A qualitatively validated mathematical-computational model of the immune response to the yellow fever vaccine

Transcription

1 Bonin et al. BMC Immunology (2018) 19:15 RESEARCH ARTICLE Open Access A qualitatively valiate mathematical-computational moel of the immune response to the yellow fever vaccine Carla R. B. Bonin 1*, Guilherme C. Fernanes 2, Rorigo W. os Santos 1 an Marcelo Lobosco 1 Abstract Backgroun: Although a safe an effective yellow fever vaccine was evelope more than 80 years ago, several issues regaring its use remain unclear. For example, what is the minimum ose that can provie immunity against the isease? A useful tool that can help researchers answer this an other relate questions is a computational simulator that implements a mathematical moel escribing the human immune response to vaccination against yellow fever. Methos: This work uses a system of ten orinary ifferential equations to represent a few important populations in the response process generate by the boy after vaccination. The main populations inclue viruses, APCs, CD8+ T cells, short-live an long-live plasma cells, B cells an antiboies. Results: In orer to qualitatively valiate our moel, four experiments were carrie out, an their computational results were compare to experimental ata obtaine from the literature. The four experiments were: a) simulation of a scenario in which an iniviual was vaccinate against yellow fever for the first time; b) simulation of a booster ose ten years after the first ose; c) simulation of the immune response to the yellow fever vaccine in iniviuals with ifferent levels of naïve CD8+ T cells; an ) simulation of the immune response to istinct oses of the yellow fever vaccine. Conclusions: This work shows that the simulator was able to qualitatively reprouce some of the experimental results reporte in the literature, such as the amount of antiboies an viremia throughout time, as well as to reprouce other behaviors of the immune response reporte in the literature, such as those that occur after a booster ose of the vaccine. Keywors: Computational vaccinology, Yellow fever, Mathematical moeling, Computational moeling, Immune system, Orinary ifferential equations Backgroun Mathematical an computational moeling is constantly evolving tool, which can be applie to many istinct research areas, such as Biology, Physics, Chemistry, Engineering, Biomechanics, Climate Moeling, tsunami an earthquake preiction, among others [1 23]. With this type of tool, the phenomenon uner stuy is represente by mathematical equations which can be solve using computational simulators. The use of such mathematicalcomputational moels can help reuce costs, time, risks an volunteers involve in the research. However, to *Corresponence: rezenebonin@gmail.com 1 Grauate Program in Computational Moeling, Feeral University of Juiz e Fora, Juiz e Fora, Brazil Full list of author information is available at the en of the article achieve these objectives, the moels must be very reliable. Since moels are always an abstraction of reality, using simplifications to eal with complexities, many factors that can contribute to the real phenomenon may be ignore. Mathematical moels have been use for many years to represent various aspects of the immune system an relate pathologies, but their application to escribe the effects of vaccines has been rather limite [24]. The term computational vaccinology has been use to refer to computer-aie vaccine esign [25 28], an its objective is to use ifferent moeling techniques to ai the evelopment an improvement of vaccines at ifferent stages of their esign processes. The Author(s). 2018Open Access This article is istribute uner the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricte use, istribution, an reprouction in any meium, provie you give appropriate creit to the original author(s) an the source, provie a link to the Creative Commons license, an inicate if changes were mae. The Creative Commons Public Domain Deication waiver ( applies to the ata mae available in this article, unless otherwise state.

