Implication of vaccination against dengue for Zika outbreak

Zika virus co-circulates with dengue in tropical and sub-tropical regions. Cases of co-infection by dengue and Zika have been reported, the implication of this co-infection for an integrated intervention program for controlling both dengue and Zika must be addressed urgently. Here, we formulate a mathematical model to describe the transmission dynamics of co-infection of dengue and Zika with particular focus on the effects of Zika outbreak by vaccination against dengue among human hosts. Our analysis determines specific conditions under which vaccination against dengue can significantly increase the Zika outbreak peak, and speed up the Zika outbreak peak timing. Our results call for further study about the co-infection to direct an integrated control to balance the benefits for dengue control and the damages of Zika outbreak.

Dengue and Zika are both vector-borne diseases in tropical and sub-tropical regions with a common vector, dengue and Zika both belong to the family Flaviviridae and genus Flavivirus. Dengue is a prevalent disease being transmitted by the bite of a mosquito infected with one of the four serotypes 1,2 while Zika is an emerging disease. Zika virus was first isolated in Uganda in 1947 3 , and there was an outbreak of Zika in Yap, Federated States of Micronesia 4 in 2007, and in French Polynesia 5 till 2013. By the end of January 2016, autochthonous circulation of Zika was reported in more than 20 countries or territories in South, Central, and North America and the Caribbean 6-12 , leading to the declaration of WHO that Zika virus is a global public health emergency 13 .
Recent clinical and experimental evidences support immunological cross-reactivity between dengue and Zika [14][15][16][17] . In particular, these evidences show that plasma to dengue was able to drive antibody-dependent enhancement of Zika infection. Co-circulation of multiple serotypes of dengue and dengue-Zika co-circulation have previously been reported in refs 18-20. In particular, co-infection of dengue and Zika were observed in two patients during the Zika outbreak in New Caledonia in 2014 18 , and in two patients during the Zika outbreak in Tuparetama of Brazil in 2015 19 . The co-circulation could be a potentially series public concern given that more than a third of the world's population lives in countries where dengue is endemic 21 , with the dengue belt covering Central America, most of South America, sub-Saharan Africa, India, and South East Asia. Relevant to this co-infection is the development of vaccine products against dengue by Sanofi Pasteur, and the clinical trials by Butantan and Takeda. Thus, it is an important urgent issue for public health decision makers to know how dengue immunization program impacts Zika transmission when co-circulation becomes wide spread. Specially, under which conditions implemented dengue immunization control programs may boost the outbreak of Zika is no longer a thought-provoking issue. Developing a framework to address this issue through a mathematical model is the main objective of this study.
Much progress has been made for modelling dengue infection dynamics including the role of cross-reactive antibodies for the four different dengue serotypes as discussed in the review paper 22 . The dengue transmission dynamics becomes very complex because of the co-circulating serotypes in many endemic areas, and the absence of long-term cross-immunity [23][24][25][26] . In 1997, Feng et al. 27 proposed a two-stain model with the vector population being subdivided into a susceptible class and two serotype-specific infectious classes and the host populations being described by the SIR-type model for each serotype. Esteva and Vargas 28 considered a further model by including an explicit state for individuals who recovered from primary infections. Nuraini et al. 29 and Sriprom et al. 30 extended Esteva's model by accounting for two separate symptomatic and asymptomatic compartments for secondary infections. A four-serotype model was considered in these papers [31][32][33] . Different from these previous studies, recently developed mathematical models have emphasized the evaluation of the impact of co-circulation of the four serotypes mainly among hosts [34][35][36][37][38][39][40] . In contrast to dengue, the epidemiology of Zika among humans remains poorly understood, despite some recent outbreaks of modelling activities [41][42][43][44] . I dz : the number of dengue-infected humans with dengue-infection age a, at time t, who are immune to Zika; R dz (t): the number of humans recovered from dengue and Zika at time t, who can neither be infected by dengue nor Zika.
