinates are given by Eqs.(3.1)±(3.5)

Then, we have seen thatR0>1 is the threshold condition that determines when the disease dies

out or remains endemic in the population. The square root ofR0 is called the basic reproductive

number of the disease.

Next, we analyze the stability properties of E0. In Appendix A, we give a proof of the local

stability of this equilibrium by using the theory of M-matrices.

To prove the global stability of E0 inX forR061, we use the following Lyapunov function:

LˆAI‡BVm‡CVl‡DV2; …3:9†

The orbital derivative ofL is given by

_

Clearly, L_ 60 in X when R061. Since the expressions inside the square brackets are greater or

equal to 0, the subset of XwhereL_ ˆ0 holds is de®ned by the equations

I ˆ0; Vmˆ0; …1ÿS†V2ˆ0; if R0 <1;

VmˆVl ˆV2 ˆ0 or I ˆVmˆV2 ˆ0 or I ˆ0;Sˆ1 if R0 ˆ1:

From inspection of system (2.1), it can be seen in any case that E0 is the only invariant set

contained inL_ ˆ0. Therefore by LaSalle±Lyapunov theorem [10], all trajectories that start inX

approachE0 when t! 1.

The characteristic polynomial of the Jacobian DF…E0† has the form r5‡a1r4‡a2r3

a5 ˆlpm…rm‡m†…r‡m†…c‡l†…1ÿR0†:

For R0 >1, this coecient is negative, and thereforeE0 is unstable.

All these results can be collected in the following theorem.

Theorem 3.1. The equilibrium pointE0 is globally asymptotically stable in X for R061, and it is

unstable forR0 >1.

4. Stability of the endemic equilibrium

In this section, we shall prove the local stability of the endemic equilibrium whenR0 >1. For

this we shall follow the method given by Hethcote and Thieme in [12], which is based on a Krasnoselskii technique [16].

A usual way to prove the local asymptotic stability of an equilibrium pointx0 of the system of

di€erential equations

x0ˆf…x† …4:1†

is proving that the linearized equation

Z0ˆDf…x0†Z

has no solutions of the form

Z…t† ˆZ0ewt …4:2†

withZ02Cnÿ f0g;w2C and Re wP0, whereCdenotes the complex numbers.

In the following, we shall work with the equivalent system to (2.1) which is obtained by taking the coordinates I;Vm;Vl;V2 and Rˆ1ÿSÿI. Substituting a solution of the form (4.2) in the

linearized equation of the endemic equilibrium, we obtain the following linear equations.

wZ1 ˆ ÿ k1V_m ÿ

‡k2V2‡c‡l

Z1‡k1…1ÿIÿR†Z2

‡k2…1ÿIÿR†Z4ÿ k1Vm ÿ

‡k2V2

Z5;

wZ2 ˆd…1ÿVmÿV

l ÿV

2†Z1ÿ …dI‡rm‡m†Z2ÿdIZ3ÿdIZ4; …4:3†

wZ3 ˆrmZ2ÿ …r‡m†Z3;

wZ4 ˆrZ3ÿpmZ4;

wZ5 ˆcZ1ÿlZ5;

whereZ1;. . .;Z5 2C, and I;Vm;Vl;V2;R are the coordinates of the endemic equilibrium.

Solving the last three equations of (4.3) forZ3;Z4andZ5, and substituting the results in the ®rst

two equations we obtain, after some manipulations the equivalent system

…1‡Gi…w††Ziˆ …HZ† _i; iˆ1;. . .;4; …4:4†

G1…w† ˆ

Also, since the coordinates of Y are positive, if Z is any solution of (4.4) then there exists a minimal positive s, depending onZ, such that

jZj 6sY; …4:6†

wherejZj ˆ …jZ 1j;. . .;jZ4j† and j jis the norm in C.

Now, we want to show that Re w<0. Deny it, we distinguish two cases:wˆ0 andw6ˆ0. In the ®rst case, the determinant of system (4.4) is given by

Dˆ …1‡G1…0††…1‡G2…0†† ÿ

one which implies thatw6ˆ0.

Assume now thatw6ˆ0, andRe wP0. LetG…w† ˆminfj1‡Gi…w†j; iˆ1;. . .;4g. It is easy to

prove that in this case j1‡Gi…w†j>1 for all i, and therefore G…w†>1. Taking norms on both

sides of (4.4), and using the fact thatHis non-negative, we obtain the following inequality:

G…w†jZj 6HjZj: …4:7†

Using (4.6) and then (4.5)

G…w†jZj 6sHY ˆsY

jZj 6 s G…w†Y

_<sY

but this contradicts the minimality of s. ThereforeRe w<0. In this way we have proved the following theorem.

Theorem 4.1. IfR0 >1,the endemic equilibriumE1 is locally asymptotically stable.

