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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 outbreaksbut in a differentiated manner with specific effects depending upon the interaction vaccine and strain type.
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Transmission Dynamics of Two Dengue Serotypes with
vaccination scenarios
N.L. Gonz´alez Morales1, M. N´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 1days.
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)RD2µRD2
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+γ2Y2µR
(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)V0δV0
d
dt V1=A1V0δV1
d
dt V2=A2V0δV2
(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
1Life expectancy for humans 70 years (25550 days)
1iIncubation period 4 to 7 days
1iDuration of disease (infectiousness) 7 to 15 days
1Life expectancy of mosquitoes 14 to 21 days
1iDuration 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
1Period 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 βirepresent 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[tµ, t] (3)
Re
02(t, µ) = T ot2[t, t +µ]/T ot2[tµ, 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 (1i, 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 1days. Both doses are applied, D1is applied to individuals entering
the naive susceptible compartment Sand D2is applied after 1days 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
170 years (25550 days) 114 days
127 days 118 days
1210 days 115 days
11180, 270 days 12180, 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[tµ, 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 99months 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 6,69,96,99 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 (pvalue < 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).
WD1D2F
0 50 100 150 200
Means of I1 for 6,6 m
cases per 100,000
● ●
WD1D2F
0 100 300 500
Means of I1 for 6,9 m
WD1D2F
0 50 100 150 200
Means of I1 for 9,6 m
cases per 100,000
● ●● ●
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
WD1D2F
0 100 200 300 400
Means of I2 for 6,6 m
cases per 100,000
WD1D2F
0 100 200 300 400
Means of I2 for 6,9 m
WD1D2F
0 100 200 300 400
Means of I2 for 9,6 m
cases per 100,000
WD1D2F
0 100 300 500
Means of I2 for 9,9 m
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+RN,
V0+V1+V2M}
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 V1evaluated 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
V1=
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=
000000001
0
0000000002
0000000000
0000000000
00000000T2β1σ1
0
000000000T2β2σ2
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+RN,
V0+V1+V2M}
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 2
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
,
V1=
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 1
q0
0 0 0 0 0 0 0 0 0 0 0 2
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 1φ1
δN (µ+γ1)(µ+φ1)0 0 1
δN (µ+γ1)01φ1
δN (µ+γ1)(µ+φ1)01
δN (µ+γ1)0 0
0 0 0 q α2φ2
δN (µ+γ2)(µ+φ2)02
δN (µ+γ2)02φ2
δN (µ+γ2)(µ+φ2)02
δN (µ+γ2)0 0
where S=µ(1p)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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36
... Lately, dengue vaccine development has dramatically accelerated as a result of the increase in the number of dengue infections, just as the prevalence of everyone of the circulating DENV-1,2,3,4 serotypes [12]. The recently licensed dengue vaccine, Dengvaxia (CYD-TDV) made by Sanofi Pasteur has been approved by regulatory authorities in more than twenty countries [11], including Indonesia and Mexico [13]. The vaccine protects against DENV-1, DENV-3 and DENV-4 but only imperfectly against DENV-2 [13]. ...
... The recently licensed dengue vaccine, Dengvaxia (CYD-TDV) made by Sanofi Pasteur has been approved by regulatory authorities in more than twenty countries [11], including Indonesia and Mexico [13]. The vaccine protects against DENV-1, DENV-3 and DENV-4 but only imperfectly against DENV-2 [13]. Thus, it may or may not be possible to have any future perfect dengue vaccine that protect against the four virus serotypes [14]. ...
... Furthermore, we also have λ * h = 0 when I * v = 0. Thus, the solutions in (13) reduce to ...
Article
Full-text available
Dengue is a mosquito-borne disease which has continued to be a public health issue in Malaysia. This paper investigates the impact of singular use of vaccination and its combined effort with treatment and adulticide controls on the population dynamics of dengue in Johor, Malaysia. In a first step, a compartmental model capturing vaccination compartment with mass random vaccination distribution process is appropriately formulated. The model with or without imperfect vaccination exhibits backward bifurcation phenomenon. Using the available data and facts from the 2012 dengue outbreak in Johor, basic reproduction number for the outbreak is estimated. Sensitivity analysis is performed to investigate how the model parameters influence dengue disease transmission and spread in a population. In a second step, a new deterministic model incorporating vaccination as a control parameter of distinct constant rates with the efforts of treatment and adulticide controls is developed. Numerical simulations are carried out to evaluate the impact of the three control measures by implementing several control strategies. It is observed that the transmission of dengue can be curtailed using any of the control strategies analysed in this work. Efficiency analysis further reveals that a strategy that combines vaccination, treatment and adulticide controls is most efficient for dengue prevention and control in Johor, Malaysia.
