In a short paper, answer the following questions: • After having reviewed two papers that make use of a mathematical/computational model, what problem from one of them is still left unsolved? Inclu

2017 IEEE International Conference on Bioinformatics and Biomedicine (BIBM) 978-1-5090-3050-7/17/$31.00 ©2017 IEEE 1425 JOURNAL OF L AT EX CLASS FILES, VOL. 6, NO. 1, OCTOBER 20171 A simpliﬁed mathematical-computational model of the immune response to the yellow fever vaccine C.R.B. Bonin a∗, G.C. Fernandes bR.W. dos Santos aand M.Lobosco a Abstract—An effective yellow fever vaccine has been available since 1937. However, some issues regarding its use remain open, such as the minimum dose that can provide immunity against the disease. Mathematical-computational tools can be useful to assist the search for answers to some of these open issues.

In this context, this study presents a simpliﬁed mathematical- computational model of the human immune response to the vaccination against yellow fever that takes into account important cells of the immune system. The model was able to qualitatively reproduce some experimental results reported in the literature, such as the amount of antibodies and viremia along time, as well as to reproduce distinct behaviors of the immune response reported in the literature. This is the ﬁrst step towards an ideal scenario where it will be possible to simulate distinct situations related to the use of the yellow fever vaccine, such as its application in immunodeﬁcient individuals, different vaccination strategies, duration of immunity and the need for a booster dose.

Index Terms—Computational vaccinology, Yellow fever, Math- ematical modeling, Computational modeling, Immune system, Ordinary Differential Equations I. I NTRODUCTION Since 1937 an effective vaccine against Yellow Fever (YF) has been available [1]. Despite this, outbreaks still occur in countries in Africa, Central and South America, where the population has low vaccination coverage. An outbreak was recorded between 2015 and 2016 in Angola. From December 5, 2015 to August 4, 2016, 3,867 suspected cases were reported, of which 879 were laboratory conﬁrmed, causing a total of 369 deaths, of which 119 were reported among conﬁrmed cases[2]. Another signiﬁcant recent outbreak oc- curred in the Democratic Republic of Congo (DRC) in 2016.

A total of 2,987 suspected cases were reported to the national surveillance system, with 81 laboratory conﬁrmed cases and 16 deaths [2].

The latest outbreak occurred in Brazil, starting in December 2016. In the period from December 2016 to February 22, 2017, a YF outbreak has affected Brazil, with 1345 suspected cases, of which 295 have been conﬁrmed, and 215 deaths reported to Brazilian Ministry of Health [3].

Although the number of YF outbreaks in Brazil between the end of 2016 and the beginning of 2017 is higher than in previous outbreaks, it is important to note that all cases are wild-type, observed in people who live in rural areas aGraduate Program in Computational Modeling, Federal University of Juiz de Fora, 36036-900, Juiz de Fora, MG, Brazil bPresidente Antnio Carlos University - Medical School, Juiz de Fora, MG, Brazil *Corresponding author. Tel.: +55 32 2102-3481. E-mail address: rezen- [email protected] (C.R.B. Bonin). or who have had contact with wilderness areas for leisure or work. Cases were considered as sylvatic transmission although some cases occurred in cities with previous recent dengue fever outbreaks, highlighting the imminent risk of YF reintroduction in urban areas with high infestation ofAedes aegypti. Unfortunately, Brazil, like other major countries in the world, has favorable conditions for this. Urbanization, population mobility in endemic areas, expansion of these areas due to climate change, and the resurgence of theAedes aegypti mosquito increase the likelihood of future YF epidemics in the absence of effective countermeasures [4]. Furthermore, the routine YF vaccination is not mandatory in larger centers in Brazil (such as Rio de Janeiro and So Paulo).

World stocks of YF vaccine may not be prepared to supply the need for vaccination if a large outbreak occurs. In fact, this has already occurred in recent outbreaks. In Kinshasa, capital of the DRC, fractional doses of the YF vaccine were administered [4]. Some studies that have been used as a basis for the implementation of this strategy are summarized in an information published by the World Health Organization (WHO) [5]. The most recent ones indicate that fractional doses have the same immunogenicity as standard doses, at least in the short and medium term [6], [7].