Mosquito population N m is divided into S m , I md , I mz , I mdz , representing the density of mosquitos who are susceptible, infected with dengue only, infected with Zika only, infected with both dengue and Zika. The transmission diagram of co-infection of dengue and Zika among humans and mosquitos is shown in Fig. 1.
We start with an intuitive view about the effects of vaccination against dengue among humans on the outbreak of Zika through a very simple static transmission model illustrated in Fig. 2. Here the susceptible humans (S) can be infected with Zika virus via three different routes, namely of Zike infections if the parameters P z and P J z are located in the red region while it can decrease the total number of Zika infections in the green region; (B) The relationship of the total number of Zika infections to the ratio P J z with or without vaccination against dengue. Here, P z = 0.3 and P v = 0.7; (C) The relationship of Δ Z to the effective coverage rate of dengue vaccine P v while the parameters P z and P J z are chosen in the red region of (A) with P z = 0.3; (D) The relationship of Δ Z to the effective coverage rate of dengue vaccine P v while the parameters P z and P J z are chosen in the green region of (A) with P z = 0.3. Other parameters in (A-D) are fixed as P d = 0.3, P dz = 0.1, S 0 = 100000.
Scientific RepoRts | 6:35623 | DOI: 10.1038/srep35623 Let the initial number of susceptible humans (S) be S 0 . If we do not inoculate against dengue, then the final average number of humans infected with Zika virus through the above three routes (i.e. I z , I dz , J d z ) can be calculated as Then, the total number of humans infected with Zika virus after vaccination against dengue should be Comparing equation (2) with equation (4), we can see that with the implementation of vaccination of dengue the final numbers of I z and I dz decrease while the final number of J d z increases. To determine whether the total number of humans infected with Zika is increased or not, we let where P J z is the ratio at which the part of the susceptible humans inoculated with dengue vaccine are infected with Zika, − + + P P P P P P (1 ) is the total ratio at which the susceptible humans are infected with Zika through the above mentioned three routes described in (1). It follows from equation (6)  , as shown in the red region of Fig. 3(A)), then the higher ratio the susceptible humans are inoculated with dengue vaccine, the more the total number of humans are infected with Zika virus compared with the case without dengue vaccination, as shown in Fig. 3  . This discussion, based on a static infection outcome analysis, suggests a likely scenario that, under certain conditions, vaccination against dengue can significantly boost the outbreak of Zika. Our analysis below is to theoretically and numerically examine these conditions with our proposed transmission dynamics model.

Model formulation
We assume a SI-type model for dengue and Zika co-infection for the mosquito population. The model equations for mosquitos give Here, Λ is the recruitment rate of mosquitos, and the definitions for other parameters are listed in Table 1. We assume SIR-type model for dengue and Zika co-infection in human population and formulate the following age-structured model to describe the dynamics of co-infection of dengue and Zika among humans: represents the recover rate at which individuals in the class J d z with Zika-infection age b recover to the compartment R dz , and γ a ( ) z d is the recover rate at which individuals in the class J a ( ) z d with dengue-infection age a move to compartment I dz , γ dz (a, b) denotes the recover rate at which individuals in the class I dz with time-since-infection a for dengue and time-since-infection b for Zika recover to the compartment R dz directly, γ dz (a) represents the recover rate at which individuals in the class I dz transit to the compartment J d z due to recovery of dengue, and γ dz (b) is the recover rate at which individuals in the class I dz transit to the compartment J z d due to recovery of Zika. The definitions for other parameters independent of infection ages are given in Table 1. Here, the condition I dz (0, 0, t) = 0 means that the susceptible individuals can not be infected with dengue and Zika in the same time.
