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Transmission Dynamics of Two Dengue Serotypes with

vaccination scenarios

N.L. Gonz´alez Morales1, M. N´u˜nez-L´opez2, J. Ramos-Casta˜neda3and J.X.

Velasco-Hern´andez1

1Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico, Boulevard

Juriquilla No. 3001, Juriquilla, 76230, M´exico

2Departamento de Matem´aticas Aplicadas y Sistemas, DMAS, Universidad

Aut´onoma Metropolitana, Cuajimalpa, Av. Vasco de Quiroga 4871, Col. Santa Fe

Cuajimalpa, Cuajimalpa de Morelos, 05300, M´exico, D.F., M´exico

3Centro de Investigaciones sobre Enfermedades Infecciosas, Instituto Nacional de

Salud P´ublica, Cuernavaca, Mexico

Abstract

In this work we present a mathematical model that incorporates two Dengue serotypes.

The model has been constructed to study both the epidemiological trends of the disease

and conditions that allow coexistence in competing strains under vaccination. We

consider two viral strains and temporary cross-immunity with one vector mosquito

population. Results suggest that vaccination scenarios will not only reduce disease

incidence but will also modify the transmission dynamics. Indeed, vaccination and

cross immunity period are seen to decrease the frequency and magnitude of outbreaks

but in a diﬀerentiated manner with speciﬁc eﬀects depending upon the interaction

vaccine and strain type.

1 Introduction

Dengue is a vector-borne disease with more than 50 million cases per year [18]. The

major vector, Aedes aegypti, is located in tropical regions, mainly in urban areas that

provide water holding containers that function as breeding sites. There are four dengue

serotypes (DEN-1, DEN-2, DEN-3 and DEN-4) that coexist in many endemic areas [23].

Dengue is an emergent infectious disease that can be very severe. Dengue Hemorrhagic

fever (DHF) is a life-threatening condition whose development is not well known [7]. One

of the main hypothesis that have been put forward to explain it is Antibody Dependent

Enhancement (ADE) whereby previous exposure to a Dengue infection may generate a

1

To appear in Mathematical Biosciences

DOI:10.1016/j.mbs.2016.10.001

very strong immune response on a secondary infection, thus triggering DHF [21]. In recent

years, the development of dengue vaccines has dramatically accelerated [16], [26] given the

frequent epidemics and morbidity and DHF mortality rates around the world. Vaccination

is a cost-eﬀective measure of control and prevention but its development is challenged by

the existence of the four viral serotypes, the possibility of ADE and therefore of DHF [24].

Previous mathematical models have incorporated the eﬀect of immunological interac-

tions between the diﬀerent dengue serotypes in disease dynamics. Infection with a par-

ticular serotype is believed to result in life-long immunity to that serotype and temporal

cross-protection to the other serotypes. There exists many diﬀerent models on Dengue

population dynamics (e.g. [35], [6], [24], [17], [15], [1], [25]). In a recent paper Coudeville

and Garnett [11], propose a compartmental, age structured model with four serotypes

that incorporates cross protection and the introduction of a vaccine. Likewise Rodriguez-

Barraquer et al. [29], use an age-stratiﬁed dengue transmission model to assess the impact

of partially eﬀective vaccines through a tetravalent vaccine with a protective eﬀect against

only 3 of the 4 serotypes. Other compartmental and agent-based models [10] have found

that vaccines with eﬃcacies of 70 −90% against all serotypes have the potential to signiﬁ-

cantly reduce the frequency and magnitude of epidemics on a short to medium term.

Many of the published mathematical models include the four dengue serotypes (e.g.

[19], [17], [25]) and deal with the full complexity of the population dynamics that this

diversity triggers. In this paper the potential impact of a vaccine is studied through the

use of a mathematical model of transmission for two dengue serotypes. In the Americas,

Dengue has a typical pattern of presenting a dominant serotype while the others circulate

at low densities and in very localized regions of the continent [13]. Dengue epidemics come

sequentially thus reducing the basic population dynamics to the competition between two

viral strains: the invading and the resident. This is the justiﬁcation of the model that

we study in this paper. On the other hand the introduction of vaccination is founded in

the imminent release of a vaccine that has the characteristic of having high eﬃcacy for

only three of the four serotypes [7]. In our setting, the vaccination programs that we study

consider the application of one or two doses in the presence of cross protection. In our model

the vaccine is assumed to confer higher protection to one serotype than to the second one.

The paper is organized as follows. In section 2 we present a mathematical model for Dengue

and the incorporation of the vaccine. In section 3 we explain the vaccination strategies. In

sections 4 and 5, we present and discuss the numerical results of the vaccination scenarios

with diﬀerent cross immunity periods. In section 6 we present a statistics summary. Finally,

in section 7 we draw some conclusions about this work.

2

2 Mathematical model

A basic model for Dengue

In this section we describe the mathematical model for dengue transmission in the presence

of vaccination and two co-circulating strains. All human newborns are susceptible to both

dengue strains.

Figure 1: Basic model without vaccination. Ssusceptible, Ciinfectious in latent stage,

Iiinfected contagious, Eitemporary cross immunity, Tisusceptibles to strain jalready

recovered from strain i,Ziinfectious with secondary infection, Yiinfectious and contagious

with a secondary infection, Rimmune to both strains.

The model that we present considers a human host population classiﬁed in compartments

according to Dengue infection status. We consider the population of individuals that are

all fully susceptible to both strains of Dengue. At time t= 0 a few infected individuals are

introduced and infection process is then triggered.

