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

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. Electronic supplementary material The online version of this article (10.1007/s11538-019-00690-1) contains supplementary material, which is available to authorized users.

The basic reproduction number R 0 for the model given in Equations (6) and (7) can be derived by considering how many infections are caused by a single infected human in an otherwise entirely disease-free population at equilibrium similarly to the approach of Massad et al. [1] for their dengue model.
Let T H→M (a) be the number of infectious mosquitoes that get infectious by biting a single newly infected human who enters an entirely disease-free population at age a. The probability that the individual is still alive and infectious at age s > a is e − s a (µ H (s)+γ H )ds = π(s) π(a) e −γ H (s−a) . Using Equation (5) the total cumulative future contribution to the force of infection for mosquitoes is therefore c ∞ a q(s) N H π(s) π(a) e −γ H (s−a) ds. The total number of exposed mosquitoes is then given by mc These infectious mosquitoes will in turn bite susceptible humans and thus cause new infections. Denote the age-distribution of newly infected humans caused by a single mosquito by T M →H (a ). This distribution depends on the density function of susceptible humansū(a ), the age-dependent biting rate q(a ), the age-dependent seroconversion rate C(a ), the transmission probability from mosquito to human b, and the expected time a mosquito remains infectious 1/µ M . The density function of unaffected humans is given byū(a ) = N H L π(a ) since apart from a single human all humans are assumed unaffected. As the mosquito lifetime is short compared with the age-dependent changes in the other quantities we may assume that all the infections occur at approximately age a . T M →H (a ) is therefore given by Hence f (a, a ) the age-distribution of infectious individuals caused by a single newly infected individual of age a entering a disease-free population at equilibrium is The basic reproduction number R 0 is given by the spectral radius of this function [2]. As f (a, a ) factorises as a function of a multiplied by a function of a its largest eigenvalue is its trace given by to that of Massad et al. [3] can be followed to get