The WHO launched in April 2017 a strategy called Elimi- nate Yellow fever Epidemics (EYE), which aims to eliminate YF epidemics in the world by 2026 [4]. Through early detection and rapid and appropriate response, it is possible to minimize suffering, damage and propagation [4]. This strategy has three goals: protect populations at risk, prevent the international spread of the disease and contain outbreaks quickly. To achieve these goals, the strategy suggests actions on different fronts, including research and development for better tools and practices.

In this scenario, mathematical and computational modeling presents itself as a tool to assist researches in vaccinology and public health. Mathematical models have been used for many years to represent various aspects of the immune system and related pathologies, but their application to describe the effects of vaccines has been rather limited [8]. The term computa- tional vaccinology has been used to refer to computer-aided vaccine design [9], [10], [11], [12]. In a previous work[13], we proposed a new use of computational vaccinology, in the clinical development stage. With the use of mathematical and computational models, it is possible to experiment,in silico, different scenarios related to vaccination, for answering important questions still open.

This work presents a simpliﬁed mathematical-computational model of the immune response to the YF vaccine. The model 1426 JOURNAL OF L AT EX CLASS FILES, VOL. 6, NO. 1, OCTOBER 20172 considers the major populations of Human Immune System (HIS) cells and molecules important in the process of im- munity acquisition, such as Antigen Presenting Cells (APCs), B and T lymphocytes, and antibodies, which are considered the main marker of immunity. The model is then evaluated using distinct scenarios, and as will be shown, it was able to qualitatively reproduce some experimental results reported in the literature.

The remaining of this work is organized as follows. Section II brieﬂy presents related work. Section III presents the simpliﬁed mathematical model and its computational imple- mentation. The results of the model are presented in Section IV. Finally, Section V presents the conclusions and plans for future works.

II. R ELATED WORK Several computational modeling techniques applied to vac- cination are analyzed and discussed by Pappalardoet al.[8].

The authors describe what mathematical and computational modeling are and how they can aid research in vaccination.

Modeling is deﬁned as human activity involving the repre- sentation, manipulation, and communication of everyday real world objects. In this study, two main types of modeling are considered: Agent-Based Models (ABM) and mathematical models. The dynamic agents of an ABM can be described as a function of time, a position and an internal state that includes the most important properties of the agent, such as age. Mathematical models are mainly based on differential equations, whether ordinary or partial, with delayed and/or stochastic equations. The studies reviewed by the authors[8] are more related to tumor modeling or to vaccines that use mechanisms other than live and attenuated virus inoculation.

In this work, we propose a mathematical-computational model of the immune response to the YF vaccine, which is based on a live, attenuated viral strain. The model used Ordinary Differential Equations (ODEs) to model the main cells and molecules related to the adaptive immune response.

Other works use computational tools to aid the vaccine design. For example, epitope-mapping algorithms are used for vaccine design since the 1980s[14]. Since then, new computational tools have been used for selection of vaccine targets[15], [16], [17], [18], [19], [20], [21], [22], [23]. Most of the works focus on using computational science to predict epitopes[24] or to develop virtual screening approaches (i.e, the identiﬁcation of relevant antigens)[25], [26], [27], [28].

This traditional use of computational vaccinology is related to the pre-clinical development. This work focus in the de- velopment of mathematical and computational models that can be used in the clinical development stage, i.e., when the vaccine is ﬁrst tested in humans. We argue that it is possible to carry some experimentsin silico, reducing the search space for experimentsin vivoorin vitro, and it is possible to eliminate, reinforce or weaken hypotheses and to direct studies, thus saving time and resources.

A mathematical model, using ODEs, of the human immune response to vaccination against both YF and smallpox was presented in a previous work [29]. The aim of the authors wereto evaluate primarily the dynamics of CD8+ T cells and not the immune response as a whole. The model proposed in this work differs from those presented by Leet al.[29] because it considers important populations at each stage of the immune response to YF vaccination, from virus inoculation to APC antigen presentation and consequent activation of lymphocytes to the generation of antibodies and memory cells.

III. M ETHODS A. Mathematical model The model proposed in this work was based on a previous work [30] and consists of a system of 10 ODEs representing some important populations in the response process generated by the body after vaccination. The main populations are the following: viruses, APCs, CD8+ T cells, short-lived and long- lived plasma cells, B cells and antibodies.

The model can estimate the concentration of all cells and molecules represented by the equations. In the next section the values of viremia and antibody concentration will be estimated and then compared to the values reported in the literature, in order to qualitatively validate the proposed model.