We call primary infections to those infections that occur in individuals with no previous

exposure to either strain; we call secondary infections to those infections that occur in in-

dividuals that have been previously exposed to one of the two strains. Let Srepresents the

3

susceptible individuals, Ci, Zithe individuals in the latent period of primary or secondary

infections for each strain, i= 1,2, respectively. Likewise, Ii, Yiare individuals with pri-

mary and secondary infections for each of the two strains respectively. Eiare individuals

in the state of temporary cross-immunity (temporary immune protection to both strains

independent of the strain causing the immediate previous infection), respectively, Tiare

susceptible population to dengue strain j(j6=i). Note that Tiindividuals have already

recovered from and infection by dengue strain i.

Infection with one serotype has been shown to provide lifelong immunity to that

serotype but short-term cross-protection to the other serotypes [8, 18]. Rrepresents the

immune population to both infections (see Fig. 1).

2.1 Incorporating the vaccine

As mentioned in the introduction Dengue is a major global public health problem aﬀecting

Asian and Latin America countries. The development of prevention and control measures

that focus on epidemiological surveillance and vector control is thus a priority.

Once the vaccine is released and applied to a target population, the proportion of the

vaccinated population (coverage) is expected to reach 89% of the 2–5 year old class and

69% in the 2–15 year old class after 5 years since the start of the application (these cov-

erage rates correspond to those achieved in Thailand using a combination of catch-up and

routine vaccination) [28]. The recommended target age according with SAGE committee

(World Health Organization) depends on the seroprevalence of the population, 9 years old

if seroprevalence is 90% in that age group; 11 to 14 years old if seroprevalence at 9 years

old is less than 90% but above 50% [33].

Currently, the vaccine Dengvaxia (CYD-TDV) by Sanoﬁ Pasteur has been approved in

Indonesia and now available in Mexico for vaccination of individuals of 9 to 45 years old

(See [36]). The vaccine produced by Sanoﬁ-Pasteur protects against serotypes 1, 3 and 4

but only imperfectly against serotype 2 [30].

We propose a vaccination model that consists of the application of the vaccine in three

strategic proﬁles: one dose vaccine application to all new recruits into the susceptible class

(S), one dose after a waiting time of six months to all susceptible individuals including

those who recovered from a previous infection by either strain (S, Ti, i = 1,2) and ﬁnally

the application of both doses.

In our model the vaccine is applied under the following assumptions (see Fig. 2).

1. Vaccination coverage. A fraction pof naive susceptible individuals is vaccinated with

a bivalent vaccine. we consider a vaccine coverage p= 0.8 [4].

4

2. Incomplete protection. A proportion p1of vaccinated individuals is susceptible to

dengue 1 and a proportion p2of vaccinated individuals is susceptible to dengue 2.

3. All susceptible individuals (S, T1, T2) are vaccinated with a dose after a waiting time

of 1/ψ days.

4. Vaccinated but unsuccessfully protected individuals can be infected and eventually

pass to the fully immune compartment R.

With the previous hypothesis, we set up the following vaccination scenarios labelled as

W, D1, D2, F :

W: No vaccine application

D1: One dose vaccine application. A proportion of p= 0.8 of all then naive susceptible

population is vaccinated. Of these vaccinated individuals, proportions pi, i = 1,2

remain susceptible (hypothesis 2).

D2: One dose vaccine application. A vaccine dose is applied after a waiting time of six

months to all susceptible individuals including those recovered from a ﬁrst infection

(S, T1, T2). Of these vaccinated individuals, proportions pi, i = 1,2 remain susceptible

(hypothesis 2).

F: Two doses vaccine application: a dose application D1with a coverage of p= 0.8 and

a dose application with a delay of six months D2(hypothesis 3). In this scenario, D2

is also applied to people vaccinated with D1.

The evaluation of the vaccination scenarios is done through simulations that incorporate

heterogeneity in serotype transmission rates. Age-stratiﬁed seroprevalence studies suggests

that the average transmission intensity and reproductive number of DENV-2 is higher than

that of other serotypes [17], [29].

The parameters p1and p2are the proportions of vaccinated individuals that fail to be

protected against serotypes 1 and 2, respectively. Therefore, 1 −p1and 1 −p2represent

the protection conferred by the vaccine.

The vaccine schedule is shown in ﬁgure 2.

5

S

I

RD1

RD2

Figure 2: Basic model with vaccination. D1and D2doses are applied sequentially; unvac-

cinated individuals follow a natural route of infection (see Figure 3). The compartment

Irepresents all infections. The dotted lines represent infections due to failure of the ﬁrst

and second vaccine application. The dashed lines represents the application of the second

dose to all susceptible individuals (including those who recovered from a ﬁrst infection).

The mathematical model with vaccination is the following

d

dt S=µ(1 −p)N−(B1+B2)S−(ψ+µ)S

d

dt RD1=µpN −(p1B1+p2B2)RD1−(ψ+µ)RD1

d

dt RD2=ψ(RD1+S) + ψ(T1+T2)−(p1B1+p2B2)RD2−µRD2

d

dt C1=B1S−(φ1+µ)C1

d

dt C2=B2S−(φ2+µ)C2

d

dt I1=φ1C1−(µ+γ1)I1

d

dt I2=φ2C2−(µ+γ2)I2

d

dt E1=γ1I1−(η1+µ)E1

d

dt E2=γ2I2−(η2+µ)E2

d

dt T1=η1E1−(σ2B2+ψ+µ)T1

d

dt T2=η2E2−(σ1B1+ψ+µ)T2

d

dt Z1=σ1B1T2−(φ1+µ)Z1+p1B1RD1+p1B1RD2

d

dt Z2=σ2B2T1−(φ2+µ)Z2+p2B2RD1+p2B2RD2

d

dt Y1=φ1Z1−(γ1+µ+m1)Y1

d

dt Y2=φ2Z2−(γ2+µ+m2)Y2

d

dt R=γ1Y1+γ2Y2−µR

(1)

6

Figure 3: Complete diagram for the model with vaccination. The compartments RD1and

RD2indicate vaccinated populations with dose D1and dose D2, respectively. The diagram

shows the transitions between states. Black lines represent natural infections. Solid red

lines indicate the transitions from those individuals eventually vaccinated. Dashed red

lines represent transitions due to failure in protection against either or both strains which

results on an inﬂow to the latent stage in secondary infections (Zi, i = 1,2). The labels

correspond to the rates in the system 1.

