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Mathematical Biosciences 167 (2000) 51±64
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In¯uence of vertical and mechanical transmission on the
dynamics of dengue disease
Lourdes Esteva a,*, Cristobal Vargas b
a
b

Departamento de Matem
aticas, Facultad de Ciencias, UNAM, M
exico, D.F. 04510, Mexico
Departamento de Matem
aticas, CINVESTAV-IPN, A.P. 14-740, M
exico, 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 transovarial or vertical transmission [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

*

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.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 2 4 - 9

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L. Esteva, C. Vargas / Mathematical Biosciences 167 (2000) 51±64

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. aegypti is 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 than A. aegypti by
combining transovarial transmission with periodic outbreaks in human or monkey populations.
It has been suggested that an important factor that makes A. aegypti such an ecient vector of
dengue fever is its habit of taking partial blood meals [9,22]. It is not uncommon that a single A.

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:

where V1 is the class of susceptible mosquitoes, Vm the class of mosquitoes that have acquired the
virus from an infectious person and can transmit it immediately. These mosquitoes are not yet
infectious because they have not gone through the entire infectious process, but since they have
the virus in their saliva, they can transmit it when they bite immediately another person; we will

refer to this kind of transmission as mechanical transmission, Vl the class of latent or exposed

L. Esteva, C. Vargas / Mathematical Biosciences 167 (2000) 51±64

53

mosquitoes, consisting of individuals which are infected but not yet infectious, V2 the class of
infectious mosquitoes, S the class of susceptible humans, I the class of infectious humans, R is the
class of removed humans.
The assumptions and parameters in the model are as follows:
(i) The human population size N, and the mosquito population size V are constant, with constant mortality rates l and m, respectively.
(ii) Newborn mosquitoes from either the susceptible or mechanical transmission class are susceptible. 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 q ˆ 1 ÿ p is infectious. In this paper we will assume that
p > 0.
(iii) 1=rm is 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 mechanically.
(iv) 1=r is the latent period in mosquitoes (about 8±10 days).
(v) 1=c is the infectious period in humans (about 4±7 days).

(vi) k1 and k2 denote the contact rates between susceptible humans and infected mosquitoes belonging to the classes Vm and V2 , respectively. In the model, k1 and k2 have the form
k1 ˆ bb1

V
;
N ‡m

k2 ˆ bb2

V
;
N ‡m

where b is the biting rate of mosquitoes (average number of bites per mosquito per day); m is
the number of alternative hosts available as blood sources; and b1 and b2 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) d is the contact rate between susceptible mosquitoes and infectious humans, and it is given
by
N

d ˆ ba
;
N ‡m
where a is the transmission probability from humans to mosquitoes.
Since both populations are constant, it is equivalent to formulate the model in terms of the
proportions V1 ; Vm ; Vl ; V2 ; S; I; R of individuals in each class. Also, since V1 ˆ 1 ÿ Vm ÿ Vl ÿ V2 and
R ˆ 1 ÿ S ÿ I, it is enough to consider the following system of equations:
S 0 ˆ l…1 ÿ S† ÿ …k1 Vm ‡ k2 V2 †S;
I 0 ˆ …k1 Vm ‡ k2 V2 †S ÿ …c ‡ l†I;
Vm0 ˆ dI…1 ÿ Vm ÿ Vl ÿ V2 † ÿ …rm ‡ m†Vm ;
Vl0 ˆ rm Vm ÿ …r ‡ m†Vl ;
V20 ˆ rVl ÿ pmV2

…2:1†

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L. Esteva, C. Vargas / Mathematical Biosciences 167 (2000) 51±64

in the region

X ˆ f…S; I; Vm ; Vl ; V2 † 2 R5‡ : 0 6 S ‡ I 6 1; 0 6 Vm ‡ Vl ‡ V2 6 1g:

3. Analysis of the model
It can be seen that the vector ®eld F, of system (2.1), on the boundary of X does 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 express S; Vm ; Vl and V2 in terms
of I.
Adding the equilibrium equations for S and I of system (2.1) we obtain
c‡l
S ˆ1ÿ
I:
…3:1†
l
From the equations for Vl and V2 :
rm
Vl ˆ
Vm ;
r‡m
rm r

Vm :
pm…r ‡ m†

V2 ˆ

…3:2†
…3:3†

Adding the last three equations of (2.1):
Vm ˆ

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 6ˆ 0. Substituting S; Vm and V2 in the second
equation of (2.1), we obtain after some manipulations the following solution

I ˆ T …R0 ÿ 1†;

…3:5†

where
dF
;
…c ‡ l†…rm ‡ m†


rrm k2
F ˆ k1 ‡
pm…r ‡ m†
R0 ˆ

…3:6†
…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 E1 ˆ …S  ; I  ; Vm ; Vl ; 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.