Equation (1) represents the vaccine virus (V).

d dtV=π vV−c v1V cv2 +V−k v1VA−k v2VT E (1) The virus can not proliferate itself, it needs to infect a cell and use it as a factory for new viruses. This is implicitly considered in the termπ vV, which represents the multipli- cation of the virus in the body, with a production rateπ v.

The term cv1V cv2+V denotes a non-speciﬁc viral clearance made by the innate immune system. This function models growth combined to the saturation phenomenon and is similar to the Hill family of equations[31]. The termk v1VAdenotes speciﬁc viral clearance due to antibody signaling, wherek v1 is the clearance rate. The termk v2VT E denotes speciﬁc viral clearance due to the induction of apoptosis of cells infected by the YF virus, wherek v2 is the clearance rate.

APCs are all cells that display antigens complexes on their surfaces, such as dendritic cells and macrophages. Two stages of APCs were considered: immature and mature. The ﬁrst stage, immature APCs (A P), is described by Equation (2).

d dtA P=α AP(A P0 −A P)−β APAP(kAP1 +tanh(V−k AP2)) (2) The termα AP(A P0 −A P)denotes the homeostasis of APCs, whereα AP is the homeostasis rate. The term β APAP(kAP1 +tanh(V−k AP2))denotes the conversion of immature APCs into mature ones. So the same term appears in Equation (3) with positive sign.β APrepresents the convertion rate and(k AP1 +tanh(V−k AP2))is a sigmoidal saturation function in the form of a hyperbolic tangent.

The Equation (3) represents the mature APCs (A PM ).

d dtA PM =β APAP(kAP1 +tanh(V−k AP2))−δ APM APM (3) 1427 JOURNAL OF L AT EX CLASS FILES, VOL. 6, NO. 1, OCTOBER 20173 The ﬁrst term, as just explained, denotes the dynamics of APCs maturation. The second term,δ APM APM , denotes the natural decay of the mature APCs, whereδ APM is the decay rate.

Equation (4) represents the population of naive CD8+ T cells (T N).

d dtT N =α TN(TN0 −T N)−π TAPM TN (4) The termα TN(TN0 −T N)represents the homeostasis of CD8+ T cells, whereα TN is the homeostasis rate. The term π TAPM TN denotes the activation of naive the CD8+ T cells, whereπ Tis the activation rate.

Equation (5) represents the effector CD8+ T cell population (T E).

d dtT E=π TAPM TN+k TEAPM TE−δ TETE (5) The termk TEAPM TErepresents the proliferation of effec- tor CD8+ T cells. The termδ TETErepresents the natural death of these cells, withδ TErepresenting its decay rate.

Equation (6) represents B cells(B), both naive and effector ones. These populations were not considered separately in order to simplify the model.

d dtB=α B(B 0−B)+π BAPM B−β SAPM B −β LAPM B−β BMAPM B(6) The termα B(B 0−B)represents the B cells homeosta- sis, whereα B is the homeostasis rate. The termπ BAPM represents the proliferation of the active B cells. The terms β SAPM B,β LAPM Bandβ BMAPM Bdenote the portions of active B cells that differentiate into short-lived plasma cells, long-lived plasma cells and memory B cells, respectively. The activation rates are respectively given byβ S,β Landβ BM.

Equation (7) represents the short-lived plasma cells (P S).

d dtP S=β SAPM B−δ SPS (7) The termδ SPS denotes the natural decay of short-lived plasma cells, whereδ Sis the decay rate.

Equation (8) represents the long-lived plasma cells (P L).

d dtP L=β LAPM B−δ LPL+γ MBM (8) The termδ LPL denotes the natural decay of long-lived plasma cells, withδ Lrepresenting the decay rate. The term, γ MBM represents the production of these cells by memory B cells, whereγ M is the production rate.

Equation (9) corresponds to memory B cells (B M).

d dtB M =β BMAPM B+k BM1 BM(1−B M kBM2 )−γ MBM (9) The termk BM1 BM(1− BMkB M2 )represents the logistic growth of memory B cells, i.e., there is a limit to this growth.k BM1 represents the growth rate, andk BM2 limits the growth.Equation (10) represents the antibodies. The termsπ AS PS andπ AL PLare the production of the antibodies by the short- lived and long-lived plasma cells, respectively. The production rates are given byπ AS andπ AL , respectively. The termδ AA denotes the natural decay of these cells, whereδ Ais the decay rate.

d dtA=π AS PS+π AL PL−δ AA(10) The model presented in this paper was based on an earlier study [30], which described a mathematical model to represent the human immune response to an infection by YF virus. So the ﬁrst difference is that this paper focus on modelling the effects of the YF vaccine administered subcutaneously.