The total human population is given by

N=S+RD1+RD2+C1+C2+I1+I2+E1+E2+T1+T2+Z1+Z2+Y1+Y2+R

The dynamics of the vector are given by

d

dt V0=q(t)−(A1+A2)V0−δV0

d

dt V1=A1V0−δV1

d

dt V2=A2V0−δV2

(2)

with

q(t) = q0(b+kcos (2πt/365))

to represent yearly seasonal forcing. V0represents the susceptible mosquitoes, and V1,V2

the number of mosquitoes infected with strains 1 and 2 respectively. A1,A2represent the

forces of infection for strains 1 and 2 in the mosquitoes, respectively. δis the mosquito

death rate. The total mosquito population is M=V0+V1+V2(see Table 1 for other

parameter deﬁnitions and values).

The forces of infection follow the proportional mixing assumption and are given by

7

Ai=αi(Ii+Yi)

Nand Bi=βiVi

M

Parameter Description Chosen values

1/µ Life expectancy for humans 70 years (25550 days)

1/φiIncubation period 4 to 7 days

1/γiDuration of disease (infectiousness) 7 to 15 days

1/δ Life expectancy of mosquitoes 14 to 21 days

1/ηiDuration of cross immunity 180 to 270 days

σiReinfection rate undetermined

αiEﬀective contact rate human-mosquito undetermined

βiEﬀective contact rate mosquito-human undetermined

miRate death of the disease undetermined

qRecruitment rate of mosquito population undetermined

pFirst dose vaccine coverage undetermined

p1Failure probability of vaccine for protection against strain 1 undetermined

p2Failure probability of vaccine for protection against strain 2 undetermined

1/ψ Period of time for application of the second dose undetermined

Table 1: Deﬁnitions and ranges of the main parameters in mathematical model with vac-

cination [11], [35].

In our model once a mosquito is infected it never recovers and it cannot be reinfected

with a diﬀerent strain of virus. Secondary infections occur only in the host.

8

2.2 The basic reproduction number

The basic reproduction number is deﬁned as the number of secondary infections that a

single infectious individual produces in a population where all host are susceptible. R0is a

threshold parameter for the model, such that if R0<1 then the Disease Free Equilibrium

is locally asymptotically stable and the disease cannot invade the population (eventually

the infection dies out), but if R0>1, then the Disease Free Equilibrium is unstable and

invasion is possible.

Applying the next generation matrix methodology [12], we obtain the basic reproduc-

tion number (without vaccination) 1:

R0=max{R01, R02}=max (sβ1α1φ1

δ(γ1+µ)(φ1+µ),sβ2α2φ2

δ(γ2+µ)(φ2+µ))

where βi/δ represent the number of eﬀective contacts mosquito-to-human during the life

time of mosquito, αi/(µ+γi) the number of eﬀective contact human-to-mosquito during

the infectious period of human and φi/(µ+φi) represents the fraction of the time that

humans spend in the incubation period of the disease.

When vaccination is introduced the vaccination reproduction number is:

Rv

0=max{Rv

01, Rv

02}=max (sβ1α1φ1(1 −p(1 −p1))

δ(γ1+µ)(φ1+µ),sβ2α2φ2(1 −p(1 −p2))

δ(γ2+µ)(φ2+µ))

where p(1 −pi) is the eﬀective coverage against each serotype. Therefore 1 −p(1 −pi) is

the proportion of susceptible individuals to serotype iafter vaccination.

On another hand, since the model (1) undergoes time-dependent vector population

size, we consider the eﬀective reproduction number to take into account the proportion of

infections generated during successive periods of time. We use the eﬀective reproduction

number approach proposed by Nold (1979) who deﬁned Rtusing the mean generation time

(see [27]):

Re

01(t, µ) = T ot1[t, t +µ]/T ot1[t−µ, t] (3)

Re

02(t, µ) = T ot2[t, t +µ]/T ot2[t−µ, t]

Where T oti=Ii+Yi, i = 1,2 account for total infections by each strain and µis the

mean generation time.

1See Appendix for details

9

In subsection 5.5 we present the numerical simulations of the eﬀective reproduction

numbers where the mean generation time (see [14] for further deﬁnitions) is 15 days ac-

cording to estimations in [2].

3 Scenarios for the dengue vaccination model

The model described in section 2 will be used to study the impact of vaccination strate-

gies for diﬀerent eﬃcacy values (pi, i = 1,2), transmission intensity (βi, i = 1,2) and cross

immunity periods (1/ηi, i = 1,2). As explained previously, we consider a population of

individuals that are all fully susceptible to both strains of Dengue. At time t= 0 a few

infected individuals are introduced. The infection process is triggered and a vaccine pro-

gram is applied.

