Bacterial Pneumonia Fate Decisions⁎

10th IFAC Symposium on Biological and Medical Systems 10th IFAC Symposium on Biological and Medical Systems São Brazil, September 3-5, 2018 10th Paulo, IFAC Symposium on Biological and Medical Systems São Paulo, Brazil, September 3-5, 2018 Available online at www.sciencedirect.com 10th Paulo, IFAC Symposium on Biological and Medical Systems São Brazil, September 3-5, 2018 São Paulo, Brazil, September 3-5, 2018

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IFAC PapersOnLine 51-27 (2018) 390–395  Bacterial Bacterial Pneumonia Pneumonia Fate Fate Decisions Decisions  Bacterial Pneumonia Fate Decisions  Bacterial Pneumonia ∗∗∗Fate Decisions Esteban A. Hernandez–Vargas Alessandro Boianelli ∗∗ ∗∗ ∗∗

Esteban A. Hernandez–Vargas ∗∗ Alessandro Boianelli ∗∗ ∗ Gustavo ∗ Esteban A. Hernandez–Vargas Alessandro Boianelli ∗∗ ∗ Gustavo Hernandez–Mejia Hernandez–Mejia ∗ ∗ ∗ Esteban A. Hernandez–Vargas Alessandro Boianelli ∗∗ Gustavo Hernandez–Mejia ∗ ∗ Gustavo Hernandez–Mejia ∗ ∗ Frankurt Institute for Advanced Studies (FIAS), for Advanced (FIAS), ∗ Frankurt Institute ∗ Ruth-Moufang-Strasse 1, 60438 FrankfurtStudies am Main, Germany. Frankurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany. ∗ ∗∗ Frankurt Institute for Advanced Studies (FIAS), Department of Drug Discovery Sciences, Boehringer Ingelheim Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany. ∗∗ ∗∗ Department of Drug Discovery Sciences, Boehringer Ingelheim ∗∗ ∗∗ Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany. Pharma GmbH & Co KG, 88397, Biberach, Germany. Department of Drug Discovery Sciences, Boehringer Ingelheim Pharma GmbH & Co KG, 88397, Biberach, Germany. ∗∗ Department of Drug Sciences, Boehringer Ingelheim Pharma GmbH & Discovery Co KG, 88397, Biberach, Germany. Pharma GmbH & Co KG, 88397, Biberach, Germany. Abstract: Abstract: The The influenza influenza A A virus virus (IAV) (IAV) infections infections provide provide aa susceptibility susceptibility to to aa secondary secondary bacterial infection mainly driven by Streptococcus pneumoniae (S. pneumoniae). lethal Abstract: The influenza A virus (IAV) infections provide a susceptibility to aThis secondary bacterial infection mainly driven by Streptococcus pneumoniae (S. pneumoniae). This lethal Abstract: Theofinfluenza A causative virusby(IAV) infections provide a susceptibility to aThis secondary synergy is one the main agents for severe respiratory diseases provoking high bacterial infection mainly driven Streptococcus pneumoniae (S. pneumoniae). synergy is one of the main causative agents for severe respiratory diseases provokinglethal high bacterial infection mainly driven byAlthough Streptococcus pneumoniae (S. pneumoniae). This in lethal rates of hospitalization and death. significant improvements have been made our synergy is one of the main causative agents for severe respiratory diseases provoking rates of hospitalization and death. Although significant improvements have been made inhigh our synergy is of oneIAV of infections, the main causative agentssignificant for severe respiratory diseases provoking high knowledge holistic understanding of the interactions between IAV, bacteria rates of hospitalization and death. Although improvements have been made in our knowledge of IAV infections, holistic understanding of the interactions between IAV, bacteria rates of hospitalization and death. Although significant improvements have beenIAV, madebacteria inthat our and immune modulation remains largely fragmented. Our steady-state analysis suggests knowledge of IAV infections, holistic understanding of the interactions between and immune modulation remains largely fragmented. Our steady-state analysis suggests that knowledge of modulation IAV infections, holistic understanding ofOur the interactions between IAV, bacteria influenza increases substrates as nutrients for bacterial growth affecting only the steady-state and immune remains largely fragmented. steady-state analysis suggests that influenza increases substrates as nutrients for bacterial growth affecting only the steady-state and