... Mobility and Dengue have received varied and intense attention in the last decade. Dengue moves with the human population along roads and highways in many parts of the world and, being transmitted by Aedes aegypti a highly adapted vector to urban environments [29,30,31,32], prevention and control constitute a major aim in many countries [33,34,35,36,37,12,38]. ...
... For Dengue, modeling has been applied to delucidate transmission as well as used as a tool to help in its control and prevention [38,43,44,45], but there is still much to do. In recent years, spreading processes on complex networks, like computer viruses, epidemics in human populations, rumors or information in social networks, have been modeled and described. ...
Article
Highlights: • We address the migration of the human population and its effect on pathogen reinfection. • We use a Markov-chain SIS metapopulation model over a network. • The contact rate is based on the infected hosts and the incidence of their neighboring locations. • We estimate from Dengue data in Mexico the dynamics of migration incorporating climate variability. Abstract: Most of the recent epidemic outbreaks in the world have as a trigger , a strong migratory component as has been evident in the recent Covid-19 pandemic. In this work we address the problem of migration of human populations and its effect on pathogen reinfections in the case of Dengue, using a Markov-chain susceptible-infected-susceptible (SIS) metapopulation model over a network. Our model postulates a general contact rate that represents a local measure of several factors: the population size of infected hosts that arrive at a given location as a function of total population size, the current incidence at neighboring locations, and the connectivity of the network where the disease spreads. This parameter can be interpreted as an indicator of outbreak risk at a given location. This parameter is tied to the fraction of individuals that move across boundaries (migration). To illustrate our model capabilities, we estimate from epidemic Dengue data in Mexico the dynamics of migration at a regional scale incorporating climate variability represented by an index based on precipitation data.
... Mobility and Dengue have received varied and intense attention in the last decade. Dengue moves with the human population along roads and highways in many parts of the world and, being transmitted by Aedes aegypti a highly adapted vector to urban environments [29,30,31,32], prevention and control constitute a major aim in many countries [33,34,35,36,37,12,38]. ...
... For Dengue, modeling has been applied to delucidate transmission as well as used as a tool to help in its control and prevention [38,43,44,45], but there is still much to do. In recent years, spreading processes on complex networks, like computer viruses, epidemics in human populations, rumors or information in social networks, have been modeled and described. ...
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Most of the recent epidemic outbreaks in the world have as a trigger, a strong migratory component as has been evident in the recent Covid-19 pandemic. In this work we address the problem of migration of human populations and its effect on pathogen reinfections in the case of Dengue, using a Markov-chain susceptible-infected-susceptible (SIS) metapopulation model over a network. Our model postulates a general contact rate that represents a local measure of several factors: the population size of infected hosts that arrive at a given location as a function of total population size, the current incidence at neighboring locations, and the connectivity of the network where the disease spreads. This parameter can be interpreted as an indicator of outbreak risk at a given location. This parameter is tied to the fraction of individuals that move across boundaries (migration). To illustrate our model capabilities, we estimate from epidemic Dengue data in Mexico the dynamics of migration at a regional scale incorporating climate variability represented by an index based on precipitation data.
... This leads to an upsurge in research on dengue virology, pathogenesis, and immunology and in development of antivirals and vaccines. Especially, mathematical models are one useful tool to investigate the cause of epidemic and to suggest the best way to control and prevent dengue [9,[16][17][18][19][20][21][22][23][24][25][26][27]. Kooi et al. [22] studied an asymmetric two-strain dengue model for predicting characteristic dynamic behavior and chaos occurring for smaller parameter regimes. ...
... Woodall et al. [16] presented a new modeling framework based on SIR model with enhancement to study cross enhancement between dengue serotypes which may be influencing the epidemic oscillations. González Morales et al. [23] studied the asymptotic and dynamical behavior of a two-strain dengue model under the application of a vaccine and indicated that 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 the strain type. Meanwhile, Anggriani et al. [26] studied the effect of reinfection with the same serotype on dengue transmission dynamics by developing a multi-strain dengue mathematical model, which suggested that reinfection with the same serotype may be one of the underlying factors causing an increase in the number of secondary infections. ...