2 Bonin et al. BMC Immunology (2018) 19:15 Page 2 of 17 In 1796 Ewar Jenner introuce vaccination against smallpox, which was a major health problem at the time. Jenner observe that milkmais were protecte from smallpox after having suffere from cowpox, an conclue that cowpox coul be use as a eliberate mechanismofprotectionagainstsmallpox[29]. Jenner inoculate an 8-year-ol boy, James Phipps, with cowpox. Subsequently, Jenner inoculate the boy again, this time with smallpox, an he i not contract the isease [29]. Jenner conclue that protection was complete. This is the key of vaccination: expose the boy to antigens from pathogens, in orer to stimulate the prouction of antiboies an efense cells against a specific isease. Much of the work on computational vaccinology is relate to the process of creating a new vaccine, such as in the selection of the best strains for use. In a previous paper [30] we propose a new use for computational vaccinology, i.e. in the clinical evelopment stage. With the use of mathematical an computational moels, it is possible to experiment, in silico, ifferent scenarios relate to vaccination, to aress important questions that remain unanswere. The Yellow Fever (YF) vaccine, available since 1937 [31], is mae from live attenuate virus still capable of triggering an immune response an inucing the prouction of antiboies an memory cells. Live-virus vaccines inuce an immune response similar to that obtaine with exposure to wil virus, but the risk of presenting characteristic symptoms of the isease an its complications, or eath, ue to vaccination, is extremely small. The YF vaccine is consiere an effective an safe vaccine, with high ocumente seroconversion rates an low rates of averse events. It has been effectively use to control a non-eraicable isease an is one of the vaccines that can benefit from the new use of computational immunology. The reason for this is manifol. Although recognize as an effective an safe vaccine, some questions remain unanswere or poorly unerstoo, an coul be reassesse using new technologies an tools. As the vaccine was evelope ecaes ago, some steps of its evelopmental processes were establishe empirically. A goo example is the optimal ose require for immunization. What is the vaccine ose with the best immunogenicity/reatogenicity ratio? There are clinical stuies esigne to evaluate this [32], but these stuies require time an resources, an there are methoological restrains to test several ifferent oses. With the use of mathematical an computational moeling techniques, it is possible to evaluate a larger spectrum of oses in a much shorter time, using far less resources. Another controversial issue is the nee for a booster ose. Using mathematical an computational moeling, it is possible to simulate, for an iniviual, what his/her antigen levels will be years in the future, in a few minutes, to assess the uration of immunity an the nee for booster ose aministration, taking into account ifferences among iniviuals an oses,to help in the esign of prospective cohort stuies. Despite being consiere a safe vaccine, there are rare serious averse events that nee to be reassesse, such as viscerotropic an neurotropic events. There are also questions regaring the safety for vaccinating specific populations such as the elerly, people living with Human Immunoeficiency Virus (HIV)/AIDS an other immunocompromise populations. Because the YF vaccine is a live-attenuate virus vaccine, there is a small but not insignificant risk of occasional higher viral replication relate either to vaccine virus attenuation aspects or an inability of the immune system to control the vaccine virus replication. Recently, YF outbreaks were recore in Angola an the Democratic Republic of Congo (DRC), with the latest outbreak still unerway in Brazil, starting in December From December 2016 to February 22, 2017, 1,345 suspecte cases were recore, of which 295 have been confirme, an 215 eaths reporte to the Brazilian Ministry of Health [33]. YF is not an eraicable isease because of its sylvatic cycle. Reporte cases in the Brazilian outbreaks were classifie as sylvatic YF, but the risk of urban YF reintrouction is imminent ue to high levels of Aees aegypti infestation in Brazilian cities where vaccination coverage is not routinely recommene. Worl stocks an YF vaccine prouction capacity are a logistic concerns which coul impact the control of isease transmission, particularly in large outbreaks. In Kinshasa, capital of the DRC, fractional oses of the YF vaccine were aministere for outbreak control [34]. Concerne about the risk of a global epiemic, the WHO launche in April 2017 a strategy calle Eliminate Yellow fever Epiemics (EYE), which aims to eliminate YF epiemics in the worl by 2026 [34]. Through early etection an rapi an appropriate response, it is possible to minimize suffering, amage an propagation [34]. This strategy has three goals: protect populations at risk, prevent the international sprea of YF an contain outbreaks quickly. To achieve these goals, the strategy suggests actions on ifferent fronts, incluing research an evelopment of better tools an practices. Assessing ata about optimal vaccine ose an uration of immunity coul help the esign of new vaccination strategies for isease control. This work, therefore, presents a first step towars an ieal scenario to simulate istinct situations relate to the use of the YF vaccine: a qualitatively valiate mathematical-computational moel of the immune response to the YF vaccine. The moel consiers the

