# The Optimal Age of Vaccination Against Dengue with an Age-Dependent Biting Rate with Application to Brazil

## Abstract

In this paper we introduce a single serotype transmission model, including an age-dependent mosquito biting rate, to find the optimal vaccination age against dengue in Brazil with Dengvaxia. The optimal vaccination age and minimal lifetime expected risk of hospitalisation are found by adapting a method due to Hethcote (Math Biosci 89:29–52). Any number and combination of the four dengue serotypes DENv1–4 is considered. Successful vaccination against a serotype corresponds to a silent infection. The effects of antibody-dependent enhancement (ADE) and permanent cross-immunity after two heterologous infections are studied. ADE is assumed to imply risk-free primary infections, while permanent cross-immunity implies risk-free tertiary and quaternary infections. Data from trials of Dengvaxia indicate vaccine efficacy to be age and serostatus dependent and vaccination of seronegative individuals to induce an increased risk of hospitalisation. Some of the scenarios are therefore reconsidered taking these findings into account. The optimal vaccination age is compared to that achievable under the current age restriction of the vaccine. If vaccination is not considered to induce risk, optimal vaccination ages are very low. The assumption of ADE generally leads to a higher optimal vaccination age in this case. For a single serotype vaccination is not recommended in the case of ADE. Permanent cross-immunity results in a slightly lower optimal vaccination age. If vaccination induces a risk, the optimal vaccination ages are much higher, particularly for permanent cross-immunity. ADE has no effect on the optimal vaccination age when permanent cross-immunity is considered; otherwise, it leads to a slight increase in optimal vaccination age.

## Keywords

Dengue Vaccination Optimal vaccination age Age-structured model Biting rate Hospitalisation## 1 Introduction

Dengue is considered the most important mosquito-borne viral disease of humans with half of the world’s population living in endemic areas and over 2 million dengue cases reported each year to the World Health Organization (2009, 2012). Due to the occurrence of asymptomatic infections and atypical clinical presentation dengue is in fact significantly under-reported so that the actual annual incidence is much higher (Gubler 2011); it has recently been estimated to be as high as 390 million cases of which approximately 100 million are symptomatic (Bhatt et al. 2013). These symptomatic cases of dengue fever (DF) are usually characterised by high fever accompanied by fatigue, rash, and headaches. If the disease manifests in one of its severe forms, i.e. dengue haemorrhagic fever (DHF) or dengue shock syndrome (DSS), symptoms can be plasma leakage and organ failure which can lead to death (Halstead 1980; Fredericks and Fernandez-Sesma 2014).

There are four distinct dengue virus serotypes DENv1–4 all of which are mainly spread by the *Aedes Aegypti* mosquito in Brazil and can cause any manifestation of dengue from an asymptomatic infection to severe dengue (SD). The coexistence of four serotypes entails the possibility of consecutive, heterologous infections which may be affected by interactions between serotypes and antibodies that were developed upon exposure to the different types. In fact, it is thought that a primary infection with any serotype leads to lifelong immunity specific to that type but protection against the other serotypes for a limited time only (Halstead 1980). Some studies have further shown that secondary infections cause 90–95% of cases of SD, with the remaining 5–10% being caused by primary infections, usually in infants between the ages of 6 and 12 months who have a low level of maternal antibodies (Leong et al. 2007; Halstead 2009; Jain and Chaturvedi 2010). Therefore, a consequence of the coexistence of several serotypes seems to be the enhancement of infection, particularly during secondary infections and during primary infections in infancy when maternal antibodies fall to low levels. This increase in infection severity is believed to be caused by a higher virulence which is in turn due to antibodies specific to the first serotype an individual was infected with or those passed on by the mother. These antibodies are cross-reactive with heterologous dengue types but non-neutralising and thus cause antibody-dependent enhancement (ADE) by binding on to the very similar dengue serotype and allowing the active virus entry into its target cells more easily (Halstead 2009; Jain and Chaturvedi 2010). Other observations regarding heterologous infections are that the sequence of serotypes with which individuals get infected influences the development of SD (Fried et al. 2010) and that two heterologous infections confer permanent cross-immunity (Gibbons et al. 2007; Anderson et al. 2013). Considering all of these complex interdependencies it is not surprising that instead of vaccines mainly vector control strategies were used to prevent the transmission of dengue in the past.