Article
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Dengue fever is a common disease which can cause shock, internal bleeding, and death in patients if a second infection is involved. In this paper, a multi-serotype dengue model with nonlinear incidence rate is formulated to study the transmission of two dengue serotypes. The dynamical behaviors of the proposed model depend on the threshold value R0n known as the reproductive number which depends on the associated reproductive numbers with serotype-1 and serotype-2. The value of R0n is used to reflect whether the disease dies out or becomes endemic. It is found that the proposed model has a globally stable disease-free equilibrium if R0n≤1, which indicates that if public health measures that make (and keep) the threshold to a value less than unity are carried out, the strategy in disease control is effective in the sense that the number of infected human and mosquito populations in the community will be brought to zero irrespective of the initial sizes of sub-populations. When R0n>1, the endemic equilibria called the co-existence primary and secondary infection equilibria are locally asymptotically stable. The effects of cross immunity and nonlinear incidence rate are explored using data from Thailand to determine the effective strategy in controlling and preventing dengue transmission and reinfection.
... Using epidemiological data from Brazil, the authors have estimated the basic reproduction number and the optimal vaccination age for each of the four dengue serotypes, showing that the optimal vaccination age varies depending on the serotypes in circulation. A mathematical model with two virus strains, vector mosquito population and TCI was proposed by González Morale et al. [81] to investigate the effects of various vaccination strategies based on vaccine efficacy values, transmission intensity, and cross-immunity period. Results have shown that the period of cross-immunity plays a crucial role for disease incidence reduction and overall disease transmission dynamics. ...
Article
Mathematical models have a long history in epidemiological research, and as the COVID-19 pandemic progressed, research on mathematical modeling became imperative and very influential to understand the epidemiological dynamics of disease spreading. Mathematical models describing dengue fever epidemiological dynamics are found back from 1970. Dengue fever is a viral mosquito-borne infection caused by four antigenically related but distinct serotypes (DENV-1 to DENV-4). With 2.5 billion people at risk of acquiring the infection, it is a major international public health concern. Although most of the cases are asymptomatic or mild, the disease immunological response is complex, with severe disease linked to the antibody-dependent enhancement (ADE) - a disease augmentation phenomenon where pre-existing antibodies to previous dengue infection do not neutralize but rather enhance the new infection. Here, we present a 10-year systematic review on mathematical models for dengue fever epidemiology. Specifically, we review multi-strain frameworks describing host-to-host and vector-host transmission models and within-host models describing viral replication and the respective immune response. Following a detailed literature search in standard scientific databases, different mathematical models in terms of their scope, analytical approach and structural form, including model validation and parameter estimation using empirical data, are described and analysed. Aiming to identify a consensus on infectious diseases modeling aspects that can contribute to public health authorities for disease control, we revise the current understanding of epidemiological and immunological factors influencing the transmission dynamics of dengue. This review provide insights on general features to be considered to model aspects of real-world public health problems, such as the current epidemiological scenario we are living in.
... They also showed that the introduction of age structure may change the dynamics of the corresponding model without age structure. Additionally, recent studies [3,15] pointed out cross immunity starts immediately after the primary infectious period and prevents individuals from becoming infected by another strain for a period ranging from 6 months to 9 months, even to lifelong. To the best of our knowledge, there is currently no work on the effect of cross immunity age on the dynamics of dengue fever model. ...
Article
Dengue fever is a typical mosquito-borne infectious disease, and four strains of it are currently found. Clinical medical research has shown that the infected person can provide life-long immunity against the strain after recovering from infection with one strain, but only provide partial and temporary immunity against other strains. On the basis of the complexity of transmission and the diversity of pathogens, in this paper, a multi-strain dengue transmission model with latency age and cross immunity age is proposed. We discuss the well-posedness of this model and give the terms of the basic reproduction number R0 = max{R1, R2 }, where Ri is the basic reproduction number of strain i (i = 1, 2). Particularly, we obtain that the model always has a unique disease-free equilibrium P0 which is locally stable for R0 < 1. And same time, an explicit condition of the global asymptotic stability of P0 is obtained by constructing a suitable Lyapunov functional. Furthermore, we also shown that if Ri > 1, the strain-i dominant equilibrium Pi is locally stable for Rj < R∗i (i, j = 1, 2, i ̸= j). Additionally, the threshold criteria on the uniformly persistence, the existence and global asymptotically stability of coexistence equilibrium are also obtained. Finally, these theoretical results and interesting conclusions are illustrated with some numerical simulations.