5. Discussion

Vertical and mechanical transmission in the vector population contribute to the spread of dengue fever. However, the actual impact of these factors on the dynamics of the disease has not been completely determinated. As a theoretical experiment, we incorporated both factors in the transmission of dengue disease. For this model the threshold condition that determines the ex-istence and stability of the endemic equilibrium is given by

R0 ˆ

dF …c‡l†…rm‡m†

>1; F ˆ k1

‡ rrmk2 pm…r‡m†

:

We want to analyze how changes in the mechanical transmission contact rate k1, and in the

proportion of infected newborns due to vertical transmissionqˆ1ÿpa€ects R0. The parameter

k1 appears in the numerator of the expression forR0. If we incrementk1by an amount >0,R0is

ampli®ed by a factor

A1…† ˆ1‡

pm…r‡m†

k1pm…r‡m† ‡k2rmr

:

We observe thatA1 grows linearly when we increment by. If for example, wen-fold the value of

k1, then R0 at most will n-fold itself.

On the other hand, qˆ1ÿp appears in the denominator of R0. If we increment q by an

amount , maintainingq‡ <1, R0 is ampli®ed by the factor

A2…† ˆ k1pm…r

‡m† ‡k2prrm pÿ

₁

k1pm…r‡m† ‡k2rmr

which behaves like an hyperbola when we increment. In fact, forqnear one, small increments on

q will produce large increments inR0.

In order to compareA1…†andA2…†we notice thatA1 ! 1when!1ÿq, therefore it is clear

that for near 1ÿq; A2 >A1. On the other hand, for near 0

A0₂…†j_ˆ0ÿA0₁…†j_ˆ0 ˆ k2rrmÿp

2_m…r‡m†

p…k1pm…r‡m† ‡k2rrm†

and this di€erence will be bigger than zero if

k2 >

p2m…m‡r†

rrm

which implies that the hyperbolaA2…†dominates the straight line A1…†. This means that for the

important parameter than mechanical transmission. For typical values: 1=rmˆ0:5 day, 1=rˆ8

days, 1=mˆ25 days, the inequality becomes k2 >0:0264p2, which is true ifk>0:0264.

The parameter rm, which controls the mechanical transmission period, appears also in the

denominator ofR0, but it is temperated by the vector life spanm, so variations of this parameter

have little e€ect onR0.

In Figs. 1 and 2 we illustrate how changes in k1 and q a€ect the temporal course of the

in-fectious humansI and infectious mosquitoesV2. We assume that the period of time that a

mos-quito can transmit the disease mechanically, 1=rmis equal to 1:30 h; for the vector life span, 1=m,

we assume a value of 20 days; and the values for the other parameters are taken as in [21]:

k2 ˆ0:75; dˆ0:375; 1=rˆ10 days, 1=cˆ3 days, 1=lˆ68 yr. The initial conditions are:

Sˆ0:1; I ˆ0:0001; Vmˆ0:000002; Vl ˆ0:0002 and V2 ˆ0:0005: In all ®guresR0 >1.

Figs. 1(a)±(d) illustrate the caseqˆ0 andk1ˆ0;0:75;1:5;2:25, respectively. When we

incre-ment the parameter k1 from 0 to 2.25, the endemic proportions I and V2, increase from

Iˆ0:00010927 and V₂ˆ0:00054424 to Iˆ0:00010942 and V₂ ˆ0:00054499, respectively. In

Fig. 1. Log10 of the temporal course of the proportions of infectious humans and infectious vectors: (a) k1ˆ0; qˆ0; R0ˆ11:21; Iˆ0:00010927; V2ˆ0:00054424; (b) k1ˆ0:75; qˆ0; R0ˆ11:26; Iˆ0:000109327; V

2 ˆ0:00054449; (c) k1ˆ1:5; qˆ0; R0ˆ11:31; Iˆ0:00010937; V2ˆ0:000544736; (d) k1ˆ2:25; qˆ0; R0ˆ 11:37; I_{ˆ0:00010942; V}

this simulations the temporal course ofIandV2 as well as the endemic proportions practically do

not change when we increasek1. We notice that we need larger values ofk1 in order to obtain a

change of one order of magnitude in the endemic proportions. We also observed a similar behavior for di€erent sets of values of the parameters.

On the other hand, in Figs. 2(a)±(d), we take k1 ˆ0 and qˆ0;0:25;0:5;0:75; respectively.

When we increment q from 0 to 0.75 the endemic proportions I and V₂ in these simulations, increase from I_{ˆ0:00010927 and V}

2 ˆ0:00054424 to I ˆ0:00017304 and V2 ˆ0:00233264;

respectively. We note that the increment in the vector population is much larger than the one in the human population.