The development of a dengue vaccine was a complicated and lengthy process; however, in December 2015 after 20 years of development Sanofi Pasteur licensed Dengvaxia, the first vaccine against dengue (Sanofi Pasteur Press Release 2015). Since then it has been licensed for the use in individuals between the ages of either 9 and 45 or 9 and 60 years in more than ten countries including Brazil (Sanofi Pasteur Press Release 2016). Even before the licensure of Dengvaxia mathematical models had been used to predict the impact vaccination could have on the spread of dengue, and considering the complicated interdependencies like ADE and short-term cross-protection there is unsurprisingly some dispute about the effects of vaccination. While there is an overall agreement that vaccination could reduce DF cases significantly (Coudeville and Garnett 2012; Knipl and Moghadas 2015), there are indications that vaccination in the presence of ADE could lead to more SD cases (Knipl and Moghadas 2015). Ferguson et al. (2016) draw the conclusion that the transmission setting plays an important role in whether dengue vaccine will be beneficial or harmful by making the assumption that vaccination acts as a silent natural infection and by using a mathematical transmission model to show that in low-transmission settings vaccination may lead to more SD cases, whereas in high-transmission settings vaccination will be favourable both for the population as a whole and for the vaccinated individual. It can therefore be said that the phenomenon of ADE in dengue infections poses a great challenge for the development of vaccines since it makes it necessary to achieve a successful immune response to all four serotypes in order to prevent the creation of enhancing antibodies (Stephenson 2005). Dengvaxia, the only available vaccine at the moment, has been shown to be at least partially effective against all dengue serotypes in several Phase III trials; however, it has been found to have different efficacies for each of the serotypes and these efficacies further seem to depend on the age at vaccination and the serostatus of the vaccine recipient (Capeding et al. 2014; Hadinegoro et al. 2015). Since the licensing of Dengvaxia concerns about its application have been raised; in particular its use in seronegative recipients has been questioned since in this group an increase in the risk of acquiring SD in a subsequent natural infection has been observed (Aguiar et al. 2016; Halstead and Russell 2016; SAGE/World Health Organization 2016). This observation seems to be in agreement with the findings of Ferguson et al. (2016) pointing towards vaccination causing ADE in the first natural infection if seronegative individuals are vaccinated.

While recently much attention has been given to the possible effects of vaccination and to the optimisation of vaccination strategies in order to employ the most cost-effective strategy of vaccination or to achieve herd immunity and eradication of the disease (Billings et al. 2008; Durham et al. 2013) the age at which vaccination should ideally take place has rarely been considered. However, mathematical modelling has in the past been used to find optimal vaccination ages for other infectious diseases such as rubella and measles (Hethcote 1988; Anderson and May 1983) and we employ a method due to Hethcote (1988) to do the same for dengue. The existence of four dengue serotypes requires us to extend this method to take account of any number of serotypes existing in one endemic area, and we further take account of the survival probability of humans which Hethcote (1988) neglected. Additionally, while in many dengue transmission models the mosquito biting rate and the human mortality rate are assumed constant, we want to model more realistic transmission dynamics by assuming an age-dependent mosquito biting rate and a step death function which is more realistic for countries like Brazil. In this paper our objective is therefore to find optimal vaccination ages for dengue when the aim of vaccination is to reduce the risk of hospitalisation due to SD. While considering a single serotype transmission model to achieve this we still take account of the possible coexistence of multiple serotypes in an endemic area and the assumptions relating to their interactions by utilising a risk function to incorporate them into the model. Additionally, we investigate the optimal vaccination age when vaccination of seronegative individuals can have negative effects.

## 2 Vaccination Model

In order to find optimal vaccination ages for dengue we model the transmission of the virus in the presence of vaccination. To do this we assume independent transmission of the four distinct dengue serotypes which allows us to use a single serotype transmission model where some of the parameters are interchangeable to describe any one of the serotypes. Considering observations regarding interactions between the serotypes such as short-term cross-immunity and ADE this is only an approximation of the real dynamics. However, since these interactions are observed only in the short term the model can be considered to be a reasonable approximation. We further incorporate a three-dose vaccination strategy based on the restrictions under which Dengvaxia is licensed in Brazil by requiring a set of matching conditions to be met in addition to the initial conditions of the transmission model. These matching conditions include parameters that are serotype specific to allow for different vaccine efficacies for each serotype and at different vaccination ages.

### 2.1 Single Serotype Transmission Model

The Ross–MacDonald model is a common way to describe vector-borne infections such as dengue (Esteva and Vargas 1998; Garba et al. 2008), and the model we use is in fact of this type with the relevant modification of considering age density functions rather than total numbers for the human population.

We assume that humans potentially progress through four different stages in their life. Every human is born passively immune due to maternal antibodies, once these antibodies decline the individual becomes susceptible to the virus so that an adequate contact with the virus leads to infection with dengue from which an individual eventually recovers. Once humans recover they are immune to the serotype that they were infected with for the remainder of their life. By taking the loss of passive immunity to be given by an age-dependent function *C*(*a*) which is estimated for each of the four dengue serotypes individually using data on the decline of maternal antibodies given by van Panhuis et al. (2011) and by taking the death rate to be the same for both passively immune and susceptible individuals we can in fact consider one compartment of unaffected which comprises both the passively immune and the susceptible. Therefore, the age densities \(U_H(a,t)\), \(I_H(a,t)\), and \(R_H(a,t)\) of ‘unaffected’, ‘infected’, and ‘recovered’ humans at time *t* are modelled. The age densities for passively immune and susceptible individuals are then given by \((1-C(a))U_H(a,t)\) and \(C(a)U_H(a,t)\), respectively, and the age density for the entire human population is \(N_H(a,t) = U_H(a,t) + I_H(a,t) + R_H(a,t)\).