... Some used a compartmental model to study vector-host interactions [11,13,14]. There are other studies that focused on dengue disease transmission and control [7,8,[15][16][17][18][19]. Also, a one-strain mathematical model has been used to study the influence of seasonality effect on the transmission dynamics of dengue disease [6,20]. ...
Article
Dengue is a mosquito-borne disease which is endemic, particularly in the tropical and subtropical regions across the globe. Most of these regions have strong seasonal patterns in climatic factors such as rainfall and temperature, which are directly linked to dengue disease transmission through the mosquito population. These climatic factors have great influence on the mosquito survival, propagation and abundance. Several mathematical models have been used to forecast dengue burden in Madeira Island if two dengue virus serotypes coexist, but do not capture the seasonality effects on the dynamics of mosquito population. Hence, this study proposes a two-strain compartmental model to forecast the impact of seasonal variation on the transmission dynamics of dengue disease if two virus serotypes coexist in the Island. We derive the basic reproduction number, 0 = max{√ 01 ,√ 0 j }, related to the model through the Next Generation Matrix operator. The diseasefree and boundary equilibrium points of the model are obtained, and we discuss the local and global stability of the disease-free equilibrium in terms of 0 . It is found that the disease free-equilibrium is locally asymptotically stable whenever both 01 , 0 j < 1, and unstable otherwise. The Comparison Theorem is used to prove the global asymptotic stability of the disease-free equilibrium. The results of our numerical simulation show that the presence of seasonal effect influences a high number of dengue infections in both human and mosquito populations.
... For Ae. aegypti, the mathematical models have proven to be useful tools to understand dengue transmission [14,36] and, recently, chikungunya [37] and Zika [38], as well as to help in planning control strategies [39]. Among the different mathematical approaches to study infectious diseases through vectors, as with the Mayaro virus, the SEIR-type (susceptible, exposed, infected and recovered) epidemiological models [40,41] and in metapopulations, have been used widely [42][43][44][45][46]. Nonetheless, none of them considers the passive vector transport (dispersion process of invasive species associated with human activities) or its dynamic populations, under different biogeographical conditions; therefore, an option is to include a meta-population model in differential equations, where each local population includes a structured model on age for the dynamic population of Ae. aegypti and an SEI/SEIR-type epidemiological model for the population of humans, as well as the passive transport of the mosquito through land cargo, which, has also been used recently to study the dynamics of Rift Valley fever transmission in the human population, with Ae. aegypti as the main vector [47]. ...
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Background: Mayaro virus (Togaviridae) is an endemic arbovirus of the Americas with epidemiological similarities with the agents of other more prominent diseases such as dengue (Flaviviridae), Zika (Flaviviridae), and chikungunya (Togaviridae). It is naturally transmitted in a sylvatic/rural cycle by Haemagogus spp., but, potentially, it could be incorporated and transmitted in an urban cycle by Aedes aegypti, a vector widely disseminated in the Americas. Methods: The Mayaro arbovirus dynamics was simulated mathematically in the colombian population in the eight biogeographical provinces, bearing in mind the vector's population movement between provinces through passive transport via truck cargo. The parameters involved in the virus epidemiological dynamics, as well as the vital rates of Ae. aegypti in each of the biogeographical provinces were obtained from the literature. These data were included in a meta-population model in differential equations, represented by a model structured by age for the dynamic population of Ae. aegypti combined with an epidemiological SEI/SEIR-type model. In addition, the model was incorporated with a term of migration to represent the connectivity between the biogeographical provinces. Results: The vital rates and the development cycle of Ae. aegypti varied between provinces, having greater biological potential between 23 °C and 28 °C in provinces of Imerí, biogeographical Chocó, and Magdalena, with respect to the North-Andean Moorland (9.33-21.38 °C). Magdalena and Maracaibo had the highest flow of land cargo. The results of the simulations indicate that Magdalena, Imerí, and biogeographical Chocó would be the most affected regarding the number of cases of people infected by Mayaro virus over time. Conclusions: The temperature in each of the provinces influences the local population dynamics of Ae. aegypti and passive migration via transport of land cargo plays an important role on how the Mayaro virus would be disseminated in the human population. Once this arbovirus begins an urban cycle, the most-affected departments would be Antioquia, Santander, Norte de Santander, Cesar (Provinces of Magdalena), and Valle del Cauca, and Chocó (biogeographical province of Chocó), which is why vector control programmes must aim their efforts at these departments and include some type of vector control to the transport of land cargo to avoid a future Mayaro epidemic.