immune modulation remains largely fragmented. Our steady-state analysis suggests that that the would reach, but more would not directly to bacterial influenza increases substrates for bacterial affecting only the steady-state that the bacteria bacteria would reach, as butnutrients more nutrients nutrients would growth not contribute contribute directly to the the bacterial influenza increases substrates as nutrients for bacterial growth affecting only the steady-state decision to colonize. Furthermore, using a Markov branching process, the hypothesis of cell to that the bacteria would reach, but more nutrients would not contribute directly to the bacterial decision to colonize. Furthermore, using a Markov branching process, the hypothesis of cell to that the bacteria would reach, but more nutrients would not however, contribute directly to the of bacterial cell heterogeneity in single bacterial infection is supported, this plays a minimal role decision to colonize. Furthermore, using a Markov branching process, the hypothesis cell to cell heterogeneity in single bacterial infection is supported, however, this plays a minimal role decision to colonize. using a Markov branchinghowever, process,this the plays hypothesis of cellrole to in case cellthe heterogeneity in Furthermore, single bacterial infection is supported, a minimal in the case of of coinfection. coinfection. cellthe heterogeneity in single bacterial infection is supported, however, this plays a minimal role in case of coinfection. © IFACof(International of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. in 2018, the case coinfection. Federation Keywords: Keywords: Modeling, Modeling, Virus, Virus, Bacteria, Bacteria, Infections, Infections, Influenza, Influenza, Markov Markov Branching Branching Process Process Keywords: Modeling, Virus, Bacteria, Infections, Influenza, Markov Branching Process Keywords: Modeling, Virus, Bacteria, Infections, cal Influenza, Markov Branching Process 1. 1. INTRODUCTION INTRODUCTION cal modeling modeling has has been been particularly particularly important important to to provide provide quantitative in IAV infectionimportant [M¨ohler et to al. provide (2005); 1. INTRODUCTION cal modelinginsights has been particularly quantitative insights in IAV infection [M¨ o hler et al. (2005); 1. INTRODUCTION cal modeling has been particularly important to provide Baccam et al. (2006); Beauchemin and Handel (2011); The influenza A virus (IAV) and Streptococcus pneumoquantitative insights in IAV infection [M¨ o hler et al. (2005); The influenza A virus (IAV) and Streptococcus pneumo- Baccam et al. (2006); Beauchemin and Handel (2011); insights inBoianelli IAV infection ohler et al. (2011); (2005); Mitchell et al. (2011); et al. [M¨ (2015, 2016)] and to niae influenza (S. pneumoniae) are responsible for a large pro- quantitative The A virus (IAV) and Streptococcus pneumoBaccam et al. (2006); Beauchemin and Handel et al. (2011); Boianelli etwith al. and (2015, 2016)] (2011); and to niae (S.ofpneumoniae) are responsible for a worldwide, large pro- Mitchell Baccam al. (2006); Beauchemin Handel capture the dynamics of the virus the immune system The influenza A virus (IAV) and Streptococcus pneumoMitchell et al. (2011); Boianelli et al. (2015, 2016)] and to portion respiratory infection-related deaths niae (S. pneumoniae) are responsible for a large pro- capture the dynamics of the virus with the immune system portion of respiratory infection-related deaths worldwide, Mitchell etal. al. (2011); of Boianelli etwith al. (2015, 2016)] system and al. to [Saenz et (2010); Miao et al. (2010); Pawelek niae[McCullers (S.ofpneumoniae) are responsible for aand large pro- capture see (2006)]. The high morbidity mortalportion respiratory infection-related deaths worldwide, the dynamics the virus the immune et al. (2010); Miao et al. (2010); Pawelek et et al. see [McCullers (2006)]. The high morbidity and mortal- [Saenz capture the dynamics of the and virus with the immune system (2012); Hernandez-Vargas Meyer-Hermann (2012); portion ofofrespiratory infection-related deaths worldwide, ity rates