Supplementary Appendix B. Serostatus-dependent Hospitalisation Risk
In the long-term follow-up of the Dengvaxia trials hospitalisation cases were reported according to serostatus at baseline for individuals in the control group and the vaccine group as shown in Table B.1. These numbers indicate a difference in the risk for vaccinated and unvaccinated seronegative and seropositive individuals. The associated risk functions need to be computed depending on these relative risks of hospitalisation and the vaccine efficacies given in Table 3. However, in this case the assumptions relating to ADE and cross-immunity require more consideration than if we assume that the risk for vaccinated and recovered individuals are the same. In this section we derive the risk functions for risky primary infections in detail, and briefly outline the derivation in the case of ADE implying risk-free primary infections. Consider primary infections to have some associated risk. According to the data presented in Table B.1 the risk of hospitalisation depends on the serostatus, the vaccination history and the age of an individual. We denote the age-classes used in Table B.1 by G 1 and G 2 .
Further we define the relative risks for individuals of age a ∈ G s who are indeed at risk bȳ Note that all of the relative risks are defined for at risk individuals. In the case of no ADE and no permanent cross-immunity any type of infection is associated with some risk, but only individuals with antibodies to no more than three serotypes are at risk of infection.
If two heterologous infections confer permanent cross-immunity only the first two infections are associated with risk. For h − * (a) and h + * (a) only successfully vaccinated individuals are considered, unsuccessfully vaccinated individuals have the same risks as unvaccinated individuals with the same serostatus. To compute the relative risks the numbers in Table B.1 therefore need to be corrected to reflect the fact that not every individual in the vaccine group was successfully vaccinated and that not all seropositives are at risk. We therefore define the following probabilities for each age-class G s : p 1 (G s ) = P (unvaccinated seropositive is at risk at the pre-vaccination steady state), p 2 (G s ) = P (initial seronegative is successfully vaccinated and at risk immediately afterwards), p 2 (G s ) = P (initial seronegative is at risk | successful vaccination against at least one serotype), where v − e (G s ), v + e (G s ) and v i e (G s ) are the serostatus-specific and serotype-specific vaccine efficacies given in Table 3.
Note that p 2 (G s ), p 2 (G s ) and p 3 (G s ) are evaluated immediately after vaccination. Triv- However, the computation of p 1 (G s ), p 2 (G s ) and p 3 (G s ) depends on whether two heterologous infections confer cross-immunity. If there is no crossimmunity the probability p 1 (G s ) can be calculated as the probability of being seropositive to at most three serotypes given that an individual is seropositive. The pre-vaccine steady-state fraction of individuals unaffected by serotype i in age-class G s can be obtained as is the pre-vaccine steady-state fraction of individuals unaffected by serotype i at age a and B 1 and B 2 are the limits of the age-class G s . Therefore In the case of cross-immunity p 1 (G s ) is the probability of an individual being seropositive to exactly one serotype given that he or she is seropositive, i.e.
Similarly for p 2 (G s ) consider the serotype-specific vaccine efficacies v i e (G s ) and note that a successfully vaccinated initial seronegative who is at risk in the case of no cross-immunity has been vaccinated against at least one and no more than three serotypes, i.e. .
In the case of cross-immunity individuals who have been vaccinated against exactly one serotype need to be considered, i.e. .
In the case of permanent cross-immunity after two heterologous infections clearly p 3 (G s ) = 0 and by definition h + * (a) = 0 since no initially seropositive individual who was successfully vaccinated is at risk. However, if there is no cross-immunity individuals who are seropositive to at most three serotypes after vaccination remain at risk. From the steady-state analysis we know that the distributions of seropositivity for individuals in age-class G S to serotype i only, and serotypes i and j only, conditional on seropositivity are given by , and respectively. An individual who was initially seropositive to serotype i and then successfully vaccinated remains at risk if the vaccine was successful against no more than two other serotypes. Similarly an individual who was seropositive to serotypes i and j remains at risk after successful vaccination against only one more serotype. The probabilities of this happening in age-class G s are given by , and respectively for the two cases. Individuals who were initially seropositive to three serotypes are no longer at risk after successful vaccination and those initially seropositive to all serotypes cannot be vaccinated. The probability p 3 (G s ) in the case of no cross-immunity is therefore given by We haveh − (a) = 1 by definition. Additionally using Table B.1 we havē since for example of the 236 initial seropositives aged 2-8 in the control group only 236p 1 (G 1 ) were at risk. Similarly only 481p 3 (G 1 ) individuals were at risk vaccinated seropositive individuals aged 2-8 years immediately after vaccination, and therefore To find the relative risk h − * (a) for the age-group 2-8 years note that out of the 330 initially seronegative individuals in the vaccine group 330v − e (G 1 ) p 2 (G 1 ) = 330p 2 (G 1 ) were successfully vaccinated and remain at risk, while 330v − e (G 1 )(1 − p 2 (G 1 )) were successfully vaccinated and are no longer at risk, and 330(1 − v − e (G 1 )) were not successfully vaccinated. The number of hospitalisation cases in unsuccessfully vaccinated individuals is therefore 330(1 − v − e (G 1 )) 5 173 with the remaining 17 − 330(1 − v − e (G 1 )) 5 173 hospitalisations having occurred in at risk successfully vaccinated initially seronegative individuals. For the age-group 9-16 years a similar argument can be followed and the relative risk for successfully vaccinated initially seronega-tives is therefore given by / 4 204 , 9 ≤ a < ∞.
Having found the relative risksh − (a),h + (a), h − * (a) and h + * (a) the pre-vaccine hospitalisation risk R(a) given in Equation (19) can now be used to determine the risk functions associated with any type of infection. Define R − (a) to be the pre-vaccination hospitalisation risk for seronegative individuals and R + (a) to be the pre-vaccination hospitalisation risk for at risk seropositive individuals. Note that R + (a) =h + (a)R − (a). Then in the case of no cross-immunity we have