The temporal courses of the proportions in all ®gures present damped oscillations. In the model

lhappens to be very small with respect to the other parameters, since the average expected life of humans is very large comparing with the length of infected period and the expected life of mosquitoes. It was proved in [5] for a simpler model, that under this situation of the parameters, there exist damped oscillations forR0 >1;regardless of the initial conditions. Since the structure

Fig. 2. Log10 of the temporal course of the proportions of infectious humans and infectious vectors: (a) k1ˆ0; qˆ0; R0ˆ11:21; Iˆ0:00010927; V2ˆ0:00054424; (b) k1ˆ0; qˆ0:25; R0ˆ14:95; Iˆ0:00011195; V

2 ˆ0:0007432; (c) k1ˆ0; qˆ0:5; R0ˆ22:42; Iˆ0:00011462; V2ˆ0:00114108; (d) k1ˆ0; qˆ0:75; R0ˆ 44:85; I_{ˆ0:00017304; V}

and the parameters in both models are similar, we expected for the model of this paper the same behavior.

The behavior mentioned above can be explained intuitively in terms of the basic reproductive number R0: Denoting by U the fraction of susceptible vectors, we have that when the initial

fraction of both type of susceptiblesS‡U satis®es R0…S‡U†>1; then it decreases and the

in-fection proportionI ‡V2 ®rst increases to a peak and then decreases because there are not

suf-®cient susceptible to be infected. When the susceptible fraction gets large enough due to births of new susceptibles, there are secondary smaller epidemics and thus, solutions spiral to the endemic equilibrium.

The period between epidemics depends on the value of the parameters and the initial condi-tions. For our parameters and initial conditions, we obtained an interepidemic period that varies from almost 6 to 4 yr. These values are comparable with the ones reported in [18] for dengue in the Americas, where they found an initial interepidemic period of 5 yr which later became of 2 yr. Of course, some other factors that are not taken into account here, as for example, population growth, weather can also modify the period of the oscillations. But for our original questions we consider that these factors are not relevant.

The simulations also show big di€erences between maximal and endemic prevalence. This pattern coincides with the epidemiological records. Where we see notorious di€erences among prevalences of di€erent years (see for example [11]).

Returning to the main questions of this paper, analysis and numerical simulations of the model suggest that interrupted feeding is not an important factor for the transmission of dengue. Nevertheless, although the model predicts that the impact of this kind of transmission is small, it is necessary to have further knowledge of the parameters involved in order to fully estimate its relevance.

On the other hand, the dynamics of dengue disease appears to be strongly in¯uenced by vertical transmission. The simulations of Fig. 2 show that this transmission favors the establishment of a constant endemic level in both populations. Also, they show an important increase in the endemic level of the vector population, in contrast with the endemic level of the human population. This last result reinforces the idea that vertical transmission can be an important mechanism that favors the maintenance of the virus in rural and forested areas with low human densities.

Acknowledgements

We are grateful to Professor Karl P. Hadeler for his very valuable comments during the preparation of the manuscript. We also want to thank two anonymous referees for their careful reading that helped us to improve the paper.

Appendix A

In this appendix we shall prove, by the use of M-matrices, that E0 is locally asymptotically

stable forR0<1. Recall that local stability is given by the eigenvalues of the Jacobian of system

DF…E0† ˆ

The eigenvalues of DF…E0† areÿland the eigenvalues of the submatrix

A11ˆ

To determine the stability properties of A11; we will use well-known results on M-matrices. Our

main reference on this topic is [2].

De®nition A.1.We say that thennmatrixAˆ ‰aijŠis a non-singular M-matrix ifaij60; i6ˆj;

and there exists a matrixBP0 and a real number s>0 such that

AˆsIÿB and s>q…B†:

The following equivalences are well known.

Proposition A.1.A is a non-singular M-matrix if and only if the real part of each of its eigenvalues is greater than 0.

Proposition A.2.A is a non-singular M-matrix if and only if all the diagonal entries are positive,and there exists a positive diagonal matrix D, such that AD is strictly diagonal dominant, that is,

aiidi> X

j6ˆi

jaijjdj; iˆ1;. . .;n:

Returning to our problem, we consider the matrix ÿA11: Its diagonal elements are positive.

According to Proposition A.2,ÿA11is a non-singular M-matrix if and only if there exist numbers

d1;d2;d3 and d4 bigger than zero such that the following inequalities are satis®ed

…c‡l†d1>k1d2‡k2d4;

k1d2‡k2d4ˆ …c‡l†R0‡A;

where

Aˆ k1

rm‡m

‡k2 pm 1

‡ r

r‡m‡

rrm

…r‡m†…rm‡m†

:

Now, sinceR0<1, we can take 0< <……c‡l†…1ÿR0††=A. Then

…c‡l†R0‡A <c‡lˆd1…c‡l†:

Therefore, system (A.3) has positive solution when R0 <1: This implies that ÿA11 is a

non-singular M-matrix for R0<1: From Proposition A.1 it follows that the eigenvalues ofA11 have

negative real part.

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