influenza are mainly attributed to secondary see [McCullers (2006)]. The high morbidity and mortal[Saenz et al. (2010); Miao et al. (2010); Pawelek et al. Hernandez-Vargas and Meyer-Hermann (2012); ity rates of influenza are mainly attributed to secondary (2012); etHernandez-Vargas al.et(2010); Miao et al.Meyer-Hermann (2010); Pawelek et al. Dobrovolny al. (2013); Hernandez-Vargas et al. (2014)]. see rates [McCullers (2006)]. highwith morbidity and mortal- [Saenz bacterial pneumonia, particularly the most commonly ity of influenza areThe mainly attributed to secondary (2012); and (2012); Dobrovolny et al. (2013); Hernandez-Vargas et al.modeling (2014)]. bacterial pneumonia, particularly with the most commonly (2012); Hernandez-Vargas and Meyer-Hermann (2012); The work of Smith et al. (2013) proposed the first ity rates of influenza are mainly attributed to secondary Dobrovolny et al. (2013); Hernandez-Vargas et al. (2014)]. identified co-pathogen, S. pneumoniae, also seen in victims bacterial pneumonia, particularly with the most commonly The work of Smith et al. (2013) proposed the first modeling identified co-pathogen, S. pandemic pneumoniae, alsomost seen in Standivictims Dobrovolny et al. (2013); Hernandez-Vargas et al.modeling (2014)]. approach to represent basic interactions between IAV bacterial pneumonia, particularly with the commonly of H1N1 [Ballinger and identified co-pathogen, S. pandemic pneumoniae, also seen in Standivictims The work of et al.the (2013) the first approach to Smith represent the basicproposed interactions between IAV of the the more-recent more-recent H1N1 [Ballinger and The work of Smith et al.the (2013) proposed the first modeling and bacterial coinfection by S. pneumoniae. To the best identified co-pathogen, S. pandemic pneumoniae, also seen in Standivictims ford (2010)]. This gram positive is a latent colonizer of of the more-recent H1N1 [Ballinger and approach to represent basic interactions between IAV bacterial coinfection by S. pneumoniae. To the best ford (2010)]. This gram positive is a latent colonizer of and approach to represent the basic interactions between IAV knowledge, the first experimental and theoretical of the more-recent pandemic [Ballinger and Standithe nasopharynx asymptomatic and ford (2010)]. ThisinH1N1 gram positive ispreschool a latent children colonizer of author and bacterial coinfection by S. pneumoniae. To the best author knowledge, the first experimental and theoretical the nasopharynx inyears asymptomatic preschool children and and bacterial coinfection by S. pneumoniae. To the best work to dissect the contributions of pro-inflammatory ford (2010)]. This gram positive is a latent colonizer of author knowledge, the first experimental and theoretical individuals over 65 of age but remains to be the most the nasopharynx in asymptomatic preschool children and work to dissect the contributions of pro-inflammatory cycyindividuals overof65in years of age but preschool remains tochildren be the most firstcandidates experimental and theoretical tokines as main for the nasopharynx asymptomatic and author common pneumonia. Experindividuals overof65community-acquired years of age but remains to be the most work toknowledge, dissect the the contributions of pro-inflammatory cytokines as main responsible responsible candidates for the the colonization colonization common cause cause community-acquired pneumonia. Expertoas dissect the contributions of pro-inflammatory cyin a secondary bacterial infection was presented by [Duviindividuals overof65 years of agethat butpreceding remains toIAV be the most work imental evidence has revealed infection common cause community-acquired pneumonia. Expertokines main responsible candidates for the colonization in a secondary bacterial infection was presented by [Duviimental evidence has revealed that preceding IAV infection tokines as main responsible candidates for the colonization gneau et al. (2016)]. common cause of community-acquired pneumonia. Experenhancesevidence all aspects of S. pneumoniae pathogenesis from gneau imental has revealed that preceding IAV infection in