L. Esteva, C. Vargas / Mathematical Biosciences 167 (2000) 51±64

55

Lemma 3.1. Let R0 defined by (3.6). If R0 6 1, 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 that R0 > 1 is the threshold condition that determines when the disease dies
out or remains endemic in the population. The square root of R0 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 in X for R0 6 1, we use the following Lyapunov function:
L ˆ AI ‡ BVm ‡ CVl ‡ DV2 ;

…3:9†

where
…R0 ‡ 1†
p
;
2
…rm ‡ m†

Aˆ

pd
;
…c ‡ l†…rm ‡ m†

Cˆ

dk2 r
;
m…c ‡ l†…rm ‡ m†…r ‡ m†

Bˆ

Dˆ

dk2
:
m…c ‡ l†…rm ‡ m†

The orbital derivative of L is given by


pd
…R0 ‡ 1†
L_ ˆ ÿ
1ÿ
…1 ÿ Vm ÿ Vl ÿ V2 † I
…rm ‡ m†
2



R0 ‡ 1
d
k2 rrm
k1 S ‡
ÿ
ÿp
Vm
2
…c ‡ l†…rm ‡ m†
pm…r ‡ m†
pdk2
ÿ
…1 ÿ S†V2
…c ‡ l†…rm ‡ m†




pd
…R0 ‡ 1†
R0 ‡ 1
1ÿ
6ÿ
I ÿp
ÿ R0 Vm
…rm ‡ m†
2
2


…1 ÿ R0 †
pd
6ÿ
I ‡ pVm :
2
rm ‡ m
Clearly, L_ 6 0 in X when R0 6 1. Since the expressions inside the square brackets are greater or
equal to 0, the subset of X where L_ ˆ 0 holds is de®ned by the equations
I ˆ 0;

Vm ˆ 0;

Vm ˆ Vl ˆ V2 ˆ 0

…1 ÿ S†V2 ˆ 0;

if R0 < 1;

or I ˆ Vm ˆ V2 ˆ 0

or I ˆ 0; S ˆ 1

if R0 ˆ 1:

From inspection of system (2.1), it can be seen in any case that E0 is the only invariant set
contained in L_ ˆ 0. Therefore by LaSalle±Lyapunov theorem [10], all trajectories that start in X
approach E0 when t ! 1.
The characteristic polynomial of the Jacobian DF …E0 † has the form r5 ‡ a1 r4 ‡ a2 r3
‡ a3 r2 ‡ a4 r ‡ a5 . After some calculations it can be seen that the independent coecient is

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L. Esteva, C. Vargas / Mathematical Biosciences 167 (2000) 51±64

a5 ˆ lpm…rm ‡ m†…r ‡ m†…c ‡ l†…1 ÿ R0 †:
For R0 > 1, this coecient is negative, and therefore E0 is unstable.
All these results can be collected in the following theorem.
Theorem 3.1. The equilibrium point E0 is globally asymptotically stable in X for R0 6 1, and it is
unstable for R0 > 1.

4. Stability of the endemic equilibrium
In this section, we shall prove the local stability of the endemic equilibrium when R0 > 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 point x0 of the system of
di€erential equations
x0 ˆ f …x†

…4:1†

is proving that the linearized equation
Z0 ˆ Df …x0 †Z
has no solutions of the form
 ˆ Z0 ewt
Z…t†

…4:2†

with Z0 2 Cn ÿ f0g; w 2 C and Re w P 0, where C denotes 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 R ˆ 1 ÿ S ÿ I. Substituting a solution of the form (4.2) in the
linearized equation of the endemic equilibrium, we obtain the following linear equations.