*L*. The age-dependent natural mortality rate \(\mu _H(a)\) and the survival probability \(\pi (a)\) for the human population are therefore given by

*q*(

*a*) depending on their age according to mosquito biting data (Massad 2015). Further if a susceptible human is bitten by an infectious mosquito, the probability of the human becoming infected with dengue is denoted by

*b*. Using these parameters and functions, and noting that the number of infectious mosquitoes is denoted by \(I_M(t)\), the force of infection for the human population is

*c*. We have already noted that an age-dependent biting rate

*q*(

*a*) is assumed and that the age density of infected humans is given by \(I_H(a,t)\) so that adequate contacts take place at a rate \(\int _0^{\infty }q(a)I_H(a,t)\frac{1}{N_H}\hbox {d}a\) and the force of infection in the mosquito population is

Description parameters and age-dependent rates used in the model

Parameter | Significance |
---|---|

| Total rate per unit time at which a single mosquito bites humans of age |

| Probability per bite that an initially susceptible human bitten by an infected mosquito becomes infected |

| Probability per bite that an initially susceptible mosquito biting an infected human becomes infected |

\(N_H\) | Total number of humans |

| Expected lifetime of humans in Brazil |

\(\mu _H(a)\) | Step death rate depending on |

\(\gamma _H\) | Per capita recovery rate of humans |

\(N_M\) | Total number of mosquitoes |

\(\mu _M\) | Natural per capita death rate of mosquitoes |

\(\tau \) | Incubation period in mosquitoes (the extrinsic incubation period) |

\(A_i\) (\(i=1,2,3\)) | Vaccination age for each of the three vaccination stages |

\(V_i\) (\(i=1,2,3\)) | Vaccinated proportion of the population for each vaccination age |

### 2.2 Vaccination Strategy

*C*(

*a*) as the seroconversion rate as was done by Hethcote (1988) and assuming a fraction \(V_i\) of the population is vaccinated at age \(A_i\) the probability of becoming immune due to vaccination at age \(A_i\) is given by \(V_iC(A_i)\). Consequently this means the probability of staying unaffected at age \(A_i\) is \(1-V_iC(A_i)\) so that there is a jump decrease in the age density of susceptible humans which leads to the matching condition

*i*. Note that the vaccinated fraction \(V_i\) can be utilised to incorporate the vaccine efficacy.

Combining the single serotype transmission model with the matching conditions obtained by including vaccination it is now possible to model each of the four dengue serotypes. The variation in the serotypes can be modelled by using different seroconversion rates *C*(*a*), as well as by substituting different values for the vaccine efficacy and therefore different fractions \(V_i\). Note that successful vaccination against a serotype is essentially a silent infection with that serotype.

## 3 Basic Reproduction Number \(R_0\)

*a*is given in years. With this biting rate, the survival probability \(\pi (a)\) as given in Eq. (2), and by setting \(C(a)\equiv 1\) as an approximation the double integrals in both expressions of \(R_0\) can be solved analytically. For Eqs. (9) and (10) one obtains

### 3.1 Serotype-Specific Basic Reproduction Numbers

Serotype-specific basic reproduction numbers

Serotype | \(R_0\) | Lower bound | Upper bound |
---|---|---|---|

DENv1 | 4.7045 | 1.2230 | 6.1777 |

DENv2 | 2.9942 | 1.3745 | 8.5133 |

DENv3 | 4.2974 | 1.4341 | 13.4129 |

DENv4 | 4.1864 | 1.8291 | 4.8711 |

## 4 Risk of Infection

Our goal is to apply the transmission model to find optimal vaccination ages. To do this we need to identify which consequences of an infection vaccination aim to minimise. This can be achieved by defining the lifetime expected risk of dengue, i.e. the total risk from infection during the lifetime of an individual by considering the expected risk from infection with any one serotype.

### 4.1 Expected Risk from Infection with Serotype *i* at Age *a*

*a*which is denoted by

*u*(

*a*) can therefore be obtained by taking the fraction of unaffected with respect to all humans of age

*a*, i.e. \(u(a) = U(a)/N(a)\), where \(N(a) = \frac{N_H}{L}\pi (a)\) is the equilibrium density with respect to age of the human population. By considering the matching conditions the fraction of successfully vaccinated individuals

*v*(

*a*) can be obtained as

*a*. We further obtain the fraction of unaffected who become infected on exposure to be

*C*(

*a*)

*u*(

*a*) and therefore the probability of infection at age

*a*as \(P(a)=\lambda (a)C(a)u(a)\).

*R*(

*a*) assigned to being infected at age

*a*that describes how undesirable an infection at that age is we can obtain the expected risk from an infection at age

*a*as

*a*might be a primary, secondary, tertiary, or quaternary infection depending on how many serotypes coexist in an endemic region and whether the infected individual was previously infected by or vaccinated against any of them. If the risk of each of these types of infection was the same the expected risk at age

*a*could be described by Eq. (16). However, since there is substantial evidence that secondary infections cause more severe infections and some indication that the sequence of serotypes plays an important role (Halstead 2009; Fried et al. 2010; Gibbons et al. 2007) the expected risk at age

*a*depends on how many and which serotypes an individual was previously infected with. Additionally it has been observed that the risk varies depending on the serostatus of a vaccine recipient (SAGE/World Health Organization 2016). We therefore need to find the probabilities of an infection with a given serotype being a primary, secondary, tertiary, or quaternary infection when previous exposure was caused by natural infection or vaccination. Note that vaccination is considered to be a silent infection. These probabilities are calculated as