a secondary bacterial infection was presented by [Duviet al. (2016)]. enhances all to aspects of S. pneumoniae pathogenesis from a secondary bacterial infection was presented by [Duviimental evidence has revealed that preceding IAV infection gneau etwork, al. (2016)]. colonization invasive pneumococcal disease (IPD), indi- in enhances all aspects of S. pneumoniae pathogenesis from In this we colonization invasive pneumococcal disease (IPD), indi- gneau al. (2016)]. In thisetwork, we consider consider deterministic deterministic and and stochastic stochastic enhances all to aspects of S. pneumoniae pathogenesis from cating a strong predisposition to lethal secondary bacterial colonization to invasive pneumococcal disease (IPD), indito continue dissecting key mechanisms during cating a strong predisposition to lethal secondary bacterial approaches this work, we consider deterministic and stochastic approaches to continue dissecting key mechanisms during colonization to predisposition invasive (IPD), indi- In infection in IAV infectedpneumococcal patients. Thedisease underlying mechacating a strong to lethal secondary bacterial In this work, we consider deterministic and stochastic IAV and S. pneumoniae coinfection. Using experimental approaches to continue dissecting key mechanisms during infection in IAV infected patients. The underlying mechaIAV and S. pneumoniae coinfection. Using experimental cating a strong predisposition to lethal secondary bacterial nisms implicated in this synergism physical mechadisrup- approaches infection in IAV infected patients. include The underlying to continue dissecting key mechanisms during data and the mathematical model from Duvigneau nisms implicated in this synergism include physical disrupIAV and S. pneumoniae coinfection. Using experimental data and the mathematical model from Duvigneau et et al. al. infection in lung IAV infected patients. The mechation ofimplicated the epithelial barrier include and underlying aberrant immunonisms in this synergism physical disrup- IAV and S. pneumoniae coinfection. Using experimental (2016), we develop a steady-state analysis, which suggests tion of the lung epithelial barrier and aberrant immunodata and the mathematical model from Duvigneau et al. we develop a steady-state analysis, which suggests nismsofimplicated in this synergism include physical disrup- (2016), logical responses causing lung injury [McCullers (2006)]. tion the lung epithelial barrier and aberrant immunodata and mathematical modelby from Duvigneau et the al. as promoted logical responses causing lung injury [McCullers (2006)]. substrates (2016), wethe develop a steady-state which only suggests as nutrients nutrients promoted analysis, by IAV IAV affect affect only the tion of responses the coinfection lung epithelial barrier and[McCullers aberrant immunoAltogether studies strongly reflect detrimental logical causing lung injury (2006)]. substrates (2016), we develop a bacteria steady-state analysis, which suggests steady-state that the would reach, but more nutrisubstrates as nutrients promoted by IAV affect only the Altogether coinfection studies strongly reflect detrimental that the bacteria would reach, but more nutrilogical responses causing lung strongly injury [McCullers (2006)]. changes in lung microenvironment primary or secondary to steady-state Altogether coinfection studies reflect detrimental substrates as nutrients promoted bythe IAV but affect only the ents would not contribute directly to decision changes in lung microenvironment primary or secondary to steady-state that the bacteria would more nutrients would not contribute directly toreach, the bacterial bacterial decision Altogether coinfection studies strongly reflect detrimental the exacerbated cytokines production. Nevertheless, it is changes in lung microenvironment primary or secondary to steady-state that the bacteria would reach, but more nutrito colonize. Moreover, we develop here a Markov branching the exacerbated cytokines production. Nevertheless, it is ents would not contribute directly to the bacterial decision colonize. Moreover, we