The previous work modeled the immune response to the YF virus from infection of epithelial cells to secretion of antibod- ies, considering various populations of cells and molecules, in different stages and compartments. There were 19 ODEs divided into two compartments: one representing the tissue where the virus proliferate and the other the lymph nodes. In order to consider all the cells and molecules, the model became complex. Another issue is related to its adjust to reproduce some behaviors described in the literature: as the number of equations and parameters increases, so does the amount of data and information needed to adjust the model.

The second difference between the two models is that this work reduces the amount of equations from 19 to 10. The reduced model described in this work considers only the main populations of cells and molecules involved in the response to the vaccine, and abstracts some details that are not crucial to represent the behavior of the immune response. For example, the distinct compartments are not represented here. Also, some populations were not considered because no experimental data is available to validate the simulations, such as the CD4+ T cells. In future, more cell or molecule can be included in the model again, if its role is important to explain or represent some behavior that the reduced model presented in this section could not represent. In fact, it is important to remember that a mathematical model is an abstraction of the reality and therefore simpliﬁcations are always necessary. This is accentuated when the target of the model is the HIS response, a complex network that involves many tissues, organs and cells and performs several processes. The level of abstraction depends on the purpose of the model. HIS can be seen at various levels, from the level of substances produced by cells, such as cytokines, to the level of cells and molecules, as in the case of the current model. It also can reach the level of population and epidemiological models that are interested in a pattern of behavior observed in a population sample.

B. Computational model For the resolution of the system of ODEs, a code was implemented using the Python programming language, which includes libraries for easily solving complex mathematical problems. The library chosen was scipy [32]. This library has a package called “integrate”. One of the functions available in this package is called “odeint”, and it is used to solve 1428 JOURNAL OF L AT EX CLASS FILES, VOL. 6, NO. 1, OCTOBER 20174 numerically a system of ODEs. The choice of the numerical method to be used is made automatically by the function based on the characteristics of the equations. The function uses an adaptive scheme for both the integration step and the convergence order. The function can solve the ODEs system using either the BDF (Backward Differentiation Formula) or the Adams method[33]. BDF is used for stiff equations and the implicit Adams method is used otherwise.

The experiments were performed using Python version 2.7.10 using the Spyder integrated development environment (IDE). The execution environment was composed by an Intel Core i5 1.6 GHz processor, with 8 GB of RAM. The system runs macOS Sierra version 10.12.5.

IV. C OMPUTATIONAL RESULTS In order to qualitatively validate our model, two experiments were carried out. The ﬁrst one simulates a scenario where an individual was vaccinated against YF for the ﬁrst time. The standard dose of the vaccine was used in this scenario. The results of the simulation were then compared to experimental data obtained in the literature.

The second scenario is based on an experimental study[6] in which smaller doses of the YF vaccine were tested. Compared to the standard dose, the experimental study reported that, to some extent, the reduction did not affect signiﬁcantly the percentage of serum-conversion. In this scenario, the compu- tational experiments is executed many times, using distinct values for the vaccine doses. For comparison purposes, the computational experiments were carried out using the same values of the experimental study[6].

All the initial values used for the variables as well as the model parameters are presented in the Appendix. The tuning of the model parameters was done manually, except forδ A, whose value was extracted from the literature.

Usually the literature reports two distinct experimental data.

The ﬁrst one is the viremia, i.e., the amount of virus present in the bloodstream. The literature reports the viremia along time.

The second data reported in the literature is the antibody levels along time. The values obtained by Equations (1) and (10) are compared to the experimental values from the literature.

A. First vaccination This section presents the computational results of a simu- lation in which an individual was ﬁrst vaccinated against YF.

Although the amount of virus particles varies depending on the vaccine lot number, ranging from 2.3 to 12 times the minimum value required by the WHO in the case of 17DD-YFV[6], in this computational experiment we used a value equal to 27,476 IU. The 17DD-YFV is the YF vaccine developed by Bio- Manguinhos / Fiocruz, one of the three producers prequaliﬁed by WHO to supply vaccines to international agencies.

Figures 1 and 2 show the comparison of the antibody curve generated as a result of 100 and 5,000 simulation days, respec- tively, with the experimental results from the literature[34].