*j*, \(j_*\), and \(\bar{j}\) indicate, respectively, a previous natural infection with serotype

*j*, successful vaccination against serotype

*j*before age

*a*, and no previous infection with or vaccination against serotype

*j*, that is, \(P_{ijk_*\bar{l}}(a)\) denotes the probability of an infection with serotype

*i*at age

*a*after a previous infection with serotype

*j*and successful vaccination against serotype

*k*but no exposure to serotype

*l*. By denoting the corresponding risk functions in a similar manner, e.g. \(R_{ijk_*\bar{l}}(a)\), the expected risk from an infection with serotype

*i*at age

*a*can then be calculated as

*R*(

*a*) describing the undesirability of acquiring dengue at age

*a*in years based on the need for hospital treatment can therefore be obtained by fitting a piecewise defined function to the data. Based on the data the function at young ages is assumed to be of the form \(k_1ae^{-k_2a}\) and at older ages it is an exponential function of the type \(l_1e^{l_2a}\). For ages above the highest recorded age the risk is taken to be constant. The pre-vaccine risk of hospitalisation is then given by:

*R*(

*a*). Similarly if there is no ADE and a secondary infection with a heterologous serotype confers permanent cross-immunity, we assume primary and secondary infections to have the same risk as before the introduction of the vaccine, and tertiary and quaternary infections to be risk-free, i.e. \(R_{ijkl}(a) = R_{ijk\bar{l}}(a) = R_{ijkl_*}(a) = R_{ijk_*l_*}(a) = R_{ij_*k_*l_*}(a) = R_{ijk_*\bar{l}}(a) = R_{ij_*k_*\bar{l}}(a) = 0\). However, results from the long-term follow-up of Dengvaxia trials show an increased risk of hospitalisation in seronegative vaccine recipients (SAGE/World Health Organization 2016; Martínez-Vega et al. 2017; Aguiar and Stollenwerk 2017). Based on these results the associated risk functions in the case of serostatus-dependent risk can be derived as outlined in Supplementary Appendix B.

### 4.2 Lifetime Expected Risk *E*

*E*as the integral over the sum of the expected risks of the different serotypes multiplied by the survival probability \(\pi (a)\) over all ages, i.e. we define

## 5 Steady-State Dynamics

In the previous section we introduced the lifetime expected risk of an infection with dengue. The definition of this risk is based on the assumption that a steady-state age distribution has been reached. We therefore need to find both the steady-state age distributions and the steady-state force of infection for the human population. At the steady state, the age distributions of the humans and the number of mosquitoes in each compartment are constant in time. The age distributions of unaffected, infected, and recovered humans are denoted by \(U(a) = \lim _{t \rightarrow \infty }U_H(a,t)\), \(I(a) = \lim _{t \rightarrow \infty }I_H(a,t)\), and \(R(a) = \lim _{t \rightarrow \infty }R_H(a,t)\), respectively. The steady-state force of infection is denoted by \(\lambda (a) = \lim _{t \rightarrow \infty }\lambda _H(a,t)\).

### 5.1 Steady-State Age Distribution

### 5.2 Steady-State Force of Infection

## 6 Results

We now want to find the optimal ages for vaccination against dengue with Dengvaxia for any combination of serotypes in an endemic region by numerically evaluating the lifetime expected risk derived in Sect. 4. This can be done by finding the serotype-specific forces of infection as given by Eq. (26) where it is important to note that since we assume independent transmission dynamics the combination of circulating serotypes does not influence the force of infection for each of the serotypes present.

Most of the parameters needed for the computation of the steady-state force of infections are already used in Sect. 3.1 to find the serotype-specific basic reproduction numbers, and the same values are used for the computation of the lifetime expected risk. The parameters that are still needed are the transmission probability from human to mosquito *c* and the fractions \(V_i\) vaccinated at the ages \(A_i\). For the probability of transmission we take \(c=1\) from the literature (Massad et al. 2010) for all serotypes. The vaccination ages \(A_i\) are such that the initial dose can be given at any age, while the second and third dose is given according to the licence of Dengvaxia, i.e. \(A_2 = A_1 + 6\) months and \(A_3 = A_1 + 12\) months. We do not restrict the initial age \(A_1\) to the age range of the licence to find the optimal vaccination age. However, at the end of this section we briefly compare the optimal age and lifetime expected risk to what can be achieved under the current restrictions of the licence in Brazil, i.e. if vaccination takes place between the ages of 9 and 45 years.