We assume that . Then formula (9) yields  Similarly, if the recover rate γ z (b) is independent of Zika-infection age b, the total number of humans infected with Zika, given by With similar calculation, we can get the derivative of the compartment I dz (t) as follows:    Also, when we assume that the recover rates γ dz (a, b), γ dz (a) and γ dz (b) are all constants, denoted by γ γ , dz dz d and γ dz z , respectively, then formula (12)  Moreover, define the total number of humans who are immune to dengue but infected with Zika as and the total number of humans who are immune to Zika but infected with dengue as . By assuming the recover rates γ a ( ) being independent of infection ages Scientific RepoRts | 6:35623 | DOI: 10.1038/srep35623  I  I  I  S  N  dI  dt  c I  p  I  S  N  c  I  I  I  N  I   dI  dt  c I  p  I  S  N  c  I  I  I  N  I   dI  dt  c  I  I  I  N  c  I  I  I  N   I  I  I  dR  dt  I  c  I  I  We call model (16) with model (7) as system S * . It follows from model (16)  and I md = I mz = I mdz = 0. Then we can show that system S * has a disease-free equilibrium, which gives µ = Λ . I  I  I  N  ( , , , , , , , , , , , , ) ( , 0, 0, 0, 0, 0, 0, 0, / , 0, 0, 0)

S I I I R R J J R S
Using the next generation matrix introduced in papers 58,59 , we can calculate the basic reproduction number for system S * , denoted by R 0 (see electronic supplementary information for details). This is the spectral radius of the next generation matrix and given by

Main Results
In this section, we carry out numerical simulations for the dynamic system S * in order to examine effect of dengue vaccination on the outbreak of Zika. In our simulations, we vary three parameters β dz , β rz and Λ , and fix all the other parameter values as follows:  Further, we examine the effects of the effective coverage rate P v on the outbreak of Zika. Fix parameters β dz = 0.05, β rz = 0.18, Λ = 10000 and let the parameter P v vary, Fig. 7(A) shows that a higher effective coverage rate of vaccination can result in a much higher peak of the outbreak of Zika. Moreover, if we choose Λ = 1000000, then we observe that with a higher rate of vaccination against dengue not only the peak of the outbreak of Zika can be significantly increased, but also the Zika outbreak peak much earlier, as shown in Fig. 7(B).
Considering the number of the accumulated Zika infections, we obtained two similar opposite results. Figure 7(C) shows that with a higher rate of vaccination against dengue the number of accumulated Zika infections will increase significantly, while Fig. 7(D) illustrates that vaccination against dengue may reduce the number of the accumulated Zika infections. In Fig. 7(D) we assumed that β rz = β z = 0.05 while in Fig. 7(C) we assumed that 0.18 = β rz > β z = 0.05 based on the emerging clinical evidence of enhancement [14][15][16][17] . Comparisons between these scenarios clearly indicate, under the conditions reflected by the parameter values, that dengue vaccination may indeed lead to significant increase in Zika infections.

Conclusion and Discussion
There are increasing evidence of co-infection of dengue and Zika. Due to similar transmission routes with the same host species, some intervention strategies such as vector control are effective for curbing both dengue and Zika. However, other interventions such as vaccination against one virus may be harmful to the control of another, specially when enhancement occurs to favor the spread of the virus not covered with vaccine. Our study examined the implication of this enhancement for Zika outbreaks when vaccination against dengue in humans is applied. We initially formulated a very simple static transmission model to give an intuitive illustration that vaccination against dengue among humans may significantly boost Zika transmission among the population. In order to theoretically verify this illustration, we then proposed a dynamic model to describe the dynamics of co-infection of dengue and Zika. More specifically, we developed a novel model with double age-structures for dengue and Zika, extending the general age-structure model [65][66][67] by incorporating compartments with specific dengue-infection and Zika-infection age. Under certain stage-specific homogenetical assumptions about the virus dynamics characteristics, we simplified our double age-structured model to an ODE model, for which the basic reproduction number can be calculated.