3 Bonin et al. BMC Immunology (2018) 19:15 Page 3 of 17 major populations of Human Immune System (HIS) cells an molecules important in the process of immunity acquisition, such as Antigen Presenting Cells (APCs), B an T lymphocytes, an antiboies, which are consiere the main marker of immunity. The moel was then evaluate using istinct scenarios, an was successful in qualitatively reproucing experimental results reporte in the literature. This work is organize as follows. First, Section Relate works presents relate works one in this fiels. Section Methos presents the mathematical an computational moels use to reprouce the immune system response to the YF vaccine. The results are then presente in Section Results an iscusse in Section Discussion, an finally SectionConclusion presentsourconclusions. Relate works The use of mathematical an computational moels to help vaccine evelopment is not new. In fact, several works use computational tools to ai vaccine esign. For example, epitope-mapping algorithms have been use for vaccine esign since the 1980s [35]. Since then, new computational tools have been use for selection of vaccine targets [36 44]. Most of the works focuses on using mathematical an computational tools to preict epitopes [45] or to evelop virtual screening approaches (i.e, the ientification of relevant antigens) [46 49]. This traitional use of computational vaccinology is relate to pre-clinical evelopment. This work focuses on the evelopment of mathematical an computational moels that can be use in the clinical evelopment stage, i.e., when the vaccine is firsttesteinhumans.wearguethatitispossibletocarry out some experiments in silico, reucing the search space for experiments in vivo or in vitro, an it is possible to eliminate, reinforce or weaken hypotheses an to propose new stuies, thus saving time an resources. Several computational moeling techniques applie to vaccination are analyze an iscusse by Pappalaro et al. [24]. The authors escribe what mathematical an computational moeling are an how they can ai research in vaccination. Moeling is efine as human activity involving the representation, manipulation, an communication of everyay real worl objects. In their review, two main types of moeling are consiere: Agent-Base Moels (ABM) an mathematical moels. Mathematical moels are mainly base on ifferential equations, whether orinary or partial, with elaye an/or stochastic equations. In this work, we try to qualitatively valiate a simplifie mathematical-computational moel of the immune response to the YF vaccine presente in a previous work [50], which is base on a live, attenuate viral strain. The moel uses Orinary Differential Equations (ODEs) to moel the main cells an molecules relate to aaptive immune response. Another work uses an ODE-base approach to moel the human immune response to vaccination against both YF an smallpox [51] using istinct ata an equations sets, one for each isease. The aim of the authors was to primarily evaluate the ynamics of CD8+ T cells, while this work will evaluate the immune response as a whole. The moel propose here iffers from that presente by Le et al. [51], since it consiers important populations at each stage of the immune response to YF vaccination, from virus inoculation to APC antigen presentation an consequent activation of lymphocytes, generation of antiboies an memory cells. Methos Mathematical moel In this section, we present the moel we propose in a previous work [50], which will be qualitatively valiate in this paper. The moel consists of a system of 10 ODEs representing important populations in the response process generate by the boy after vaccination. The main populations are viruses, APCs, CD8+ T cells, short-live an long-live plasma cells, B cells an antiboies. Only populations relate to the YF vaccine are moele. For example, only B an T cells whose receptors can recognize the YF virus are consiere in the moel. Equation 1 represents the vaccine virus (V ). t V = π vv c v1v c v2 + V k v1va k v2 VT E (1) The virus can not proliferate by itself, it nees to infect a cell an use it as a factory for new viruses. This is implicitly consiere in the term π v V, which represents the multiplication of the virus in the boy, with a prouction rate of π v.theterm c v1v c v2 +V enotes a non-specific viral clearance by the innate immune system. This function is similar to the Hill family of equations [52]. ( ) x k 2 y = k 1 k k xk 2 The above equation is a generalization of the hyperbolic saturation function. The parameter k 1 scales the maximum value to which the function is asymptotic, k 2 is a shape parameter an k 3 is analogous to the half-saturation constant. If k 2 = 1, the Michaelis-Menten function is prouce [53]. The term k v1 VA enotes specific viral clearance ue to antiboy signaling, where k v1 is the clearance rate. The term k v2 VT E enotes specific viral clearance ue to the inuction of apoptosis of cells infecte by the YF virus, where k v2 is the clearance rate. APCs are all cells that isplay antigens complexes on their surfaces, such as enritic cells an macrophages. Two stages of APCs were consiere: immature an

4 Bonin et al. BMC Immunology (2018) 19:15 Page 4 of 17 mature. The first stage, immature APCs (A P ), is escribe by Eq. 2. t A ( P = α AP (A P0 A P ) β AP A P kap1 + tanh ( )) V k AP2 The term α AP (A P0 A P ) enotes the homeostasis of APCs, where ( α AP is the homeostasis rate. The term β AP A P kap1 + tanh ( )) V k AP2 enotes the conversion of immature APCs into mature ones. Therefore, the same term appears in Eq. 3 with positive sign. The ( constant β AP represents the conversion rate an kap1 + tanh ( )) V k AP2 is a sigmoial saturation function in the form of a hyperbolic tangent. Equation 3 represents the mature APCs (A PM ). (2) t A PM = β AP A P ( kap1 + tanh ( V k AP2 )) δapm A PM The first term, as explaine, enotes the ynamics of APCs maturation. The secon term, δ APM A PM, enotes the natural ecay of the mature APCs, where δ APM is the ecay rate. Equation 4 represents the population of naïve CD8+ T cells (T N ). t T N = α TN (T N0 T N ) π T A PM T N (4) The term α TN (T N0 T N ) represents the homeostasis of CD8+ T cells, where α TN is the homeostasis rate. The term π T A PM T N enotes the activation of naïve the CD8+ T cells, where π T is the activation rate. Therefore, the same term appears in Eq. 5 with positive sign. Equation 5 represents the effector CD8+ T cell population (T E ). t T E = π T A PM T N + k TE A PM T E δ TE T E (5) The term k TE A PM T E represents the proliferation of effector CD8+ T cells. The term δ TE T E represents the natural eath of these cells, with δ TE representing its ecay rate. Equation 6 represents B cells (B), both naïve an effector ones. These populations were not consiere separately in orer to simplify the moel. t B = α B(B 0 B) + π B A PM B β S A PM B β L A PM B β BM A PM B (6) The term α B (B 0 B) represents the B cells homeostasis, where α B is the homeostasis rate. The term π B A PM represents the proliferation of the active B cells. The terms β S A PM B, β L A PM B an β BM A PM B enote the portions of active B cells that ifferentiate into short-live plasma cells, long-live plasma cells an memory B cells, (3) respectively. These terms will appear with positive sign in Eqs. (7), (8) an(9). The activation rates are respectively given by β S, β L an β BM. Equation 7 represents the short-live plasma cells (P S ). t P S = β S A PM B δ S P S (7) The term δ S P S enotes the natural ecay of short-live plasma cells, where δ S is the ecay rate. Equation 8 represents the long-live plasma cells (P L ). t P L = β L A PM B δ L P L + γ M B M (8) The term δ L P L enotes the natural ecay of long-live plasma cells, with δ L representing the ecay rate. The term γ M B M represents the prouction of these cells by memory B cells, where γ M is the prouction rate. Equation 9 correspons to memory B cells (B M ). t B M = β BM A PM B + k BM1 B M ( 1 B M k BM2 ) γ M B M The term k BM1 B M (1 B M represents the logistic growth of memory B cells, i.e., there is a limit to this growth. The constants k BM1 an k BM2 represent the growth rate an limits, respectively. Equation 10 represents the antiboies. The terms π AS P S an π AL P L are the prouction of the antiboies by shortlive an long-live plasma cells, respectively. The prouction rates are given by π AS an π AL,respectively.The term δ A A enotes the natural ecay of these cells, where δ A is the ecay rate. t A = π ASP S + π AL P L δ A A (10) The moel presente in this paper was base on an earlier stuy [54], which escribe a mathematical moel to represent the human immune response to an infection by YF virus. Therefore, the first ifference is that this paper focus on moelling the effects of the YF vaccine aministere subcutaneously. The previous work [54] moeletheimmuneresponse to the YF virus from infection of epithelial cells to secretion of antiboies, consiering various populations of cells an molecules, in ifferent stages an compartments. There were 19 ODEs ivie into two compartments: one representing the tissue where the virus proliferates an the other the lymph noes. In orer to consier all the cells an molecules, the moel became complex. Another issue is relate to its ajustment to reprouce some behaviors escribe in the literature: as the number of equations an parameters increases, so oes the amount of ata an information neee to ajust the moel. The secon ifference between the two moels is k BM2 ) (9)