Vaccine efficacy | Age independent (%) | Under 9 years (%) | 9 years or older (%) |
---|---|---|---|

| |||

DENv1 | 54.7 | 46.6 | 58.4 |

DENv2 | 43.0 | 33.6 | 47.1 |

DENv3 | 71.6 | 62.1 | 73.6 |

DENv4 | 76.9 | 51.7 | 83.2 |

| |||

Seropositive | 78.2 | 70.1 | 81.9 |

Seronegative | 38.1 | 14.4 | 52.5 |

Optimal vaccination age \(A_1\) in months with the corresponding minimal lifetime expected risk *E* of hospitalisation due to dengue for unchanged risk in seronegative recipients and relative risks according to serostatus

3rd and 4th infections | Hospitalisation with serostatus-independent risk | Hospitalisation with serostatus-dependent risk | ||||||
---|---|---|---|---|---|---|---|---|

1st infection risky | 1st infection risk-free | 1st infection risky | 1st infection risk-free | |||||

\(A_1\) |
| \(A_1\) |
| \(A_1\) |
| \(A_1\) |
| |

| ||||||||

DENv1 | 14 | 7.12 | – | 0.00 | – | 16.84 | – | 0.00 |

DENv2 | 9 | 8.51 | – | 0.00 | – | 24.54 | – | 0.00 |

DENv3 | 14 | 4.13 | – | 0.00 | 258 | 17.25 | – | 0.00 |

DENv4 | 23 | 3.99 | – | 0.00 | 206 | 17.20 | – | 0.00 |

DENv12 | 11 | 15.64 | 76 | 11.57 | 126 | 40.81 | 147 | 33.26 |

DENv13 | 14 | 11.25 | 58 | 9.02 | 108 | 33.81 | 119 | 25.11 |

DENv14 | 18 | 11.20 | 63 | 8.33 | 111 | 33.77 | 120 | 24.69 |

DENv23 | 14 | 12.69 | 70 | 9.91 | 133 | 41.44 | 149 | 33.26 |

DENv24 | 17 | 12.66 | 70 | 9.24 | 135 | 41.66 | 149 | 33.10 |

DENv34 | 17 | 8.20 | 48 | 6.39 | 116 | 34.08 | 123 | 24.43 |

DENv123 | 14 | 19.80 | 48 | 17.33 | 94 | 56.53 | 106 | 54.75 |

DENv124 | 17 | 19.80 | 48 | 16.60 | 95 | 56.97 | 108 | 54.46 |

DENv134 | 17 | 15.34 | 42 | 13.00 | 94 | 51.20 | 102 | 45.80 |

DENv234 | 17 | 16.81 | 48 | 14.29 | 101 | 58.89 | 111 | 54.89 |

DENv1234 | 17 | 23.94 | 38 | 21.23 | 77 | 77.20 | 91 | 78.69 |

| ||||||||

DENv1 | 13 | 2.66 | – | 0.00 | – | 5,703 | – | 0.00 |

DENv2 | 14 | 2.70 | – | 0.00 | – | 37,169 | – | 0.00 |

DENv3 | 14 | 1.95 | – | 0.00 | – | 8,780 | – | 0.00 |

DENv4 | 17 | 1.92 | – | 0.00 | – | 9,296 | – | 0.00 |

DENv12 | 11 | 3.78 | 28 | 2.77 | 298 | 25,204 | 298 | 25,203 |

DENv13 | 12 | 3.66 | 28 | 2.67 | 289 | 10,679 | 289 | 10,678 |

DENv14 | 13 | 3.64 | 28 | 2.64 | 293 | 12,223 | 293 | 12,223 |

DENv23 | 12 | 3.60 | 28 | 2.63 | 307 | 39,766 | 307 | 39,765 |

DENv24 | 14 | 3.58 | 35 | 2.59 | 311 | 48,007 | 311 | 48,007 |

DENv34 | 14 | 3.37 | 28 | 2.43 | 307 | 19,181 | 307 | 19,181 |

DENv123 | 11 | 4.04 | 21 | 3.02 | 248 | 45.59 | 248 | 45.56 |

DENv124 | 11 | 4.03 | 24 | 3.01 | 254 | 50.91 | 254 | 50.88 |

DENv134 | 11 | 3.99 | 24 | 2.97 | 238 | 19.81 | 238 | 19.79 |

DENv234 | 11 | 3.95 | 24 | 2.94 | 265 | 75.17 | 265 | 75.14 |

DENv1234 | 10 | 4.19 | 21 | 3.16 | 156 | 1.69 | 156 | 1.68 |

### 6.1 Constant Efficacy

*E*is plotted against the vaccination age \(A_1\) in months at which the first dose is administered. Since the goal is to minimise this risk the optimal vaccination age is the age \(A_1\) with the lowest lifetime expected risk

*E*. Note that in Figs. 2, 3, 4 and 5 the results shown on the left of Table 4 are presented, i.e. the vaccine efficacy is constant and only depends on the serotype, while the hospitalisation risk only depends on the age at infection but not the serostatus.

However, if we consider the first infection to be risk-free, we can see that the lifetime expected risk decreases with vaccination age for all serotypes. DENv2 always poses the highest risk due to the low efficacy for this serotype. With a step death function it makes sense to consider vaccination only within the age range 0–*L* years, where \(L = 73.8\) years is the maximum human lifetime. Therefore, vaccination is not recommended if only one serotype exists but immunity caused by natural infection or vaccination leads to ADE. This can also intuitively be concluded, since if there is only one serotype, but primary infections are risk-free, there is no need to vaccinate at all. Now if instead of assuming symptomatic third and fourth infections we assume secondary infections to confer permanent cross-immunity, the corresponding results are shown in Fig. 2b. While the overall conclusions are very similar to the symptomatic case the lifetime expected risk is lower in general since a natural infection that occurs after successful vaccination against two other serotypes now no longer contributes to the lifetime expected risk. The optimal vaccination age decreases slightly to between 13 and 17 months in the case of risky primary infections. In this case it is particularly noticeable that the effects of the differences in basic reproduction number and efficacy of the serotypes are less pronounced if there is permanent cross-immunity. This is due to the fact that independent of circulating serotype successful vaccination against two serotypes means the natural infection will not be risky. For risk-free primary infections vaccination is again not recommended.