We also numerically investigated the dynamics of model S * and obtained some observations which are in agreement with the conclusions from the analysis of our static transmission model in Section 2. Figure 4 shows that vaccination against dengue among humans may result in the total number of humans infected with Zika virus decline or increase, depending on the parameter Λ , the recruitment rate of mosquitos. In particular, it significantly enlarges the peak of the outbreak of Zika when Λ is relatively large. It follows from Figs 5 and 6 that this enlarged outbreak of Zika by vaccination against dengue is due to multiple factors. Vaccination against dengue can reduce the numbers of I z and I dz while it always increases the number of J d z . Thus the balance of increase in the number of J d z and decrease in the number of I z and I dz determines whether the total number of infected with Zika increase or not. Further, we observed that a higher rate of vaccination against dengue can also results in a higher and earlier peak of the outbreak of Zika, as shown in Fig. 7(A,B). Comparing Fig. 7(B) with Fig. 7(A), we observe that the conclusion that vaccination against dengue can boost Zika outbreak remains true for a wide range of mosquito index values (when the recruitment rate of mosquito decreases from 1000000 to 10000). This conclusion is also shown in Fig. S2 (electronic supplementary information) when the mosquito mortality rate μ m varies. Comparison between Fig. 7(B) and Fig. 7(A) however also shows that reducing the mosquito indices can significantly decrease the magnitude of Zika outbreak as the number of Zika cases at the peak time can be reduced substantially. Therefore, given the simultaneous impact on both dengue and Zika outbreaks, vector control should be always implemented regardless of the availability of vaccine. Figure 7(C,D) further confirm that the accumulated Zika infections may be greater for a greater rate of vaccination of dengue vaccine to human. Sensitive analyses show that parameters β z , β dz , Λ and μ m can significantly affect the outbreak of Zika, in terms of both the accumulated Zika infections and the daily number of Zika infections (see electronic supplementary information for details).
Most existing studies on the multi-serotype models of vector-host transmission of dengue focus on the importance of subsequent infections with different dengue serotypes. It was assumed that the patients can be subsequently infected by another serotypes after recovering from one serotype. In our consideration of dengue-Zika co-infection, we extended these models by adding a new compartment of humans as well as mosquitos infected by both of Zika and dengue simultaneously. From our numerical analysis, the parameter β dz (i.e. the transmission rate of the compartment of mosquitos infected with dengue and Zika to susceptible humans), which is related to the newly added compartment I mdz , can have important influences on the dynamics of the co-infection model. For the models of co-infection of HIV with TB and HCV, a SI-type model is usually assumed as the basic model for each disease. In comparison with these models, our model with SIR-type for humans is different to handle the asymmetric vector-host interaction as discussed in ref. 27, and to allow recovered (or vaccinated) individuals from one virus to have higher risk of infection by another. Our analysis indicates that with a big recruitment rate of mosquitos Λ vaccination against dengue among humans can significantly boost the Zika outbreak (as shown in Fig. 6(H)), and cause the Zika outbreak peak coming early with a bigger mosquito to humans transmission rate β rz and lower β dz (as shown in Fig. 7(B)). It is important to note that a safe, effective and affordable dengue vaccine against the four strains offers an important tool to reach the WHO goal of reducing dengue morbidity by at least 25% and mortality by at least 50% by 2020 68 . The first dengue vaccine, Dengvaxiar(CYD-TDV) (developed by Sanofi Pasteur), was licensed in Mexico in 2015 69 ; and two dengue vaccine candidates (developed by Butantan and Takeda) entered the Phase III trails in early 2016 [70][71][72] . Our study should not serve as a discouragement to the development of these dengue vaccine products, but rather we determine conditions under which dengue vaccination can contribute to the prevention and control of dengue without inducing significant increase in Zika infection.
Most published works focus on the benefits of the control strategies (such as treatments for only one or both diseases) to both diseases involved in the co-infection. For example, Derouich and Boutayeb 73 considered a model of two subsequent infections of dengue at separate time intervals with continuous vaccination. They concluded that vaccination can be a control strategy for dengue. However, with consideration of co-infection and the current development of dengue vaccine, our results suggest that additional study on co-infection is urgently and critically needed.