5 Bonin et al. BMC Immunology (2018) 19:15 Page 5 of 17 that the moel reprouce here [50] reuces the number of equations from 19 to 10. The reuce moel reprouce in this work consiers only the main populations of cells an molecules involve in the response to the vaccine, an abstracts some etails that are not crucial to represent the behavior of the immune response, such as the representation of istinct compartments. In aition, some populations were not consiere because no experimental ata are available to valiate the simulations, such as CD4+ T cells. In the near future, more cells or molecules can be reintrouce in the moel if their roles are important to explain or represent behaviors that the reuce moel [50] coul not represent. Table 1 summarizes the main ifferences between the moel presente in previous work [54] an the one evaluate in this work. It is important to remember that a mathematical moel is an abstraction of reality an therefore simplifications are always necessary. This is accentuate when the target of the moel is the HIS response, a complex network that involves many tissues, organs an cells an that performs several processes. The level of abstraction epens on the purpose of the moel. HIS can be seen at various levels, from the level of substances prouce by cells, such as cytokines, to the level of cells an molecules, as in the case of the simplifie moel [50]. It also can reach the level of an entire population, as in the case of the epiemiological moels. The use of a simplifie moel oes not imply that it can reprouce only a limite number of scenarios. The point is that some of the aspects not irectly inclue in the moel may be inirectly present, as constants. As such, the choice of istinct values for some constants may represent istinct behaviors in the system. Computational moel For the resolution of the ODEs system, a coe was implemente using Python programming language, which inclues libraries for easily solving complex mathematical problems. The library chosen was SciPy [55]. This library has a package calle integrate. One of the functions available in this package is calle oeint, an it is use to numerically solve a system of ODEs. The choice of the numerical metho to be use is mae automatically by the function base on the characteristics of the equations. The function uses an aaptive scheme for both the integration step an the convergence orer. The function can Table 1 Main ifferences between the moels Previous Number of equations Number of parameters Number of compartments 2 1 Number of populations consiere 10 8 Current solve the ODEs system using either the Backwar Differentiation Formula (BDF) or the Aams metho [56]. BDF is use for stiff equations an the implicit Aams metho is use otherwise. The experiments were performe using Python version using the Spyer Integrate Development Environment (IDE). The execution environment was compose by an Intel Core i5 1.6 GHz processor, with 8 GB of RAM. The system runs macos Sierra version Results In orer to qualitatively valiate our moel, four experiments were carrie out. The first one simulates a scenario where an iniviual was vaccinate against YF for the first time. The stanar ose of the vaccine was use in this scenario. The results of the simulation were then compare to experimental ata obtaine from the literature. The secon scenario assesses the immune response following the aministration of a booster ose ten years after the first ose. The stanar ose of the vaccine was also use in this scenario. Although there are no experimental ata from the literature that coul be use for comparison purposes, there is research reporting that the expecte behavior is an iniviual to present a lower viremia, an to raise antiboies levels to levels higher than those obtaine after the aministration of the first ose [57]. The thir scenario simulates the immune response to the YF vaccine in iniviuals with ifferent levels of naïve CD8+ T cells prior to vaccination. This simulation aims to evaluate the importance of this population of cells in the control of viremia an in the prouction of antiboies. The fourth scenario is base on an experimental stuy [32] in which istinct oses of the YF vaccine were teste. Compare to the stanar ose, the experimental stuy reporte that, to some extent, the reuction i not significantly affect the percentage of sero-conversion. In this scenario, computational experiments are execute several times, using istinct values for the vaccine oses. For comparison purposes, the computational experiments were carrie out using the same values of the experimental stuy [32]. In aition to evaluating the response of the moel when ifferent oses are aministere, we performe a sensitivity analysis of the parameters relate to virus ynamics. The main results of this analysis are shown in subsection Sensitivity analysis. All the initial values use for the variables as well as the moel parameters are presente in Tables 2 an 3. The parameters were ajuste, except for δ A, whose value was extracte from the literature. In general, the literature reports two istinct sets of experimental ata. The first one is viremia along time, i.e., the amount of virus present in the bloostream. The secon ataset reporte in the literature is the antiboy levels