ÿ
wZ1 ˆ ÿ k1 Vm ‡ k2 V2 ‡ c ‡ l Z1 ‡ k1 …1 ÿ I  ÿ R †Z2


ÿ
‡ k2 …1 ÿ I  ÿ R †Z4 ÿ k1 Vm ‡ k2 V2 Z5 ;
wZ2 ˆ d…1 ÿ Vm ÿ Vl ÿ V2 †Z1 ÿ …dI  ‡ rm ‡ m†Z2 ÿ dI  Z3 ÿ dI  Z4 ;

…4:3†

wZ3 ˆ rm Z2 ÿ …r ‡ m†Z3 ;
wZ4 ˆ rZ3 ÿ pmZ4 ;
wZ5 ˆ cZ1 ÿ lZ5 ;
where Z1 ; . . . ; Z5 2 C, and I  ; Vm ; Vl ; V2 ; R are the coordinates of the endemic equilibrium.
Solving the last three equations of (4.3) for Z3 ; Z4 and Z5 , and substituting the results in the ®rst
two equations we obtain, after some manipulations the equivalent system
 ;
…1 ‡ Gi …w††Zi ˆ …H Z†
i
where Z ˆ …Z1 ; Z2 ; Z3 ; Z4 † and

i ˆ 1; . . . ; 4;

…4:4†

L. Esteva, C. Vargas / Mathematical Biosciences 167 (2000) 51±64





w‡c‡l
ÿ
1


;
w ‡ k1 Vm ‡ k2 V2
G1 …w† ˆ
…c ‡ l†
w‡l



1
rm
rrm

w ‡ dI 1 ‡
‡
;
G2 …w† ˆ
…rm ‡ m†
w ‡ r ‡ m …w ‡ pm†…w ‡ r ‡ m†
w
;
G3 …w† ˆ
r‡m
w
G4 …w† ˆ
;
pm
and H is the matrix
3
2
k1 …1 ÿ I  ÿ R †
k1 …1 ÿ I  ÿ R †
0
0
7
6
c‡l
c‡l
7
6
7
6 d…1 ÿ V  ÿ V  ÿ V  †
7
6
m
l
2
0
0
0
7
6
rm ‡ m
7:
6
7
6
rm
7
6
0
0
0
7
6
r‡m
7
6
5
4
r
0
0
0
pm
Note that the matrix H has non-negative entries, and Y  ˆ …I  ; Vm ; Vl ; V2 † satis®es
Y  ˆ H Y  :

57

…4:5†

Also, since the coordinates of Y are positive, if Z is any solution of (4.4) then there exists a
 such that
minimal positive s, depending on Z,
 6 sY  ;
jZj
…4:6†
 ˆ …jZ1 j; . . . ; jZ4 j† and j
j is the norm in C.
where jZj
Now, we want to show that Re w < 0. Deny it, we distinguish two cases: w ˆ 0 and w 6ˆ 0. In
the ®rst case, the determinant of system (4.4) is given by



…1 ÿ I  ÿ R †
k2 rrm
d…1 ÿ Vm ÿ Vl ÿ V2 †
k1 ‡
D ˆ …1 ‡ G1 …0††…1 ‡ G2 …0†† ÿ
c‡l
…rm ‡ m†
pm…r ‡ m†


I Vm
ˆ …1 ‡ G1 …0††…1 ‡ G2 …0†† ÿ   ˆ G1 …0† ‡ G2 …0† ‡ G1 …0†G2 …0† > 0;
Vm I


since G1 …0† and G2 …0† are positive. Then, for w ˆ 0, the only solution of system (4.4) is the trivial
one which implies that w 6ˆ 0.
Assume now that w 6ˆ 0, and Re w P 0. Let G…w† ˆ minfj1 ‡ Gi …w†j; i ˆ 1; . . . ; 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 that H is non-negative, we obtain the following inequality:
 6 H jZj:

G…w†jZj
…4:7†
Using (4.6) and then (4.5)
 6 sH Y  ˆ sY 
G…w†jZj
which implies

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L. Esteva, C. Vargas / Mathematical Biosciences 167 (2000) 51±64