The result of the simulation is presented in separate ﬁgures in order to better observe the increase of the antibody level in the ﬁrst days after vaccination. The levels of antibodies Fig. 1. Antibody curve obtained by the model (line) and experimental data extracted from the literature[34] (dot).

Fig. 2. Antibody curve obtained by the model (line) and experimental data extracted from the literature[34] (dots).

obtained from the literature[34] are in GMT (Geometric Mean Titers) and refer to time intervals after vaccination. The time values used in the graph were obtained by averaging the times of each interval. For example, the ﬁrst point was the 30-45 day post-vaccination interval, the value used was 37 days, the corresponding antibody level was8762.8IU / mL. From a qualitative point of view, one can observe that the values obtained from the computational experiments are very close to the experimental results. Also, the literature reports that the antibody concentration in the bloodstream peaks at about two weeks after vaccination[35], a value close to the one obtained in the computational experiments.

Figure 3 shows the viremia curve obtained by the simula- tion of the model in comparison with the experimental data obtained from the literature [6]. As one can observe, in the simulation the peak viremia value occurs in the ﬁfth day, which is consistent with the literature, which reports that it occurs be- tween four and six days after vaccination[36], as well as with the experimental results[6]. Also, the literature reports that ten days after vaccination, viremia is undetectable[36], which is consistent with the computational results. For some patients, 1429 JOURNAL OF L AT EX CLASS FILES, VOL. 6, NO. 1, OCTOBER 20175 Fig. 3. Viremia curve obtained by model (line) and experimental data obtained from the literature[6](dots). Each dot in time scale represents a distinct patient.

however, the viremia can be detectable, as the experimental results shows[6].

B. Dose-Response An experimental work[6] has reported that “doses from 27,476 IU to 587 IU induced similar seroconversion rates and neutralizing antibodies geometric mean titers (GMTs)”.

Based on this study, a second scenario studies the results of our model when different dose values are administered. The values used in the simulation are the same to those used by the experimental work[6]: 31 IU, 158 IU, 587 IU, 3,013 IU, 20,447 IU and 27,476 IU.

Figure 4 and 5 show the viremia curves obtained by the model for distinct vaccine doses. Figure 5 uses a smaller scale to allow the visualization of the simulated viremia curve obtained after administration of the dose with 587 IU (represented by diamonds).

As one can observe, all doses greater than 3,013 IU produce high levels of viremia. Although the viremia increases with the use of doses with higher concentrations, the antibody response presents a very subtle difference, as one can observe in Figure 6. The 587 IU dose, which presented a much smaller, unremarkable viremia in Figure 4 compared to the doses with higher concentrations, was also able to induce an antibody response similar to those induced by formulations with higher concentrations.

Figure 7 presents the antibody curves obtained by simu- lating 4,000 days after vaccination. The results suggest that the duration of immunity does not appear to be affected by the use of vaccine formulations with distinct concentrations:

all doses above 587 IU present similar results. Although the reference paper has studied the duration of immunity for a smaller period of time, approximately 10 months after vaccination[6], its conclusions were similar to the one obtained by the computational experiments: “GMTs of each group were not statistically different from the reference vaccine.”.

The computational results are also in agreement with other studies. The ﬁrst one[37] concluded: “there was no correlation between the level and duration of detectable 17D viremia Fig. 4. Viremia curves obtained by the model when distinct initial values of V (vaccine virus) are used. The values represent distinct vaccine doses. For doses equal to 31 IU and 158 IU, the viremia was equal to zero.

Fig. 5. Viremia curves obtained by the model when distinct initial values of V (vaccine virus) are used. The values represent distinct vaccine doses.

The scale was changed in order to better visualize the viremia induced after administration of a dose with 587 IU. For doses equal to 31 IU and 158 IU, the viremia was equal to zero.

and the postvaccination nAb level”. Another paper presents a similar conclusion[38]: ”it was also found that the serological response was not related to virus dose as the titres obtained with high or low doses of virus was at the same level”.

It is noteworthy that doses using 31 IU and 158 IU did not produce viremia nor antibody titers, so the curves are superimposed on the x-axis.

C. Discussion In this work we consider that the vaccine does not cause adverse events, such as Yellow fever vaccine-associated vis- cerotropic disease (YEL-AVD) and Yellow fever vaccine- associated neurotropic disease (YEL-AND), because they are rare.