6 Bonin et al. BMC Immunology (2018) 19:15 Page 6 of 17 Table 2 Moel variables an initial values Variable Description Initial value V Vaccine virus 27,476 A P Immature APCs 10 3 a A PM Mature APCs 0 T N Naïve CD8+ T cells 10 3 a T E Effectors CD8+ T cells 0 B B cells 10 3 a P S Short-live plasma cells 0 P L Long-live plasma cells 0 B M Memory B cells 0 A Antiboies 0 Values marke with a were base on [68] along time. Therefore, in orer to valiate the moel, the values obtaine by Eqs. 1 an (10) willbecompareto experimental values foun in the literature. First vaccination This section presents the computational results of a simulation in which an iniviual was vaccinate against YF for the first time. In this computational experiment, a value equal to 27,476 International Units (IU) was use as the stanar amount of virus particles present in the vaccine. This value is set as the initial conition of the virus population represente in Eq. 1 by V (all initial conitions are presente in Table 2). In fact, in the case of the 17DD-YFV the amount of virus particles varies epening on the vaccine lot number, ranging from 2.3 to 12 times the minimum value require by the WHO [32]. The 17DD-YFV is the YF vaccine evelope by Bio-Manguinhos/Fiocruz, one of the three proucers prequalifie by the WHO to supply vaccines to international agencies. Figures 1 an 2 show the comparison of the antiboy curve generate as a result at 100 an 4,000 simulation ays, respectively, with the experimental results from the literature [58]. Theresultofthesimulationispresentein separate figures in orer to better observe the increase of the antiboy level in the first ays after vaccination. The levels of antiboies obtaine from the literature [58] are in Geometric Mean Titers (GMT) an refer to time intervals after vaccination. The time values use in the graph were obtaine by averaging the times of each interval. For example, the first point was the ays post-vaccination interval, the value use was 37 ays, the corresponing antiboy level was 8,762.8 IU/mL. Figure 3 shows the viremia curve obtaine by the simulation of the moel in comparison with the experimental ata obtaine from the literature [32]. Booster ose The aministration of a booster ose was simulate 10 years after the aministration of the first ose. The simulation is quite simple. As in the previous scenario, the initial value of V was set to 27,476 to simulate the aministration of the first ose. Then, the simulation is execute until ay number 3,650, when the value of variable V is set again to 27,476. The ifference from the beginning of the simulation is that this time antiboies an memory cells that were prouce after the first ose are present. Figures 4 an 5 present the antiboy curves. The results of the booster ose simulation were shifte to facilitate its Fig. 1 Antiboy curve obtaine by the moel (line) an experimental ata extracte from the literature [58] (ots)

7 Bonin et al. BMC Immunology (2018) 19:15 Page 7 of 17 Fig. 2 Antiboy curve for the first 100 ays obtaine by the moel (line) an experimental ata extracte from the literature [58] (ots) comparison to the results of the first ose. The soli line curve represents the response to the first ose, while the ashe curve represents the response to the booster ose. Figure 6 showstheviremiacurves15aysafterthe aministration of the vaccine. Naïve T CD8+ levels The clearing of the intracellular pathogen via CD8+ cytotoxic T lymphocytes appears to be important for recovering from primary viral infection. Base on this observation foun in the literature [57], the authors ecie to compare the immunological response given by the simulation of the moel with ifferent levels of CD8+ T cells in orer to evaluate the impact of this population of lymphocytes on viral clearance. Figure 7 shows the viremia curves for ifferent initial CD8+ T cell values, an Figs. 8 an 9 show antiboy levels. Dose-response An experimental work [32] reportethat osesfrom 27,476 IU to 587 IU inuce similar seroconversion rates an neutralizing antiboies geometric mean titers (GMTs). Base on this stuy, a secon scenario analyzes the results of our moel when ifferent ose values are aministere. The values use in the simulation Fig. 3 Viremia curve for all perio obtaine by moel (line) an experimental ata obtaine from the literature [32] (ots).each ot in time scale represents a istinct patient