develop here a Markov branching changes in lung microenvironment primary or secondary still largely whether these work the cytokines production. Nevertheless, it to is to ents would not contribute directly to the bacterial decision process for infections and Simulations still exacerbated largely unknown unknown whether these cytokines cytokines work alone alone colonize. Moreover, we develop here a Markov branching process for single single infections and coinfections. coinfections. Simulations theinexacerbated cytokines production. Nevertheless, it is to or synergism as friend or foe the coinfected host still largely unknown whether thesetocytokines work alone to colonize. Moreover, wethat develop here a Markov branching support the hypothesis cell to cell heterogeneity process for single infections and coinfections. Simulations or in synergism as friend or foe to the coinfected host the hypothesis that cell to cell heterogeneity is is stillinlargely unknown whether thesetocytokines work alone [Scheller et al. (2011)]. or synergism as friend or foe the coinfected host support process for single infections Simulations important in single infection but play [Scheller et al. (2011)]. support the thatand cellcoinfections. to cell is in hypothesis single bacterial bacterial infection butheterogeneity play a a minimal minimal or in synergism as friend or foe to the coinfected host important [Scheller et al. (2011)]. support the hypothesis that remaining cell to cell heterogeneity is role during coinfections. The parts of the paper In the absence of pieces of the puzzle, mathematical modimportant in single bacterial infection but play a minimal during coinfections. The remaining parts of the paper [Scheller et al. (2011)]. In thecan absence ofaspieces of the puzzle, mathematical mod- role important in single bacterial infection but play a minimal are organized as follows. Mouse experiments settings are role during coinfections. The remaining parts of the paper eling serve a framework to test in silico hypothIn the absence of pieces of the puzzle, mathematical mod- are organized as follows. Mouse experiments settings are eling can serveof aspieces a framework to test in silico hypothduring in coinfections. remaining partsmodel of theand paper introduced Section 2. The The deterministic its In the absence of the puzzle, mathematical mod- role esis tailor experimental settings. Mathematieling can serve future as a framework to test in silico hypothare organized as follows. Mouse experiments settings esis and and tailor future experimental settings. Mathematiintroduced in Section 2. Section The deterministic model and are its organized as follows. Mouse experiments settings are analysis are presented in 3. The Markov branching elingand cantailor serve future as a framework to test in silico hypoth- are esis experimental settings. Mathematiintroduced in Section 2. The deterministic model and its analysis are presented in Section 3. The Markov branching  This work was supported by the Alfons und Gertrud Kasselintroduced in Section in 2.are The deterministic model and its process and simulations discussed in Section 4. Discusesis and tailor experimental settings. Mathemati This analysis are presented Section 3. The Markov branching work was future supported by the Alfons und Gertrud Kasselprocess and simulations are discussed in Section 4. Discus Stiftung and was the BoehringerbyIngelheim Stiftung. Corresponding analysis are presented inare Section 3. The sions the paper 5. process and discussed in Markov Section branching 4. DiscusThis work the Alfons und Gertrud KasselStiftung and the supported Boehringer Ingelheim Stiftung. Corresponding sions end end thesimulations paper in in Section Section 5.  author: [email protected] This work was supported by the Alfons und Gertrud KasselStiftung and the Boehringer Ingelheim Stiftung. Corresponding process and are discussed in Section 4. Discusauthor: [email protected] sions end thesimulations paper in Section 5. Stiftung and the Boehringer Ingelheim Stiftung. Corresponding author: [email protected] sions end the paper in Section 5.