We now consider an endemic area with two co-circulating serotypes. The results are presented in Fig. 3. We shall start by considering all infections to be equally risky, i.e. the results presented in the graph at the top of Fig. 3a. We can see that similarly to one serotype existing very low vaccination ages are obtained which lie between 11 and 18 months. The effect of the vaccine efficacy is also similar to the case of a single serotype since the low efficacy of DENv2 leads to a higher lifetime expected risk whenever this serotype is present at young vaccination ages, while the combination of DENv3 and 4 leads to the lowest lifetime expected risk due the high combined efficacy. These observations are to be expected after considering the single serotype scenario and also apply in the case of risk-free primary infections as can be seen in the bottom graph. Assuming risk-free primary infections, however, leads to a lower lifetime expected risk and an optimal vaccination age between 48 and 76 months, i.e. much higher than for risky primary infections. The decrease in lifetime expected risk is caused by primary infections not contributing to the risk. The increase in vaccination age is due to fewer potentially risky infections at young ages, so that it is better to vaccinate later when maternal antibodies have declined. Comparing the lifetime expected risk of risky and risk-free primary infections shows that for risk-free primary infections there is a slightly wider range in which near optimal vaccination is possible.

Next we shall assume instead that tertiary and quaternary infections are asymptomatic, i.e. risk-free, as shown in Fig. 3b. As was the case for a single serotype it can immediately be seen that different efficacies and basic reproduction numbers have less effect in this case. Also the effect of assuming risk-free primary infections as opposed to risky ones becomes more pronounced, with a much wider range in which vaccination is near optimal in the former case and an increase in the optimal vaccination age from between 11 and 14 months to between 28 and 35 months. It can therefore be noted that assuming risk-free primary infections in general increases the optimal vaccination age, while assuming asymptomatic, i.e. risk-free, tertiary, and quaternary, infections decreases the optimal age particularly if primary infections are considered risk-free. For risk-free primary infections the optimal age increases since the first infection does not need to be prevented; in fact, if vaccination is only successful against one serotype, it is better to wait. On the other hand if third and fourth infections are asymptomatic vaccinating after a secondary infection will be useless so that in this case the optimal ages decrease.

We now increase the number of serotypes in an endemic area to three. The results for this scenario are presented in Fig. 4. Again we start by assuming that all infections are equally risky, i.e. symptomatic; then, from the top of Fig. 4a we can see that the optimal vaccination ages are between 14 and 17 months which is similar to the cases of one or two coexisting serotypes. Again it can be seen that for low vaccination ages the combination of DENv1, 3, and 4 has the lowest lifetime expected risk since this combination has the highest combined efficacy, and the combination of DENv1, 2, and 3 leads to the highest lifetime expected risk since the combined efficacy is lowest. Therefore, the same observations are made as for one or two coexisting serotypes. This is also true with respect to risk-free primary infections and symptomatic third and fourth infections as shown at the bottom of Fig. 4a. Again the optimal vaccination ages increase to between 42 and 48 months, and there is a wider range in which near optimal vaccination is possible for the reasons previously discussed. For asymptomatic third and fourth infections as shown in Fig. 4b the vaccination ages are 11 months for risky primary infections and between 21 and 24 months for risk-free primary infections. Again the graphs clearly show that the differences in efficacy and basic reproduction number are less decisive for the lifetime expected risk than in the case of symptomatic tertiary and quaternary infections. Similar to the case of two coexisting serotypes it can also be seen that the age range in which near optimal vaccination is possible increases more significantly for asymptomatic third and fourth infections if risk-free primary infections are considered than in the case of symptomatic ones.

Finally consider an endemic area with all four dengue serotypes DENv1–4 coexisting. The results for this case are presented in Fig. 5. The optimal vaccination age obtained for symptomatic tertiary and quaternary infections is 17 months in the case of risky primary infections and 38 months for risk-free ones. For asymptomatic third and fourth infections the corresponding optimal vaccination ages are 10 and 21 months. The lifetime expected risk is lower for risk-free primary infections as was the case for less co-circulating serotypes. The assumption of risk-free primary infections leads to a higher optimal vaccination age than for risky primary infection with a wider range in which near optimal vaccination is possible, and assuming permanent cross-immunity after two heterologous infections results in a significant decrease in optimal vaccination age. The results for four serotypes are therefore as expected from considering less coexisting serotypes.

### 6.2 Serostatus-Dependent Risk

We will briefly consider the consequences of a serostatus-dependent risk on the basis of all four serotypes coexisting as shown in Fig. 6.