8 Bonin et al. BMC Immunology (2018) 19:15 Page 8 of 17 Fig. 4 Antiboy curves 50 ays after the first vaccination (full blue line) an after the booster ose (ashe green line) are the same to those use by the experimental work [32]: 31 IU, 158 IU, 587 IU, 3,013 IU, 20,447 IU an 27,476 IU. Figures 10 an 11 show the viremia curves obtaine by the moel for istinct vaccine oses. Figure 11 uses a smaller scale to allow the visualization of the simulate viremia curve obtaine after aministration of the ose with 587 IU (represente by iamons). Figure 12 presents the antiboy curves generate by the computational moel uring a 50-ay perio for ifferent oses of the vaccine, while Fig. 13 presents the antiboy curves obtaine by simulating 4,000 ays after vaccination. Doses using 31 IU an 158 IU i not prouce viremia nor antiboy titers, so the curves are superimpose on the x-axis. Sensitivity analysis The sensitivity analysis ientifies the impact cause by the variation of parameters an initial conitions of the mathematical moel in the epenent variables [59]. If a small change in a parameter is responsible for a rastic change in the result of the problem, it means that the problem is sensitive to that particular parameter. Otherwise, this parameter has a low impact on the moel. This analysis is use to help the unerstaning of the mathematical Fig. 5 Antiboy curves 250 ays after the first vaccination (full blue line) an after the booster ose (ashe green line)

9 Bonin et al. BMC Immunology (2018) 19:15 Page 9 of 17 Fig. 6 Viremia curves 15 ays after the first vaccination (full blue line) an after the booster ose (ashe green line) moel, since it allows the ientification of the most relevant parameters, that is, the values of these parameters must be carefully efine. A brute-force approach was use to examine the influence of all parameters of the moel. The parameter values were varie from -10% to + 10% (in 5% intervals) from their original values. The original values are presente in Table 3 an were obtaine after ajustment using experimental ata from the literature [32, 58, 60, 61]. For each parameter, the curves that simulate the level of antiboies an viremia were evaluate, since they are the main populations of interest an on which there is experimental ata. Only the parameters to which the moel was most sensitive will be presente in this section. As expecte, the moel was sensitive to most of the parameters of the equation escribing the virus ynamics (Eq. 1). The antiboy curves were not significantly affecte by them, therefore only the viremia curves will be presente. Figure 14 shows the istinct viremia curves obtaine for ifferent values of π v. This parameter represents the viral replication rate an, as one coul expect, the moel was very sensitive to it. The more the virus can multiply, the more ifficult it is for the HIS to contain it an the higher the viremia level is. Figures 15 an 16 present the viremia curves obtaine by simulation of the moel for ifferent values of k v1 an k v2, respectively. These parameters represent the Fig. 7 Viremia curves for ifferent initial conitions of CD8+ T cells

10 Bonin et al. BMC Immunology (2018) 19:15 Page 10 of 17 Fig. 8 Curves of antiboy levels obtaine by the 50-ay simulation of the moel, for ifferent initial values of CD8+ T cells neutralization rates of the YF virus per unit of neutralizing antiboies (k v1 )ancd8+tcells(k v2 ). If we consier that the parameter k v1 represents the ability of antiboies to neutralize the YF virus, its value can be unerstoo as its affinity/specificity to the YF virus an, if it is more specific an can neutralize the virus better, viral replication will be better controlle an viremia will be lower. It is easy to unerstan why the moel, especially the viremia curve, is so sensitive to parameter k v2.itrepresents the ability of CD8+ T cells to inuce apoptosis of an infecte cell. Thus, the higher this ability, the fewer the number of infecte cells. Since YF viruses use infecte cells to reprouce themselves, the viremia level is reuce. Discussion As can be observe from Figs. 1 an 2, from a qualitative point of view, the values obtaine from the computational experiments are very close to the experimental results. Also, the literature reports that the antiboy concentration in the bloostream peaks at about two weeks after vaccination [62], a value close to the one obtaine in the computational experiments. Figure 3 shows that, in the simulation, the peak viremia value occurs on the fifth ay, consistent with the literature, which reports that it occurs between four an six ays after vaccination [63], as well as with experimental results [32]. In aition, the literature reports that ten ays after vaccination, viremia is unetectable [63], which is consistent with the computational results. For some patients, Fig. 9 Curves of antiboy levels obtaine by the 4,000-ay simulation of the moel, for ifferent initial conitions of CD8+ T cells

11 Bonin et al. BMC Immunology (2018) 19:15 Page 11 of 17 Fig. 10 Viremia curves obtaine by the moel when istinct initial values of V (vaccine virus) are use. The values represent istinct vaccine oses. For oses equal to 31 IU an 158 IU, viremia was equal to zero however, viremia can be etectable, as one experimental result has shown [32]. Figures 4 an 5 show that the behavior escribe in the literature [57] resembles that obtaine by the simulation of the moel: the neutralizing antiboies levels are slightly increase after the booster ose. The viremia curves shown in Fig. 6 emonstrate that viremia reaches much lower levels after the aministration of the booster ose than those seen after the aministration of the first ose. Although the viremia level is lower after the aministration of the booster ose than the first ose, it is not possible to say that this level is below the threshol of etectable viremia as escribe in the literature: viremia has not been ocumente in persons receiving a booster ose of YF vaccine [64]. This occurs because of the use of istinct units to measure viremia, an the fact that it is not trivial to convert one unit to another. For this reason, this work consiers only qualitative results, an not quantitative ones. This result still nees to be quantitatively compare to experimental ata in orer to better valiate our moel, but the qualitative behavior presente is satisfactory since the level Fig. 11 Viremia curves obtaine by the moel when istinct initial values of V (vaccine virus) are use. The values represent istinct vaccine oses. The scale was change to better illustrate the viremia inuce after aministration of a ose with 587 IU. For oses equal to 31 IU an 158 IU, viremia was equal to zero