author: [email protected] 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018, 2018 IFAC 390Hosting by Elsevier Ltd. All rights reserved. Copyright 2018 responsibility IFAC 390Control. Peer review©under of International Federation of Automatic Copyright © 2018 IFAC 390 10.1016/j.ifacol.2019.02.001 Copyright © 2018 IFAC 390

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2. MOUSE INFECTION EXPERIMENTS

3. DETERMINISTIC MATHEMATICAL MODELING

Mouse infection experiments are taken from previous work in Duvigneau et al. (2016). Female C57Bl/6J wildtype mice were purchased at the age of 8 to 9 weeks. The influenza A virus strain A\PR8\34 (H1N1) [McCullers and Rehg (2002)] and the encapsulated Streptococcus pneumoniae strain TIGR4 (T4) were employed. The measurements of the time points 0 h, 1.5 h, 6 h, 18 h, 26 h and 31 h were reported using 6 mice each time point.

Here, we consider the mathematical model of S. pneumoniae infection (B) proposed in [Duvigneau et al. (2016)], the ordinary differential equation (ODE) writes as follows:   B dB = rB 1 − (1) − cb B, dt KB

For every time point, the following infection scenarios are considered: single bacterial infection (T4) and coinfection (IAV+T4) starting at day 7 post IAV infection. Every group consisted of at least 30 mice. A schematic representation of the experiments can be seen in Fig. 1.

(a) Experimental setup Sub-lethal IAV infection (A/PR8/34)

Mild S.Pneumoniae infection (T4) Sample collection after bacterial infection

day 0

day 7

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(b) Infection scenarios (i) T4

(ii) IAV + T4

As reported by Smith et al. (2013), we assumed that only a proportion of the bacterial inoculum may reach the lung since some bacteria can be removed by mucociliary mechanisms. Thus, we considered 1000 CFU/mL S. pneumoniae as inoculum to fit the model parameters in the equation (1). Bacteria (log 10 CFU/mL lung homogenate)

Before infection, day 0, all mice were narcotized with a ketamine solution. The IAV and IAV+T4 groups were intranasally infected with 25 µL of a diluted virus suspension. The T4 group was administered with 25 µL PBS (Phosphate Buffered Saline). Body weight and health status were monitored during the next 7 days. Mice that lost more than 25% of body weight were sacrificed.

where r is the bacterial proliferation rate with a maximum carrying capacity KB . The growth rate (r =1.13 h−1 ) and the carrying capacity (KB =2.3×108 CFU/ml) corresponding to single S. pneumoniae infection were taken from [Smith et al. (2013)]. Phagocytosis of the bacteria is considered by cb = 1.28 for single T4 infection, and cb = 0.72 for the coinfected group as presented by [Duvigneau et al. (2016)].

IAV and bacterial coinfection Single bacterial infection

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Fig. 2. Simulation results for single bacterial infection and coinfection. Empty circles represent the experimental data taken from [Duvigneau et al. (2016)]. Coinfection takes places 7 days after influenza infection.

GM-CSF

Fig. 1. Experimental setting scheme. C57Bl/6J wildtype mice were nasally infected with a sub-lethal dose of IAV (A/PR8/34). After, at day 7, the animals were infected with the T4 strain of S. pneumoniae. Blood, bronchoalveolar lavages (BAL), and lung homogenates samples were taken from all animals at 1.5, 6, 26 and 31 hours (a). The infection scenarios are single bacterial infection (T4) and coinfection (IAV+T4) (b). Seven days post influenza infection, a bacterial solution was needed. Preparing this solution, one aliquot was diluted with PBS to a concentration of approximately 4×107 colony-forming units per µL (CFU/µL). 10-fold dilutions were plated out onto blood agar plates to estimate this concentration. Mice of the IAV+T4 and T4 groups were oropharyngeal challenged with 25 µL bacterial solution (approximately 106 CFU) while mice in the IAV group got 25 µL PBS. 391

Fig. 2 reveals that 18 hours post second infection (hpsi) the bacterial burden of coinfected mice are elevated in comparison to single bacteria infected mice. The bacterial burden of T4 infected mice is not existent, although some of the single infected mice have slightly elevated CFU counts in bronchoalveolar lavage (BAL) and lung samples. This dichotomy effect was previously remarked by Smith et al. (2013) that attributed this effect to the heterogeneity as it exists in humans. The peak for the CFU-counts of T4 infected mice is at 18 hpsi. Following this time point the mean value for the bacterial burden of the T4 infected group decreases while the measured values for coinfected mice are increasing. Results in Fig. 2 highlight that in time points, after 18 hpsi, almost all mice of the coinfected group developed high bacterial titer in the blood, which is an indicator for septicemia. Also, many mice cleared the infection in the group infected only with bacteria.