... This finding is in agreement with a statement by WHO in [5] . A compartmental deterministic model capturing the coexistence of two virus serotypes has been formulated to assess various vaccination scenarios in [30,31] . ...
Article
This paper presents a two-strain compartmental dengue model with variable humans and mosquitoes populations sizes. The model incorporates two control measures: Dengvaxia vaccine and insecticide (adulticide) to forecast the transmission and effective control strategy for dengue in Madeira Island if there is a new outbreak with a different virus serotype after the first outbreak in 2012. The basic reproduction number, R0=max{R01,R0j}, associated with the model is computed using the next generation matrix operator. The disease-free equilibrium is found to be locally asymptotically stable when both R01,R0j<1, but unstable otherwise. The global asymptotic stability of the model is derived using the comparison theorem. Sensitivity analysis is carried out on the model parameters. The results of the analysis show that mosquito biting and death rates are the most sensitive parameters. Three strategies: the use of Dengvaxia vaccine only, the use of adulticide only, and the combination of Dengvaxia vaccine and adulticide, are considered for the control implementation under two scenarios (less and more aggressive cases). The numerical results show that a strategy which is based on Dengvaxia vaccine and adulticide is the most effective strategy for controlling dengue disease transmission in both scenarios among the considered strategies.
Article
The study examines the population-level impact of temperature variability and immigration on malaria prevalence in Nigeria, using a novel deterministic model. The model incorporates disease transmission by immigrants into the community. In the absence of immigration, the model is shown to exhibit the phenomenon of backward bifurcation. The disease-free equilibrium of the autonomous version of the model was found to be locally asymptotically stable in the absence of infective immigrants. However, the model exhibits an endemic equilibrium point when the immigration parameter is greater than zero. The endemic equilibrium point is seen to be globally asymptotically stable in the absence of disease-induced mortality. Uncertainty and sensitivity analysis of the model, using parameter values and ranges relevant to malaria transmission dynamics in Nigeria, shows that the top three parameters that drive malaria prevalence (with respect to [Formula: see text]) are the mosquito natural death rate ([Formula: see text]), mosquito biting rate ([Formula: see text]) and the transmission rates between humans and mosquitoes ([Formula: see text]). Numerical simulations of the model show that in Nigeria, malaria burden increases with increasing mean monthly temperature in the range of 22–28[Formula: see text]. Thus, this study suggests that control strategies for malaria should be intensified during this period. It is further shown that the proportion of infective immigrants has marginal effect on the transmission dynamics of the disease. Therefore, the simulations suggest that a reduction in the fraction of infective immigrants, either exposed or infectious, would significantly reduce the malaria incidence in a population.