We study the long term dynamics under the following vaccination scenarios:

•Without vaccination (W)

•One dose application only (D1)

•One dose application only with a delay of six months (D2)

•Application of both doses (F)

Dose (D1) is applied to all individuals entering the fully naive susceptible compart-

ment, a second dose (D2) is applied to susceptible individuals of all types (S, T1, T2) after

a waiting time of 1/ψ days. Both doses are applied, D1is applied to individuals entering

the naive susceptible compartment Sand D2is applied after 1/ψ days to all susceptible

individuals S, T1, T2and also to those individuals vaccinated with the dose D1.

We remark that in our model the vaccine has lower eﬃcacy against the serotype with

the highest transmission intensity. Thus, p1< p2and β1< β2.

For all scenarios the reproductive number for each serotype assumes R02 > R01. Like-

wise, the eﬃcacy of the vaccine against serotype 2 is lower than that for serotype 1. This

implies that there is a higher probability of infection from serotype 2 than from serotype

1.

Table 3 shows the baseline parameter values for all simulations.

For the simulations we have chosen cross immunity periods (180 and 270 days) of both

serotypes based on the reported information in [34], [11]. To study the eﬀect of this cross

immunity periods on the asymptotic dynamics, we show four scenarios for each one of

10

Parameter W D1D2F

p0 0.8 0 0.8

p10 0.3 0.3 0.3

p20 0.4 0.4 0.4

ψ0 0 1/(0.5×365) 1/(0.5×365)

Table 2: Parameter values for the diﬀerent vaccination scenarios

Parameter Chosen values Parameter Chosen values

1/µ 70 years (25550 days) 1/φ14 days

1/φ27 days 1/γ18 days

1/γ210 days 1/δ 15 days

1/η1180, 270 days 1/η2180, 270 days

σ10.5 σ20.5

α10.2 α20.2

β10.5 β21.0

m10m20

Table 3: Parameters of the population dynamics of dengue [3].

the cross immunity periods of 180-270 days and 270-270 days. As previously stated, the

vaccine scenarios are W, D1, D2and F.

The numerical simulations were obtained using Python. The results are shown after

running a transient period of 137 years. After the transient, we assessed the impact of

vaccination on the incidence of both serotypes along 50 years.

In the numerical results we show the dynamics corresponding to primary Iiand sec-

ondary infections Yiof strains i= 1,2. Recall that Sis the compartment of susceptible

individuals without previous infection, Ticorresponds to susceptible individuals recovered

from infection by strain iand prone to acquire dengue strain j(with j6=i) and Eiindi-

viduals in the state of temporary cross-immunity after infection by strain i.

In the following sections we use the term strong strain to designate strain 2 which has

the highest reproductive number.

11

4 Vaccination scenarios with cross immunity periods of 180

and 270 days for each strain

4.1 Without Vaccine Scenario W

In Figure 4 we see in primary infections that both strain outbreaks exhibit desyncrhonized

behaviour. The frequency of the outbreaks by the strong strain (I2) is considerably higher

(one order of magnitude) than those of the other.

In secondary infections, although the outbreaks Y2reduce their frequency, the propor-

tion of infected individuals Y1increases about ten times compared to its proportion in

primary infections. Besides, the highest peaks of T2(susceptibles to strain 1 only) trigger

the outbreaks Y1.

Figure 4: Vaccine Scenario W. Numerical results for primary (Ii) and secondary (Yi)

infections, individuals in the temporal cross immunity state (Ei) and susceptibles (S, Ti, i =

1,2). Parameter values: p= 0, pi= 0, ψ = 0

12

4.2 One dose Vaccine Scenario D1

In primary and secondary infections there is only one outbreak by the weak strain I1, while

the proportion of infections I2decreases after the highest peak around 10 years and reaches

more regular oscillatory pattern after 41 years.

On the long term, the eﬀect of the ﬁrst vaccine application is the prevention of outbreaks

by the weak strain and the appearance of yearly outbreaks in primary and secondary

infections by the strong strain (I2, Y2). In this scenario the vaccine protects eﬀectively

against strain 1 but it fails to protect against the strong strain, allowing yearly outbreaks.

Figure 5: Vaccine Scenario D1. Numerical results for primary (Ii) and secondary (Yi)

infections, individuals in the temporal cross immunity state (Ei), susceptibles (S, Ti), i=

1,2 and vaccinated individuals (RD1). Parameter values: p= 0.8, p1= 0.3, p2= 0.4,

ψ= 0.

13

4.3 One dose Vaccine Scenario D2

In this scenario, only the dose (without the application of the ﬁrst dose) with a delay of 6

months is applied.

In primary infections there is only one negligible outbreak by the weak strain I1, while

the proportion of infections by the strong strain I2reaches regular yearly outbreaks.

The vaccine eﬀect has the following characteristics: ﬁrst, it diminishes completely the

outbreaks by strain 1 in both levels of infection, but it fails to prevent outbreaks by the

strong strain. Second, unlike the scenario D1, scenario D2reduces to almost zero the pool

of susceptible individuals (Ti, i = 1,2) prone to acquire infection by both strains strain.

However, in this scenario the vaccine also fails to protect against the strong strain, allowing

yearly outbreaks Y2.

Figure 6: Vaccine Scenario D2. Numerical results for primary (Ii) and secondary (Yi)

infections, individuals in the temporal cross immunity state (Ei), susceptibles (S, Ti), i=

1,2 and vaccinated individuals (RD2). Parameter values: p= 0, p1= 0.3, p2= 0.4,

ψ= 1/(0.5×365).