12 Bonin et al. BMC Immunology (2018) 19:15 Page 12 of 17 Fig. 12 Antiboy curves generate by the computational moel. The moel simulates the antiboy concentrations uring a 50-ay perio for ifferent oses of the vaccine. For oses equal to 31 IU an 158 IU, the antiboy curves were equal to zero of antiboies an/or memory cells was able to contain viral replication more efficiently than it was observe for a naïve iniviual. Furthermore, the viremia level was almost 4 times lower for the booster ose than for the first ose. As show in Fig. 7, as the number of CD8+ T cells is reuce, the viremia increases an lasts longer, reinforcing the importance of CD8+ T cells in the control of viral replication. Figures 8 an 9 show that this variation in initial CD8+ T cell values i not significantly affect antiboy prouction nor uration of immunity. As observe in the Figs. 10 an 11, allosesgreater than 3,013 IU prouce high levels of viremia. Although the viremia increases with the use of oses with higher concentrations, the antiboy response presents a very subtle ifference, as can be observe in Fig. 12. The 587 IU ose, which presente a much smaller, unremarkable viremia (Fig. 10) compare to the oses with higher concentrations, was also able to inuce an antiboy response similar to that inuce by formulations with higher concentrations. Fig. 13 Antiboy curves generate by the computational moel. The moel simulates the antiboy concentrations uring a 4,000-ay perio for ifferent oses of the vaccine. For oses equal to 31 IU an 158 IU, the antiboy curves were equal to zero

13 Bonin et al. BMC Immunology (2018) 19:15 Page 13 of 17 Table 3 Moel parameters Parameter Equation Description Value π V 1 Viral replication rate 4.0 ( ay 1) c v1 1 Maximum viral clearance rate by the innate system ( virus titer ay 1) c v2 1 Half saturation constant (virus titer) k v1 1 YFV neutralization rate per unit of neutralizing antiboies ( ay 1 A 1) ( ) k v2 1 YFV neutralization rate per unit of CD8+ T cells ay 1 T 1 E αa P 2 Homeostasis rate of immature APCs ( ay 1) β AP 2,3 APC maturation rate ( ay 1) k AP1 2,3 With β AP, efines the minimum prouction rate of A PM 1.0 (imensionless) k AP2 2,3 Defines the saturation point of A PM (virus titer) δ APM 3 Death rate of mature APCs ( ay 1) α TN 4 Homeostasis rate of CD8+ T cells ( ay 1) π T 4,5 Activation rate of naïve CD8+ T cells ( ay 1) k TE 5 Replication rate of effector CD8+ T cells ( ay 1) δ TE 5 Death rate of effector CD8+ T cells ( ay 1) α B 6 Homeostasis rate of B cells 6.0 ( ay 1) π B 6 Replication rate of active B cells ( ay 1) β S 6,7 Differentiation rate of active B cells in short-live plasma cells ( ay 1) β L 6,8 Differentiation rate of active B cells in long-live plasma cells ( ay 1) β BM 6,9 Differentiation rate of active B cells in memory B cells ( ay 1) δ S 7 Death rate of short-live plasma cells 2.0 ( ay 1) δ L 8 Death rate of long-live plasma cells ( ay 1) γ M 8,9 Differentiation rate of memory B cells in long-live plasma cells ( ay 1) k BM1 9 Proliferation rate of memory B cell ( ay 1) k BM2 9 Maximum growth constant 10.0 (B M ) π AS 10 Antiboy secretion rate (short-live plasma cell) ( ay 1) π AL 10 Antiboy secretion rate (long-live plasma cell) ( ay 1) δ A 10 Antiboy eath rate ( ay 1) a The value marke with a was extracte from [66, 67] apu [68] Fig. 14 Sensitivity analysis of the parameter π v in the viremia curves 15 ays after vaccination

14 Bonin et al. BMC Immunology (2018) 19:15 Page 14 of 17 Fig. 15 Sensitivity analysis of the parameter k v1 (neutralization rate of YF virus per unit of neutralizing antiboies) in viremia curves 15 ays after vaccination Figure 13 suggests that the uration of immunity oes not appear to be affecte by vaccine formulations with istinct concentrations: all oses above 587 IU present similar results. For now, it seems that yellow fever vaccinecanbeuseinmuchlowerosesthanusual:the computational experiments inicate that vaccine formulations with 587 IU can prouce the same seroconversion rates than the 27,476 IU formulation, in accorance to the experimental ata [32]. Although the reference paper investigate the uration of immunity for a smaller perio of time, approximately 10 months after vaccination [32], its conclusions were similar to those obtaine by the computational experiments: GMTs of each group were not statistically ifferent from the reference vaccine". The computational results are also in agreement with other stuies. One paper [57] conclue: there was no correlation between the level an uration of etectable 17D viremia an the postvaccination nab level". Another paper presents a similar conclusion [65]: the serological response was not relate to virus ose as the titres obtaine with high or low oses of virus was at the same level". The results of the simulations for these four scenarios have shown that the moel was able to reprouce, from a qualitative perspective, clinical results reporte in the literature, espite all simplifications [50]. Some aspects not irectly inclue in the moel may be inirectly present, as constants. Therefore, the choice of istinct values for Fig. 16 Sensitivity analysis of the parameter k v2 (neutralization rate of YF virus per unit of CD8+ T cell) in viremia curves 15 ays after vaccination