 6
jZj

s 
Y < sY 
G…w†

but this contradicts the minimality of s. Therefore Re w < 0.
In this way we have proved the following theorem.
Theorem 4.1. If R0 > 1, the endemic equilibrium E1 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 existence and stability of the endemic equilibrium is given by


dF
rrm k2
> 1; F ˆ k1 ‡
:
R0 ˆ
…c ‡ l†…rm ‡ m†
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 transmission q ˆ 1 ÿ p a€ects R0 . The parameter
k1 appears in the numerator of the expression for R0 . If we increment k1 by an amount  > 0, R0 is
ampli®ed by a factor
pm…r ‡ m†
A1 …† ˆ 1 ‡
:
k1 pm…r ‡ m† ‡ k2 rm r
We observe that A1 grows linearly when we increment by . If for example, we n-fold the value of
k1 , then R0 at most will n-fold itself.
On the other hand, q ˆ 1 ÿ p appears in the denominator of R0 . If we increment q by an
amount , maintaining q ‡  < 1, R0 is ampli®ed by the factor


k2 prrm
1
A2 …† ˆ k1 pm…r ‡ m† ‡
p ÿ  k1 pm…r ‡ m† ‡ k2 rm r
which behaves like an hyperbola when we increment . In fact, for q near one, small increments on
q will produce large increments in R0 .
In order to compare A1 …† and A2 …† we notice that A1 ! 1 when  ! 1 ÿ q, therefore it is clear
that for  near 1 ÿ q; A2 > A1 . On the other hand, for  near 0
k2 rrm ÿ p2 m…r ‡ m†
A02 …†jˆ0 ÿ A01 …†jˆ0 ˆ
p…k1 pm…r ‡ m† ‡ k2 rrm †
and this di€erence will be bigger than zero if
k2 >

p2 m…m ‡ r†
rrm

which implies that the hyperbola A2 …† dominates the straight line A1 …†. This means that for the
rank of parameters for which the above inequality holds, vertical transmission is a much more

L. Esteva, C. Vargas / Mathematical Biosciences 167 (2000) 51±64

59

important parameter than mechanical transmission. For typical values: 1=rm ˆ 0:5 day, 1=r ˆ 8
days, 1=m ˆ 25 days, the inequality becomes k2 > 0:0264p2 , which is true if k > 0:0264.
The parameter rm , which controls the mechanical transmission period, appears also in the
denominator of R0 , but it is temperated by the vector life span m, so variations of this parameter
have little e€ect on R0 .
In Figs. 1 and 2 we illustrate how changes in k1 and q a€ect the temporal course of the infectious humans I and infectious mosquitoes V2 . We assume that the period of time that a mosquito can transmit the disease mechanically, 1=rm is 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 ˆ 0:75; d ˆ 0:375; 1=r ˆ 10 days, 1=c ˆ 3 days, 1=l ˆ 68 yr. The initial conditions are:
S ˆ 0:1; I ˆ 0:0001; Vm ˆ 0:000002; Vl ˆ 0:0002 and V2 ˆ 0:0005: In all ®gures R0 > 1.
Figs. 1(a)±(d) illustrate the case q ˆ 0 and k1 ˆ 0; 0:75; 1:5; 2:25, respectively. When we increment the parameter k1 from 0 to 2.25, the endemic proportions I  and V2 , increase from
I  ˆ 0:00010927 and V2 ˆ 0:00054424 to I  ˆ 0:00010942 and V2 ˆ 0:00054499, respectively. In

Fig. 1. Log10 of the temporal course of the proportions of infectious humans and infectious vectors: (a)
k1 ˆ 0; q ˆ 0; R0 ˆ 11:21; I  ˆ 0:00010927; V2 ˆ 0:00054424; (b) k1 ˆ 0:75; q ˆ 0; R0 ˆ 11:26; I  ˆ 0:000109327;
V2 ˆ 0:00054449; (c) k1 ˆ 1:5; q ˆ 0; R0 ˆ 11:31; I  ˆ 0:00010937; V2 ˆ 0:000544736; (d) k1 ˆ 2:25; q ˆ 0; R0 ˆ
11:37; I  ˆ 0:00010942; V2 ˆ 0:00054498:

60

L. Esteva, C. Vargas / Mathematical Biosciences 167 (2000) 51±64

Fig. 2. Log10 of the temporal course of the proportions of infectious humans and infectious vectors: (a)
k1 ˆ 0; q ˆ 0; R0 ˆ 11:21; I  ˆ 0:00010927; V2 ˆ 0:00054424; (b) k1 ˆ 0; q ˆ 0:25; R0 ˆ 14:95; I  ˆ 0:00011195;
V2 ˆ 0:0007432; (c) k1 ˆ 0; q ˆ 0:5; R0 ˆ 22:42; I  ˆ 0:00011462; V2 ˆ 0:00114108; (d) k1 ˆ 0; q ˆ 0:75; R0 ˆ
44:85; I  ˆ 0:00017304; V2 ˆ 0:00233264:

this simulations the temporal course of I and V2 as well as the endemic proportions practically do
not change when we increase k1 . We notice that we need larger values of k1 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 ˆ 0 and q ˆ 0; 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  ˆ 0:00010927 and V2 ˆ 0:00054424 to I  ˆ 0:00017304 and V2 ˆ 0:00233264;
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
l happens 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 for R0 > 1; regardless of the initial conditions. Since the structure

L. Esteva, C. Vargas / Mathematical Biosciences 167 (2000) 51±64

61

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 susceptibles S ‡ U satis®es R0 …S ‡ U† > 1; then it decreases and the infection proportion I ‡ 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 conditions. 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 for R0 < 1. Recall that local stability is given by the eigenvalues of the Jacobian of system
(2.1) at E0 :

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L. Esteva, C. Vargas / Mathematical Biosciences 167 (2000) 51±64

2

ÿl
0
6 0 ÿ…c ‡ l†
6
DF …E0 † ˆ 6
d
6 0
4 0
0
0
0

ÿk1
k1
ÿ…rm ‡ m†
rm
0

3
0
ÿk2
0
k2 7
7
0
0 7
7:
ÿ…r ‡ m†
0 5
r
ÿpm

The eigenvalues of DF …E0 † are ÿl and the eigenvalues of the submatrix
3
2
0
k2
ÿ…c ‡ l†
k1
6
d
ÿ…rm ‡ m†
0
0 7
7:
A11 ˆ 6
4
0
rm
ÿ…r ‡ m†
0 5
0
0
r
ÿpm

…A:1†

…A:2†

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 the n  n matrix A ˆ ‰aij Š is a non-singular M-matrix if aij 6 0; i 6ˆ j;
and there exists a matrix B P 0 and a real number s > 0 such that
A ˆ sI ÿ B

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,
X
jaij jdj ; i ˆ 1; . . . ; n:
aii di >
j6ˆi

Returning to our problem, we consider the matrix ÿA11 : Its diagonal elements are positive.
According to Proposition A.2, ÿA11 is 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 > k1 d2 ‡ k2 d4 ;
…rm ‡ m†d2 > dd1 ;

…A:3†

…r ‡ m†d3 > rm d2 ;
pmd4 > rd3 :
Let
d1 ˆ 1;

d2 ˆ

d‡
;
rm ‡ m

d3 ˆ

rm d2 ‡ 
;
r‡m

d4 ˆ

rd3 ‡ 
;
pm

where 0 < : Obviously, the last three inequalities of Eq. (A.3) hold. To prove the ®rst inequality
we observe that

L. Esteva, C. Vargas / Mathematical Biosciences 167 (2000) 51±64

63

k1 d2 ‡ k2 d4 ˆ …c ‡ l†R0 ‡ A;
where


k1
k2
r
rrm
Aˆ
‡
:
‡
1‡
r ‡ m …r ‡ m†…rm ‡ m†
rm ‡ m pm
Now, since R0 < 1, we can take 0 <  < ……c ‡ l†…1 ÿ R0 ††=A. Then
…c ‡ l†R0 ‡ A < c ‡ l ˆ d1 …c ‡ l†:
Therefore, system (A.3) has positive solution when R0 < 1: This implies that ÿA11 is a nonsingular M-matrix for R0 < 1: From Proposition A.1 it follows that the eigenvalues of A11 have
negative real part.

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