14

4.4 Both Vaccine doses Scenario F

In the scenario where both doses of the vaccine are applied, the long term eﬀect is the

prevention of outbreaks of primary and secondary infections by the weak strain (I1, Y1),

while the proportion of primary infections by the strong strain I2reaches regular oscilla-

tions after about 41 years. Compared to scenarios D1and D2, in scenario F, the vaccine

delays the occurrences of the outbreaks (I2, Y2) for the ﬁrst 25 years.

On the other hand, in this scenario, the vaccine fails to prevent outbreaks by the strong

strain (Y2) despite the negligible pool of susceptibles to acquire either strain as a secondary

infection (Ti, i = 1,2).

Figure 7: Vaccine Scenario F. Numerical results for primary (Ii) and secondary (Yi)

infections, individuals in the temporal cross immunity state (Ei), susceptibles (S, Ti), and

vaccinated individuals (RDi, i = 1,2). Parameter values: p= 0.8, p1= 0.3, p2= 0.4,

ψ= 1/(0.5×365).

15

5 Vaccination scenarios with cross immunity periods of 270

days for both infections

In this section the simulation results, where the cross immunity periods are 270 days for

both the weak and the strong serotypes [34], are presented for the previous four vaccine

scenarios (W, D1, D2, F )

5.1 Without Vaccine Scenario W

As before, we present scenario Was a baseline for the other cases.

In this scenario, the eﬀect of considering temporal cross immunity of 270 days for both

strains results in desynchronized dynamics and less frequent outbreaks than those occur-

ring when the cross immunity periods are 180 −270 days.

In primary infections there are four large outbreaks by the weak strain I1while the

outbreaks of the strong one I2occur with higher frequency. In secondary infections there

are also four outbreaks of the weak strain Y1of considerably higher proportion than of

those by the strong strain Y2. For this cross immunity periods, unlike the 180 −270 days

of cross protection case, the proportions of primary and secondary infections are about the

same order of magnitude.

Also, the outbreaks in secondary infections occur when the highest pool of recovered

from primary infections (T1, T2) are reached. The highest proportion of susceptibles is that

of T2, which promotes higher outbreaks by strain 1 in secondary infections (Y1).

16

Figure 8: Vaccine Scenario W. Numerical results for primary (Ii) and secondary (Yi)

infections, individuals in the temporal cross immunity state (Ei) and susceptibles (S, Ti, i =

1,2). Parameter values: p= 0, pi= 0, ψ = 0.

5.2 One dose Vaccine Scenario D1

In the scenario where the ﬁrst dose of the vaccine is applied, the long term eﬀect on the

disease is, on one hand, the prevention of outbreaks by the the weak strain in primary infec-

tions with only one large outbreak about 6 years after vaccine implementation. Whereas,

infections by the strong strain tends to reach a regular oscillatory pattern after about 27

years. This eﬀect is seen in both levels of infection (I2, Y2).

On the other hand, the vaccine eﬀectively protects against strain 1 but it fails in

protection against the strong strain. Thus, in scenario D1, despite the increment of the

pool of susceptibles (T2) to acquire strain 1, the outbreaks by this strain (Y1) are prevented,

unlike the yearly outbreaks occurrence by strain 2 (Y2) despite of the negligible pool of

susceptibles (T1).

17

Figure 9: Vaccine Scenario D1. Numerical results for primary (Ii) and secondary (Yi)

infections, individuals in the temporal cross immunity state (Ei), susceptibles (S, Ti, i =

1,2), and vaccinated individuals (RD1). Parameter values: p= 0.8, p1= 0.3, p2= 0.4,

ψ= 0.

5.3 One dose Vaccine Scenario D2

In this scenario the vaccine dose is applied with a delay of 6 months. In this case, the long

term eﬀect is the prevention of primary and secondary outbreaks I1, Y1, while in both levels

of infection I2, Y2tend to yearly cyclic outbreaks after about 11 years of the vaccination

program is implemented.

On one side, the scenario D2leads to a faster regularization of the dynamics of infections

by the strong strain and also diminishes the susceptible pool Ti, i = 1,2 compared to

scenario D1.

In contrast, the delay in the dose application undergoes a failing in protection against

the strong strain. Thus, there are yearly outbreaks by strain 2 although the pool of

susceptibles (T1, T2) tends to zero in the ﬁrst 5 years of the vaccination campaign.

18

Figure 10: Vaccine Scenario D2. Numerical results for primary (Ii) and secondary (Yi)

infections, individuals in the temporal cross immunity state (Ei), susceptibles (S, Ti, i =

1,2), and vaccinated individuals (RD2). Parameter values: p= 0, p1= 0.3, p2= 0.4,

ψ= 1/(0.5×365).

5.4 Both Vaccine doses Scenario F

In the scenario where both doses of the vaccine are applied, the long term eﬀect is the

prevention of outbreaks by the weak strain (I1,Y1). In contrast, the eﬀect on secondary

infections by the strong strain is the regularization of its dynamics in primary infection pro-

ducing yearly outbreaks about 25 years. This strategy has a better eﬀect on the reduction

of the proportions of primary than in secondary infections.

Besides, in secondary infections, the both vaccine doses application fails to protect

against the strong strain since there are yearly outbreaks by strain 2 although the pool of

susceptibles (T1, T2) tends to zero in the very ﬁrst years.

19

Figure 11: Vaccine Scenario F. Numerical results for primary (Ii) and secondary (Yi)

infections, individuals in the temporal cross immunity state (Ei, i = 1,2), susceptibles

(S, Ti), and vaccinated individuals (RDi, i = 1,2). Parameter values: p= 0.8, p1= 0.3,

p2= 0.4, ψ = 1/(0.5×365).

Finally, in Figure 12 we present the available pool of susceptible individuals in each

vaccination scenario.