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Background For decades, human infections with Zika virus (ZIKV), a mosquito-transmitted flavivirus, were sporadic, associated with mild disease, and went underreported since symptoms were similar to other acute febrile diseases endemic in the same regions. Recent reports of severe disease associated with ZIKV, including Guillain-Barré syndrome and severe fetal abnormalities, have greatly heightened awareness. Given its recent history of rapid spread in immune naïve populations, it is anticipated that ZIKV will continue to spread in the Americas and globally in regions where competent Aedes mosquito vectors are found. Globally, dengue virus (DENV) is the most common mosquito-transmitted human flavivirus and is both well-established and the source of outbreaks in areas of recent ZIKV introduction. DENV and ZIKV are closely related, resulting in substantial antigenic overlap. Through a mechanism known as antibody-dependent enhancement (ADE), anti-DENV antibodies can enhance the infectivity of DENV for certain classes of immune cells, causing increased viral production that correlates with severe disease outcomes. Similarly, ZIKV has been shown to undergo ADE in response to antibodies generated by other flaviviruses. However, response to DENV antibodies has not yet been investigated. Methodology / Principal Findings We tested the neutralizing and enhancing potential of well-characterized broadly neutralizing human anti-DENV monoclonal antibodies (HMAbs) and human DENV immune sera against ZIKV using neutralization and ADE assays. We show that anti-DENV HMAbs, cross-react, do not neutralize, and greatly enhance ZIKV infection in vitro . DENV immune sera had varying degrees of neutralization against ZIKV and similarly enhanced ZIKV infection. Conclusions / Significance Our results suggest that pre-existing DENV immunity will enhance ZIKV infection in vivo and may increase disease severity. A clear understanding of the interplay between ZIKV and DENV will be critical in informing public health responses in regions where these viruses co-circulate and will be particularly valuable for ZIKV and DENV vaccine design and implementation strategies. Author Summary Recent reports of severe disease, including developmental problems in newborns, have greatly heightened public health awareness of Zika virus (ZIKV), a mosquito-transmitted virus for which there is no vaccine or treatment. It is anticipated that ZIKV will continue to spread in the Americas and globally in regions where competent mosquitoes are found. Dengue virus (DENV), a closely related mosquito-transmitted virus is well-established in regions of recent ZIKV introduction and spread. It is increasingly common that individuals living in these regions may have had a prior DENV infection or may be infected with DENV and ZIKV at the same time. However, very little is known about the impact of DENV infections on ZIKV disease severity. In this study, we tested the ability of antibodies against DENV to prevent or enhance ZIKV infection in cell culture-based assays. We found that DENV antibodies can greatly enhance ZIKV infection in cells. Our results suggest that ZIKV infection in individuals that had a prior DENV infection may experience more severe clinical manifestations. The results of this study provide a better understanding of the interplay between ZIKV and DENV infections that can serve to inform public health responses and vaccine strategies.
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Dengue has become the most rapidly expanding mosquito-borne infectious disease on the planet, surpassing malaria and infecting at least 390 million people per year. There is no effective treatment for dengue illness other than supportive care, especially for severe cases. Symptoms can be mild or life-threatening as in dengue hemorrhagic fever and dengue shock syndrome. Vector control has been only partially successful in decreasing dengue transmission. The potential use of safe and effective tetravalent dengue vaccines is an attractive addition to prevent disease or minimize the possibility of epidemics. There are currently no licensed dengue vaccines. This review summarizes the current status of all dengue vaccine candidates in clinical evaluation. Currently five candidate vaccines are in human clinical trials. One has completed two Phase III trials, two are in Phase II trials, and three are in Phase I testing. Copyright © 2015. Published by Elsevier Ltd.
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Quantifying the attack ratio of disease is key to epidemiological inference and Public Health planning. For multi-serotype pathogens, however, different levels of serotype-specific immunity make it difficult to assess the population at risk. In this paper we propose a Bayesian method for estimation of the attack ratio of an epidemic and the initial fraction of susceptibles using aggregated incidence data. We derive the probability distribution of the effective reproductive number, R t , and use MCMC to obtain posterior distributions of the parameters of a single-strain SIR transmission model with time-varying force of infection. Our method is showcased in a data set consisting of 18 years of dengue incidence in the city of Rio de Janeiro, Brazil. We demonstrate that it is possible to learn about the initial fraction of susceptibles and the attack ratio even in the absence of serotype specific data. On the other hand, the information provided by this approach is limited, stressing the need for detailed serological surveys to characterise the distribution of serotype-specific immunity in the population.
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Dengue virus has four serotypes and is endemic globally in tropical countries. Neither a specific treatment nor an approved vaccine is available, and correlates of protection are not established. The standard neutralization assay cannot differentiate between serotype-specific and serotype cross-reactive antibodies in patients early after infection, leading to an overestimation of the long-term serotype-specific protection of an antibody response. It is known that the cross-reactive response in patients is temporary but few studies have assessed kinetics and potential changes in serum antibody specificity over time. To better define the specificity of polyclonal antibodies during disease and after recovery, longitudinal samples from patients with primary or secondary DENV-2 infection were collected over a period of 1 year. We found that serotype cross-reactive antibodies peaked 3 weeks after infection and subsided within 1 year. Since secondary patients rapidly produced antibodies specific for the virus envelope (E) protein, an E-specific ELISA was superior compared to a virus particle-specific ELISA to identify patients with secondary infections. Dengue infection triggered a massive activation and mobilization of both naïve and memory B cells possibly from lymphoid organs into the blood, providing an explanation for the surge of circulating plasmablasts and the increase in cross-reactive E protein-specific antibodies.