By comparing Figs. 5 and 6 we can see that serostatus-dependent risk has a significant effect on the lifetime expected risk. Particularly at young ages, vaccination with a vaccine that induces an additional risk in seronegative recipients leads to a higher lifetime expected risk. Once the vaccination age increases to ages in which few individuals are seronegative the effect is much less pronounced. The optimal vaccination ages independent of the assumptions to ADE and permanent cross-immunity are therefore higher with a vaccine-induced risk. For symptomatic third and fourth infections it can further be seen that the optimal vaccination age increases slightly if primary infections are assumed risk-free. If post-secondary infections are asymptomatic, there is hardly any difference between risky and risk-free primary infections. In this case vaccinating at early ages increases the lifetime expected risk significantly independent of whether ADE is considered or not. Independent of whether primary infections are risky or not it is therefore ideal to vaccinate after the first natural infection occurred if post-secondary infections are asymptomatic. This has two reasons: Firstly, seropositive individuals are not exposed to an increased risk by vaccination so that it is better to wait for more individuals to have had a natural infection. Secondly, if a seropositive individual is successfully vaccinated against any serotype, they were not infected with permanent cross-immunity and will protect them against all future infections. From Fig. 6b it can be seen that the optimal vaccination age is 13 years for permanent cross-immunity independent of the assumption relating to ADE; 13 years is therefore the age at which most primary infections but few secondary infections will have occurred.

### 6.3 Age-Dependent Vaccine Efficacy

So far only results for constant vaccine efficacies were presented. However, as mentioned before, Hadinegoro et al. (2015) found the vaccine efficacies to increase for recipients aged 9 years or older. This phenomenon is believed to be caused by the serostatus of the recipient rather than their age and the constant efficacy can therefore be considered to be more accurate. Additionally, if the vaccine does not induce a higher risk in seronegative recipients, the optimal ages in the case of age-dependent vaccine efficacy are in fact very similar to those presented in Table 4 for constant vaccine efficacy. We will therefore only briefly highlight the differences between constant and age-dependent vaccine efficacy results by considering a single serotype in existence and no serostatus-dependent risk.

### 6.4 Licence Restrictions

The current licence of Dengvaxia only allows vaccination of individuals aged between 9 and 45 years in Brazil. However, when vaccination does not induce an additional risk the optimal vaccination age for the risk of hospitalisation lies between 9 and 76 months for almost any scenario as shown in Table 4. The only exception is one circulating serotype and risk-free primary infections in which case the optimal vaccination policy was not to vaccinate. The optimal vaccination ages are therefore significantly below the permitted age range if there is no serostatus-dependent risk. In the case of serostatus-dependent risk only very few optimal vaccination ages are below 9 years. However, the optimal vaccination ages in this case are much closer to the permitted age range than when the vaccine is not assumed to induce any additional risk. If the age at which the first dose of the vaccine can be administered is restricted such that all doses are given to individuals aged 9–45 years, the optimal vaccination age is 108 months in all cases in which vaccination is recommended independent of the assumption regarding the serostatus-dependent risk. However, restricting the vaccination age as required by the licence leads to a significant increase in lifetime expected risk compared to what could be achieved if vaccination was possible at younger ages. For cases in which vaccination is recommended the increase in the lifetime expected risk compared to its optimum as given in Table 4 lies between approximately 7% and 630% without a vaccine-induced risk and below 7% when such a risk is considered. The percentage increase is higher the further from the optimal age vaccination takes place under the age restriction.

## 7 Discussion

In this paper we found the optimal vaccination age for dengue vaccination in Brazil according to a routine vaccination calendar in endemic regions with any number and combination of dengue serotypes. We assumed independent transmission dynamics for the different serotypes and derived both an exact and an approximate expression for the basic reproduction number of our model. From data we then found serotype-specific basic reproduction numbers, an age-dependent mosquito biting rate, and an age-dependent risk function describing the undesirability of getting an infection in terms of the risk of hospitalisation. The vaccine efficacy in the transmission model was assumed serotype specific, results corresponding to constant efficacy were presented in detail while those corresponding to age-dependent efficacy were presented only for the risk of hospitalisation in an area with one serotype and briefly addressed otherwise. In addition we discussed the effect of an increased risk in seronegative recipients. Risk-free primary infections corresponding to ADE and permanent cross-protection after two heterologous infections were considered. For optimal vaccination ages that did not match the current age range of 9–45 years for which Dengvaxia can be used in Brazil we further determined the optimal vaccination age under these constraints.

The optimal vaccination ages which we obtained for constant efficacy in the case of hospitalisation and hospitalisation with a serostatus-dependent risk are given in Table 4 alongside the minimal lifetime expected risk. We found that optimal vaccination ages if there is no serostatus-dependent risk are very low. For risky primary infections the optimal age is between 9 and 23 months, while for risk-free primary infections it is between 21 and 76 months when vaccination is recommended. The increase in optimal vaccination age if primary infections are considered risk-free can be explained by considering that in this case vaccination is beneficial only after a primary but before a secondary infection. On the one hand this is because at younger ages it may be given to children who are still protected by maternal antibodies and will thus not be effective. On the other hand if vaccination is successful only against one serotype, the first natural infection will correspond to a secondary infection and thus result in a higher risk. This is also the reason why vaccination is not recommended if only a single serotype is in circulation and primary infections are risk-free, i.e. dengue infections are subject to ADE. The assumption of asymptomatic tertiary and quaternary infections, i.e. secondary infections effectively conferring permanent cross-immunity, resulted in lower optimal vaccination ages than those found for corresponding results with symptomatic infections. This change is larger when the assumption is that primary infections are risk-free since in this case vaccination is only beneficial if it prevents secondary infections.