20

Figure 12: Susceptible population under vaccination scenarios. Numerical results for sus-

ceptible individuals for diﬀerent cross immunity scenarios. Ssusceptible naive individuals

(susceptible to both strains), SD1susceptible pool of individuals after the application of

the ﬁrst dose, SD2susceptible pool of individuals left after the application of one dose after

6 months in the susceptible stage. SFsusceptible pool of individuals after the application

of both doses. Horizontal axes is in days; vertical axes is the proportion of the population.

5.5 Eﬀective Reproduction Numbers

We present the numerical simulations for the eﬀective reproduction numbers for cross im-

munity periods of 9 months for both strains since this case is representative of the regular

behaviour that the application of the vaccine induces in each of the scenarios (D1, D2, F ).

The vaccine regularizes the outbreaks after about a 400 weeks transient.

As an approach, we use the deﬁnition of eﬀective reproduction number given in (3),

section (2.2):

Re

0i(t, µ) = T oti[t, t +µ]/T oti[t−µ, t]i= 1,2

It is noteworthy that the eﬀective reproductive numbers which accounts for infections

occurred in periods of µ= 15 days shows regular cyclic peaks (new infections) even after

de application of vaccine. The vaccine then reduces prevalence /incidence in general and

eliminates outbreaks by the weaker strain but in a very regular fashion as can be appreciated

from the dynamics of the eﬀective reproduction number regardless of the chosen strategy

(D1,D2or F. See ﬁgure 13).

21

Figure 13: Eﬀective reproduction numbers: Re

0i(t, µ), i = 1,2for 9−9months of cross

immunity periods. In the horizontal axis is indicated the number of periods of length

µ= 15 days. Top left: Without vaccination (W). Top right: One dose vaccine (D1).

Bottom left: One dose vaccine with a delay of 6 months. Bottom right: Both doses (F).

6 Summary statistics

In this section we present summary statistics based in our numerical simulations.

Means were obtained every six months over a period of 50 years for the variables: total

infections by each serotype (T oti=Ii+Yi, i = 1,2), total infections (T otal =T ot1+T ot2)

and susceptible individuals. The vaccine proﬁles and cross immunity periods are as in the

previous sections.

6.1 Total infections

An ANOVA was performed for the means of total infections by each strain T oti, i = 1,2,

total infections (T=T ot1+T ot2) and Susceptibles along 50 years using as factors: the

vaccine scenarios (W, D1, D2, F ) denoted as proﬁles and the cross immunity periods de-

noted as crossimm.periods 2.

The reduction of the mean of serotype 1 total infections (T ot1) is the result of either the

application of only the secondary dose or the application of both vaccine doses (Top Figures

14). While for the reduction of the mean of total infections by serotype 2 (T ot2), only the

primary dose application is necessary (Top right Figure 14). Primary dose application

2In this section, cross immunity periods of 180 −180,180 −270,270 −180,270 −270 days for both

serotypes are labelled as 6 −6,6−9,9−6,9−9 months.

22

reduces the six month means of total infections (Bottom left Figure 14).

0 5 10 15 20 25

Tot1=I1+Y1

crossimm.periods

mean of cases per 100,000

66 69 96 99

profile

W

D1

D2

F

35 40 45 50

Tot2=I2+Y2

crossimm.periods

mean of cases per 100,000

66 69 96 99

profile

D2

F

W

D1

40 45 50 55 60 65

Total=Tot1+Tot2

crossimm.periods

mean of cases per 100,000

66 69 96 99

profile

W

D2

F

D1

500 1500 2500

Susceptibles

crossimm.periods

mean of cases per 100,000

66 69 96 99

profile

W

D1

D2

F

Figure 14: Interaction plots of the eﬀect of vaccine proﬁles and cross immunity periods

on the means every six month along 50 years. Top left: Total infections by serotype 1.

Top right: Total infections by serotype 2. Bottom left: Total infections. Bottom right:

Susceptibles. Only the eﬀect of vaccine proﬁle on the reduction of means of T oti, i = 1,2

and of those of Susceptibles is statistically signiﬁcant (p−value < 0.05). Both factors,

taken independently, are statistically signiﬁcant in the reduction of total infections (T otal =

T ot1+T ot2).

23

6.2 Primary and secondary infections

The six month means over a 50 years period were computed and an ANOVA performed.

In this case, the eﬀect of the vaccine proﬁle (as one the factors) is statistically signiﬁcant

for both primary and secondary infections by each serotype.

This reduction of the mean for both primary and secondary infections by serotype 1

results from the application of either only the secondary dose or both vaccine doses (Top

Figures 14). In contrast, primary infections by serotype 2 is reduced only when both

vaccine doses are applied. Note that the reduction of the mean of secondary infections by

serotype 2 is achieved by the primary dose alone.

02468

Primary infections I1

crossimm.periods

mean of cases per 100,000

66 69 96 99

profile

W

D1

D2

F

0 5 10 15 20

Secondary infections I1

crossimm.periods

mean of cases per 100,000

66 69 96 99

profile

W

D1

D2

F

5 10 15 20 25

Primary infections I2

crossimm.periods

mean of cases per 100,000

66 69 96 99

profile

W

D2

D1

F

10 20 30 40

Secondary infections I2

crossimm.periods

mean of cases per 100,000

66 69 96 99

profile

D2

F

D1

W

Figure 15: Interaction plots of the eﬀect of vaccine proﬁles and cross immunity periods on

the means every six month along 50 years. Top left: Primary infections by serotype 1. Top

right: Secondary infections by serotype 1. Bottom left: Primary infections by serotype 2.