15 Bonin et al. BMC Immunology (2018) 19:15 Page 15 of 17 a constant may represent istinct behaviors in the system. For example, one paper [58] pointsoutthat The ecreasing tren in antiboy titres with the time since vaccination appeare strongly moifie by age". In our moel, the effects of age in the prouction of antiboies coul be reprouce increasing or ecreasing the values use for the antiboy secretion rate (Eq. 10, π AS an π AL ). For this reason, our moel oes not nee to inclue age as one of its parts. The same applies for other aspects of the immune system that are not irectly inclue in the set of equations. In this work we consier that the vaccine oes not cause averse events, such as Yellow fever vaccineassociate viscerotropic isease (YEL-AVD) an Yellow fever vaccine-associate neurotropic isease (YEL-AND), ue to their rarity. This moel was evelope an ajuste base on the immune system response to the YF vaccine, but it shoul be note that the concept presente in the mathematical moel is generic enough to represent the action of other iseases or vaccines in the HIS. For this, changes in both initial conitions an parameters values are probably neee. Obtaining experimental ata to ajust an valiate the moel is not a trivial task. Stuies on the uration of immunity are ifficult to interpret because ifferent groups use istinct methos to evaluate seroprotection. There is no well-establishe serological value of protection in humans an cellular immunity ata are very scarce. Also,asstateabove,thevaluesreporteforviremiause istinct units, which cannot be converte into other units ue to the ifferent methos use to obtain such ata. These factors mae it ifficult to obtain experimental ata compatible with the stanars an units use in the moel presente in this work, an consequently to use more stuies available in the literature to ajust an valiate it. Although the results foun are qualitatively in agreement with the few experimental ata foun in the literature, more tests an refinement of the moel may be neee to ajust it. To o so, experimental ata to better valiate the simulate scenarios nees to be obtaine, in particular for booster ose an CD8+ T cells. With this, it woul be possible to also evaluate the moel from a quantitative perspective an, if necessary, to better ajust it. With more ata available, the moel may be improve, making it more reliable an sufficiently accurate to be use to help answer open questions about YF vaccine. One of the next steps in this work is to reintrouce CD4+ T lymphocytes in the moel. This coul be important to simulate the effects of the YF vaccine in immunosuppresse iniviuals, such as people living with HIV. Since many of the YF enemic countries are in Africa, where the HIV infection rates are also high, the investigation of the best YF vaccination scheme for these iniviuals is relevant, since they have, in general, fewer CD4+ T lymphocytes, which are important to the activation of other lymphocytes an consequently to the prouction of antiboies. This population eserves special attention because the YF vaccine is mae with live virus, therefore the risk of systemic lethal infection exists. Conclusion This work presente the qualitative valiation of a reuce mathematical-computational moel to represent the immune response to the YF vaccine using four istinct scenarios. The first one simulates the immune response to the aministration of the stanar ose of the 17DD- YFV. The secon one simulates the immune response to istinct oses of vaccine. The thir scenario simulates the aministration of a booster ose ten years after the first ose. Finally, we evaluate the impact of changing the CD8+ T cells values. The results of a sensitivity analysis of the moel was also shown. Two populations, virus an antiboies, were the main focus of the simulations because more experimental ata are available an qualitative behaviors are escribe in the literature for these populations. The results of the simulations were collecte an compare to the values reporte in the literature. From a qualitative point of view, the results obtaine by the computational moel satisfactorily reprouce the clinical results. Abbreviations ABM: Agent-base moels; APCs: Antigen presenting cells; BDF: Backwar ifferentiation formula; DRC: Democratic Republic of Congo; EYE: Eliminate yellow fever epiemics; GMT: Geometric mean titers; HIS: Human immune system; HIV: Human immunoeficiency virus; IDE: Integrate evelopment environment; IU: International units; ODEs: Orinary ifferential equations; YF: Yellow fever; YEL-AND: Yellow fever vaccine-associate neurotropic isease; YEL-AVD: Yellow fever vaccine-associate viscerotropic isease Funing Authors woul like to thank the support from CNPq, FAPEMIG, an UFJF. Availability of ata an materials Data sharing not applicable to this article as no atasets were generate or analyse uring the current stuy. Moel parameters an initial conitions use in simulations are inclue in this publishe article. Coe to solve the mathematical moel can be mae available upon request to the authors. Ethics approval an consent to participate Not applicable Authors contributions Conception an esign of the mathematical moel: all authors. Computational implementation of the mathematical moel: CRBB. CRBB has also been involve in rafting the manuscript. All authors rea, revise an approve the final manuscript. Competing interests The authors eclare that they have no competing interests. Publisher s Note Springer Nature remains neutral with regar to jurisictional claims in publishe maps an institutional affiliations.

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