Dengvaxia was first licensed in December 2015. Since the introduction of Dengvaxia in endemic countries concerns about its application in seronegative recipients have been raised due to an increase in SD cases in those recipients, and the need for pre-vaccination screening has been discussed (Halstead and Aguiar 2016; Halstead 2016; Aguiar et al. 2016; Flasche et al. 2016). Considering the increased hospitalisation risk in seronegative recipients that was observed in the long-term follow-up of Dengvaxia trials (SAGE/World Health Organization 2016; Martínez-Vega et al. 2017; Aguiar et al. 2016), we considered serostatus-dependent risk by calculating relative risks according to serostatus as described in Supplementary Appendix B. The optimal vaccination ages and lifetime expected risks are presented in Table 4. With an imperfect vaccine and an increased risk in breakthrough infections of initially seronegative recipients it can be counterproductive to vaccinate seronegative recipients. It is therefore not surprising that in the case of serostatus-dependent risk the optimal vaccination ages were found to be much higher in every case. This is due to the fact that ideally vaccination takes place after most primary infections occurred to prevent exposing vaccinated individuals to any additional risk. More circulating serotypes generally reduce the average age of a primary infection; therefore, the optimal vaccination age decreases slightly as the number of serotypes increases. Considering that vaccination is already carried out at an age at which few individuals are seropositive it is not surprising that the assumption of ADE has little effect on the optimal vaccination age. Most of the obtained optimal vaccination ages in the case of a serostatus-dependent risk are in the permitted age range of Dengvaxia. Again, for risk-free primary infections and only a single serotype in existence it is best not to vaccinate. In addition some combinations of a single serotype in existence with risky primary infections are also better not targeted by vaccination.

The vaccine efficacy of Dengvaxia was found to be age dependent in several Phase Three trials (Capeding et al. 2014; Hadinegoro et al. 2015), and we therefore repeated all simulations using the age-dependent efficacies and presented the results for cases with one serotype and no serostatus-dependent risk. For the remaining cases the results were briefly discussed. The optimal vaccination ages are found to be very similar to those presented in Table 4 with the exceptions of most combinations of two coexisting serotypes if primary infections are risk-free and tertiary and quaternary infections symptomatic. In these cases the optimal vaccination age was found to be 108 months, i.e. 9 years. These are in fact the only scenarios for which we found the optimal vaccination age to be in the age range for which Dengvaxia is licensed in Brazil if there is no serostatus-dependent risk.

If the vaccination age is restricted to between the ages of 9 and 45 years, the optimal vaccination age is 108 months for all cases in which vaccination is recommended, irrespective of all other assumptions. This vaccination age is the earliest possible age at which vaccination is permitted under the licence restrictions and thus as soon as possible after the actual optimal vaccination age. The lifetime expected risk at this age is significantly higher than at the optimum. In general the later vaccination can take place after the optimal vaccination age the higher the percentage increase from the minimal lifetime expected risk is.

The most likely scenario in the majority of endemic areas in Brazil, and in fact in the world, is that all four dengue serotypes coexist. Based on the evaluation of nearly 7 million cases leading to hospitalisation in Brazil (Burattini et al. 2016) our conclusion is that primary infections are in fact risky. This does not mean that ADE does not play an important role in the risk of dengue, but there is a trade-off between the probability that secondary infections are more risky than primary ones and reducing the risk of getting a secondary infection. Therefore, results considering risky primary infections may be more relevant. Since third and fourth infections are rarely reported (Fried et al. 2010; Gibbons et al. 2007) it seems reasonable to assume asymptomatic third and fourth infections. We will therefore briefly discuss the optimal vaccination ages we obtained in the case of all four serotypes coexisting when primary infections are considered risky, and third and fourth infections asymptomatic. Assuming a constant efficacy the optimal vaccination age was found to be 10 months. This is, however, significantly below the licensed age range of 9–45 years for Dengvaxia. If vaccination is only possible within this age range, it should be carried out at 108 months which will lead to an increase in lifetime expected risk of roughly 620%. If the vaccine increases the risk in seronegative recipients, the optimal vaccination age was found to be 156 months, i.e. 13 years, and therefore within the permitted age range.

## Notes

### Acknowledgements

DG and SM are grateful to the University of Strathclyde for support for a Ph.D. studentship. DG is grateful to the Leverhulme Trust for support from a Leverhulme Fellowship (RF-2015-88) and the British Council Malaysia for funding from the Dengue Tech Challenge (Application Reference DTC 16022). EM and DG are grateful to the Science Without Borders Program for a Special Visiting Fellowship (CNPq Grant 30098/2014/7).

## Supplementary material

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