Bottom right: Secondary infections by serotype 2.

24

A summary of these results are shown in Figures (16) and (17).

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WD1D2F

0 50 150 250

Means of I1 for 9,9 m

Figure 16: Eﬀect of vaccine proﬁles and cross immunity periods on the means of total

infections by serotype 1. These means were obtained every six month along 50 years (cases

per 100,000) for four pairs of cross immunity periods. Top left: six months for both

serotypes. Top right: Six and nine months for serotype 1 and 2. Bottom left: Nine and six

months for serotype 1 and 2. Bottom right: Nine months for both serotypes.

25

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Figure 17: Eﬀect of vaccine proﬁles and cross immunity periods on the means of total

infections by serotype 2. These means were obtained every six month along 50 years (cases

per 100,000) for four pairs of cross immunity periods. Top left: six months for both

serotypes. Top right: Six and nine months for serotype 1 and 2. Bottom left: Nine and six

months for serotype 1 and 2. Bottom right: Nine months for both serotypes.

7 Conclusions

We have numerically explored the asymptotic and dynamical behaviour of a two-strain

Dengue model under the application of a vaccine.

The model incorporates heterogeneity regarding transmission of both strains and ef-

ﬁcacy of the vaccine against each one. In particular, the vaccine is assumed to have a

lower eﬃcacy against the serotype with the highest transmission intensity (strain 2 in the

model). This assumption implies that a large numbers of hosts might be well protected

against the weaker serotype (strain 1) but not against the stronger serotype (due to its

higher transmission rate).

26

In contrast to Coudeville and Garnett [11], we compare the eﬀect of each dose applica-

tion assuming a ﬁxed coverage of 80%. The target population is composed of susceptible

individuals to which one out of three possible vaccination scenarios is applied. These are:

a one dose vaccine application (D1) at t= 0 to individuals entering to the susceptible com-

partment; a one dose vaccine application with a delay of six months (D2) to all susceptible

individuals (S, Ti, i = 1,2); and the application of both doses of the vaccine (F). Each

vaccine proﬁle is applied taking two combinations of cross immunity periods: 180-270 and

270-270 days for each strain respectively.

In the baseline scenario W, the pool of susceptibles (T1, T2) remaining after a primary

infections directly drives the size and frequency of outbreaks in secondary infections.

In scenario D1, the vaccine eﬀectively prevents outbreaks by the weak strain. Whereas,

in scenarios D2and F, the vaccine reduces the pool of susceptibles to acquire a secondary

infection by either strain but fails to prevent outbreaks by the strong strain in secondary

infections, allowing yearly outbreaks (Y2).

The statistical analysis also indicates that the application of the ﬁrst vaccine dose con-

siderably reduces (around 85%) the average incidence of strain 1 infections, whereas it only

reduces around 9% the mean incidence by strain 2 for the two cross immunity combinations.

The other vaccine proﬁles, although eﬀective against strain 1, lead to an increase in

the mean incidence of secondary infections by strain 2. These cases could present clinically

riskier secondary infections. In general the overall eﬀect of the single vaccine application

after 6 months (D2) in the susceptible class (S, Ti, i = 1,2) or the application of both

doses (F), is the prevention of outbreaks by the weak strain together with the stabilization

of recurrent outbreaks by the stronger strain. Thus, both vaccination proﬁles although

considerably reduce the pool of susceptibles also produce increments in the proportion of

secondary infections by the strong serotype [28].

Based on our results, the period of cross-immunity plays a crucial role in each of the

scenarios. For the scenario without vaccination, with the longest period of cross-immunity

for both strains, the frequency of the outbreaks decreases.

Moreover, despite an increase in secondary infections by serotype 2 for the single vaccine

application after 6 months in the susceptible class or the application of both doses to all

individuals, the largest overall reduction in incidence of both strains occurs when the cross

immunity period is 270 days for the strong strain. And, with equal cross immunity periods

(270 days) the yearly outbreaks appear faster of the outbreaks by strain 2 compared to the

other cases.

On the long term, the three vaccination strategies seem to reduce the proportion of

primary infections by both strains, Fis the most favourable scenario since it also reduces

27

the pool of susceptibles to acquire a secondary infections. This reduction is statistically

signiﬁcant in the means of proportions of total infections by each serotype (T oti=Ii+Yi).

In all vaccine scenarios, the vaccine induces periodic yearly outbreaks of the strong strain.

Acknowledgements

This work was conducted as a part of the grant PAPIIT (UNAM) IA101215; support from

LAISLA-UNAM project is also acknowledged. N.L.G-M acknowledges the support from a

CONACYT doctoral fellowship.

28

A Basic Reproduction Number R0

The basic reproduction number is deﬁned as the number of secondary infections that a

single infectious individual produces in a population where all hosts are susceptible. It

provides an invasion criterion for the initial spread of the virus in a susceptible population.

A.1 Reproduction Number without vaccination

The set bounded by the total host and vector population

Ω = {(S, C1, C2, I1, I2, E1, E2, T1, T2, Z1, Z2, Y1, Y2, R, V0, V1, V2) :

S+C1+C2+I1+I2+E1+E2+T1+T2+Z1+Z2+Y1+Y2+R≤N,

V0+V1+V2≤M}

the disease-free equilibrium is given by E∗

0= (S∗,0,0,0,0,0,0,0,0,0,0,0,0,0, V ∗

0,0,0).

According to the notation of P. van den Driessche and Watmough [12], we calculate the

matrices Fand V−1evaluated in E∗

0.