Although YF has been used in this study, it must be stressed that the concept presented in the mathematical model is generic enough to represent the action of other diseases or vaccines in the human immune system. However, the 1430 JOURNAL OF L AT EX CLASS FILES, VOL. 6, NO. 1, OCTOBER 20176 Fig. 6. Antibody curves generated by the computational model. The model simulates the antibody concentrations during a period of 50 days for different doses of the vaccine. For doses equal to 31 IU and 158 IU, the antibody curves were equal to zero.

Fig. 7. Antibody curves generated by the computational model. The model simulates the antibody concentrations during a period of 4,000 days for different doses of the vaccine. For doses equal to 31 IU and 158 IU, the antibody curves were equal to zero.

initial conditions and parameters were adjusted to describe the Human Immune System (HIS) response to the YF vaccine. The immune response to a distinct vaccine would probably require changes in both initial conditions and parameters values.

The obtainment of experimental data to adjust and validate our model was not a trivial task. Studies on the duration of immunity are difﬁcult to interpret because different groups use distinct methods to evaluate seroprotection. There is no well-established serological value of protection in humans and cellular immunity data is very scarce. Also, the values reported for viremia use distinct units, which cannot be converted into distinct ones due to the different methods used to obtain data.

These factors made it difﬁcult to obtain experimental data compatible with the standards and units used in the model presented in this work, and consequently to use more studies available in the literature to adjust and validate our model.

Since the model parameters represent some biological pro- cesses, such as activation, neutralization and decay, after themodel is validated it will be possible to predict which effect a given value change in a parameter will cause in the model as a whole.

V. C ONCLUSION This work presented a reduced mathematical-computational model to represent the immune response to the YF vaccine.

In other to validate the model, two distinct scenarios were simulated. The ﬁrst one simulates the immune response to the administration of the standard dose of the 17DD-YFV. The second one simulates the immune response to distinct doses of vaccine. Two key values, viremia and antibody level, were collected and compared to the values reported in the literature.

From a qualitative point of view, the results obtained by the computational model reproduced the clinical results that can be found in the literature.

As future works, we would like to improve the qualitative results obtained from our model. Additional computational experiments, such as the effects of a) a booster dose and b) a reduction in the population of TCD8+ naive. Also, a sensitivity analysis will be performed to identify sensitive parameters and to identify connections between change in parameters values and computational results.

A PPENDIX A I NITIAL CONDITION AND PARAMETERS TABLES TABLE I M ODEL PARAMETERS Parameter Value πV 4.0(day −1) c v1 2×10 3(virus titer×day −1) c v2 3×10 1(virus titer) k v1 4.875×10 −4 (day −1 ×A −1) k v2 1.5694×10 −3 (day −1 ×T −1 E) αA P 2.5×10 −3 (day −1) β AP 3.0×10 −1 (day −1) k AP1 1.0(dimensionless) k AP2 2×10 2(virus titer) δ APM 5.38×10 −1 (day −1) α TN 2.17×10 −4 (day −1) π T 1×10 −2 (day −1) k TE 1×10 −5 (day −1) δ TE 1×10 −1 (day −1) α B 6.0(day −1) π B 1.77×10 −3 (day −1) β S 6.72×10 −1 (day −1) β L 8.05×10 −3 (day −1) β BM 1×10 −3 (day −1) δ S 2.0(day −1) δ L 2.22×10 −4 (day −1) γ M 1.95×10 −6 (day −1) k BM1 1×10 −5 (day −1) k BM2 10.0(B M) π AS 5×10 −1 (day −1) π AL 1.7×10 −1 (day −1) δ A 4×10 −2 (day −1)* *Value extracted from [39] and [40]apud[41] 1431 JOURNAL OF L AT EX CLASS FILES, VOL. 6, NO. 1, OCTOBER 20177 TABLE II M ODEL VARIABLES AND INITIAL VALUES Variable Description Initial valueVVaccine virus 27476 A P Immature APCs10 3* A PM Mature APCs 0 T N Naive CD8+ T cells10 3* T E Effectors CD8+ T cells 0 BB cells10 3* P S Short-lived plasma cells 0 P L Long-lived plasma cells 0 B M Memory B cells 0 AAntibodies 0 *Values based on [41] ACKNOWLEDGMENT The authors would like to express their thanks to CAPES, CNPq, FAPEMIG and UFJF for funding this work.

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