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In anticipation of this changing landscape, the Mexican Federal Ministry of Health (FMoH), in partnership with the Carlos Slim Health Institute, undertook the development of a national strategy for the introduction of a dengue vaccine. This exercise aimed to establish evidence-based policy recommendations to enable the early adoption of a dengue vaccine in Mexico incorporating evidence-based innovative strategies and approaches. The resulting recommendations were presented to national public health authorities for their use in the decision-making process. The recommendations will be shared with other countries, with a goal of developing a regional strategy for the introduction and use of dengue vaccines.
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Dengue vaccine development efforts have focused on the development of tetravalent vaccines. However, a recent Phase IIb trial of a tetravalent vaccine indicates a protective effect against only 3 of the 4 serotypes. While vaccines effective against a subset of serotypes may reduce morbidity and mortality, particular profiles could result in an increased number of cases due to immune enhancement and other peculiarities of dengue epidemiology. Here, we use a compartmental transmission model to assess the impact of partially effective vaccines in a hyperendemic Thai population. Crucially, we evaluate the effects that certain serotype heterogeneities may have in the presence of mass-vaccination campaigns. In the majority of scenarios explored, partially effective vaccines lead to 50% or greater reductions in the number of cases. This is true even of vaccines that we would not expect to proceed to licensure due to poor or incomplete immune responses. Our results show that a partially effective vaccine can have significant impacts on serotype distribution and mean age of cases.
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Dengue virus has traditionally caused substantial morbidity and mortality among children less than 15 years of age in Southeast Asia. Over the last 2 decades, a significant increase in the mean age of cases has been reported, and a once pediatric disease now causes substantial burden among the adult population. An age-stratified serological study (n = 1,736) was conducted in 2010 among schoolchildren in the Mueang Rayong district of Thailand, where a similar study had been conducted in 1980/1981. Serotype-specific forces of infection (λ(t)) and basic reproductive numbers (R0) of dengue were estimated for the periods 1969–1980 and 1993–2010. Despite a significant increase in the age at exposure and a decrease in λ(t) from 0.038/year to 0.019/year, R0 changed only from 3.3 to 3.2. Significant heterogeneity was observed across subdistricts and schools, with R0 ranging between 1.7 and 6.8. These findings are consistent with the idea that the observed age shift might be a consequence of the demographic transition in Thailand. Changes in critical vaccination fractions, estimated by using R0, have not accompanied the increase in age at exposure. These results have implications for dengue control interventions because multiple countries in Southeast Asia are undergoing similar demographic transitions. It is likely that dengue will never again be a disease exclusively of children.
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A precise definition of the basic reproduction number, R o , is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if R o < 1, then the disease free equilibrium is locally asymptotically stable; whereas if R o > 1, then it is unstable. Thus, R o is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super-and sub-threshold endemic equilibria for R o near one. This criterion, together with the definition of R o , is illustrated by treatment, multigroup, staged progression, multistrain and vector-host models and can be applied to more complex models. The results are significant for disease control.
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Dengue virus (DENV) is a significant cause of morbidity and mortality in tropical and subtropical regions, causing hundreds of millions of infections each year. Infections range from asymptomatic to a self-limited febrile illness, dengue fever (DF), to the life-threatening dengue hemorrhagic fever/dengue shock syndrome (DHF/DSS). The expanding of the habitat of DENV-transmitting mosquitoes has resulted in dramatic increases in the number of cases over the past 50 years, and recent outbreaks have occurred in the United States. Developing a dengue vaccine is a global health priority. DENV vaccine development is challenging due to the existence of four serotypes of the virus (DENV1-4), which a vaccine must protect against. Additionally, the adaptive immune response to DENV may be both protective and pathogenic upon subsequent infection, and the precise features of protective versus pathogenic immune responses to DENV are unknown, complicating vaccine development. Numerous vaccine candidates, including live attenuated, inactivated, recombinant subunit, DNA, and viral vectored vaccines, are in various stages of clinical development, from preclinical to phase 3. This review will discuss the adaptive immune response to DENV, dengue vaccine challenges, animal models used to test dengue vaccine candidates, and historical and current dengue vaccine approaches.