F=

0 0 0 0 0 0 0 0 δβ1

M0

0 0 0 0 0 0 0 0 0 δβ2

M

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 T2β1σ1

M0

0 0 0 0 0 0 0 0 0 T2β1σ1

M

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 V0α1

N0 0 0 V0α1

N0 0 0

0 0 0 V0α2

N0 0 0 V0α2

N0 0

,

29

V−1=

1

µ+φ10 0 0 0 0 0 0 0 0

01

µ+φ20 0 0 0 0 0 0 0

φ1

(µ+γ1)(µ+φ1)01

µ+γ10 0 0 0 0 0 0

0φ2

(µ+γ2)(µ+φ2)01

µ+γ20 0 0 0 0 0

0 0 0 0 1

µ+φ10 0 0 0 0

0 0 0 0 0 1

µ+φ20 0 0 0

0 0 0 0 φ1

(µ+γ1)(µ+φ1)01

µ+γ10 0 0

0 0 0 0 0 φ2

(µ+γ2)(µ+φ2)01

µ+γ20 0

0 0 0 0 0 0 0 0 1

δ0

0 0 0 0 0 0 0 0 0 1

δ

By construction F V −1is the next-generation matrix and set R0=ρ(F V −1) where ρ

denotes the spectral radius of a matrix.

F V −1=

00000000Sβ1

Mδ 0

000000000Sβ2

Mδ

0000000000

0000000000

00000000T2β1σ1

Mδ 0

000000000T2β2σ2

Mδ

0000000000

0000000000

V0α1Φ1

N(µ+γ1)0V0α1

N(µ+γ1)0V0α1Φ1

N(µ+γ1)0V0α1

N(µ+γ1)0 0 0

0V0α2Φ2

N(µ+γ2)0V0α2

N(µ+γ2)0V0α2Φ2

N(µ+γ2)0V0α2

N(µ+γ2)0 0

where Φ1=φ1

µ+φ1, Φ2=φ2

µ+φ2,S=N,V0=q

δand thus, the basic reproduction number is

R0=max{R01, R02}=max (sβ1α1φ1

δ(γ1+µ)(φ1+µ),sβ2α2φ2

δ(γ2+µ)(φ2+µ))

This expression is a generalization of the Ross-Macdonald basic reproductive number to

the case of two strains, frequency-dependent contact rates and variable population size in

both host and vector.

30

A.2 Reproduction Number with vaccination

The set bounded by the total host and vector population with vaccination

Ω = {(S, Rp, Rs, C1, C2, I1, I2, E1, E2, T1, T2, Z1, Z2, Y1, Y2, R, V0, V1, V2) :

S+Rp+Rs+C1+C2+I1+I2+E1+E2+T1+T2+Z1+Z2+Y1+Y2+R≤N,

V0+V1+V2≤M}

The disease-free equilibrium is given by E∗

0= (S∗, R∗

p,0,0,0,0,0,0,0,0,0,0,0,0,0,0, V ∗

0,0,0).

F=

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 S β1

M0

0 0 0 0 0 0 0 0 0 0 0 Sβ2

M

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 p1Rpβ1

M0

0 0 0 0 0 0 0 0 0 0 0 p2Rpβ2

M

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 V0α1

N0 0 0 V0α1

N0 0 0

0 0 0 0 0 V0α2

N0 0 0 V0α2

N0 0

,

V−1=

1

µ+ψ0 0 0 0 0 0 0 0 0 −p1Rpβ1

V0δ(µ+ψ)−p2Rpβ2

V0δ(µ+ψ)

ψ

µ2+µψ

1

µ0 0 0 0 0 0 0 0 −p1Rpψβ1

V0δµ(µ+ψ)−p2Rpψβ2

V0δµ(µ+ψ)

0 0 1

µ+φ10 0 0 0 0 0 0 0 0

0 0 0 1

µ+φ20 0 0 0 0 0 0 0

0 0 φ1

(µ+γ1)(µ+φ1)01

µ+γ10 0 0 0 0 0 0

0 0 0 φ2

(µ+γ2)(µ+φ2)01

µ+γ20 0 0 0 0 0

0 0 0 0 0 0 1

µ+φ10 0 0 0 0

0 0 0 0 0 0 0 1

µ+φ20 0 0 0

0 0 0 0 0 0 φ1

(µ+γ1)(µ+φ1)01

µ+γ10 0 0

0 0 0 0 0 0 0 φ2

(µ+γ2)(µ+φ2)01

µ+γ20 0

0 0 0 0 0 0 0 0 0 0 1

δ0

0 0 0 0 0 0 0 0 0 0 0 1

δ

31

the next generation matrix is given by

F V −1=

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 Sβ1

q0

0 0 0 0 0 0 0 0 0 0 0 Sβ2

q

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 p1Rpβ1

q0

0 0 0 0 0 0 0 0 0 0 0 p2Rpβ2

q

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 qα1φ1

δN (µ+γ1)(µ+φ1)0 0 qα1

δN (µ+γ1)0qα1φ1

δN (µ+γ1)(µ+φ1)0qα1

δN (µ+γ1)0 0

0 0 0 q α2φ2

δN (µ+γ2)(µ+φ2)0qα2

δN (µ+γ2)0qα2φ2

δN (µ+γ2)(µ+φ2)0qα2

δN (µ+γ2)0 0

where S=µ(1−p)N

µ+ψ,Rp=µpN

µ+ψ, and thus, the basic reproduction number with vaccination

Rv

0=max{Rv

01, Rv

02}=max (sβ1α1φ1(1 −p(1 −p1))

δ(γ1+µ)(φ1+µ),sβ2α2φ2(1 −p(1 −p2))

δ(γ2+µ)(φ2+µ))

32

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