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3.1 Steady-state and stability analysis To study the model parameters that determine the decision of bacteria to successfully colonize the host lung tissue, we followed a mathematical reasoning from dynamical systems. To this end, the bacteria equilibrium points (B ∗ ) were derived by setting the right term of equation (1) equal zero:   B 0 = rB 1 − (2) − cb B. KB The first equilibrium (B (1) ) represented that there was no colonization of the bacteria: B (1) = 0. (3)

The second equilibrium point (B (2) ) was written as follows:  cb  B (2) = KB 1 − . (4) r The second equilibrium is biologically meaningful (positive definite) if the condition r > cb is satisfied. This means that the bacterial proliferation rate needs to be higher than its clearance. While this condition can be even followed by intuition, the hypothesis that influenza increases substrates as nutrients for bacterial growth (KB ) affects only the steady-state that the bacteria will reach, but more nutrients do not contribute directly to the bacterial decision to colonize as can be observed in (4). To evaluate if the increase in substrates modulated by influenza infection could alternatively alter the stability properties of the system (1), the stability analysis was derived from the Jacobian of the equation (1) which reads as follows: r ∗ ∂f1 = (r − cb ) − 2 B . (5) ∂B KB For the first equilibrium point, the Jacobian is: ∂f1 (1) = (r − cb ). (6) ∂B Thus, the equilibrium point B (1) will be stable if the condition cb > r. This implies that the bacteria will not colonize the host since the system (1) has only one equilibrium point that is zero and is attracted to it independently of the initial inoculation. If the r > cb , then the equilibrium point B (1) will be unstable, meaning that bacteria will grow independently of inoculation. Furthermore, the equilibrium point B (2) will be biologically significant (bacterial colonization). By doing some algebra, the Jacobian can be written as: ∂f1 (2) = (cb − r). (7) ∂B From (7), the equilibrium point B (2) is biologically meaningful and attracting if r > cb . 3.2 Nutrient availability modulation. Experimental evidence by Siegel et al. (2014) established that higher rates of disease during coinfection could stem from increased sialic acid availability that further supports bacterial colonization and proliferation. From a mathematical point of view, nutrient sources can be represented by the current capacity KB in equation (1). By the previous 392

steady-state and stability analysis, which is independent of parameter fitting procedures, it can be inferred that the bacterial nutrient source (KB ) determines the size of the initial bacterial colony but not the decision to grow. Of note, this approach is also valid assuming more complex logistic  growth terms.  For instance, the mathematical term B rB 1 − KB (1+ΨV previously proposed by Smith et al. ) (2013) to integrate the increase in the carrying capacity, KB (1 + ΨV ), may provide equivalent conclusions. 4. MARKOV BRANCHING PROCESS Here, we evaluated the probability of extinction by developing a stochastic Markov branching process analog to the model (1). To this end, assuming B << KB in (1), the model can be treated as a birth and death process with rate r and cb respectively. Consequently, the probability of extinction using the parameters can be computed using the estimated parameters for the single S. pneumoniae infection and coinfection. According to the Markov branching process formulation [Harris (2002)], each bacterial cell lives a life-span following an exponential density distribution i.e a = r + cb . At the end of its life, each bacterial cell can generate two new bacterial cells with rate r or die with rate cb , independently from the other bacteria cells. The bacterial offspring generated by one bacterial cell is the probability generating function (pgf) defined as: fB (s) =

∞ 

pk s k ,

(8)

k=0

where pk is the probability to generate k offspring cells and s is an auxiliary variable. The pk is computed as the ratio between the rate of having k offpring cells to the sum of the rates of the all possible events. Thus, the birth and death process leads to the simplification of (8) as follows: fB (s) = p0 + p2 s2 = (cB + rs2 )a−1 ,

(9)

where p0 = cb /a and p2 = r/a, with p0 + p2 = 1. For the complete description of the birth and death process, we defined the global probability generating function for one bacteria cell: ∞  P (B(t) = k|B(0) = 1)sk . (10) FB (s, t) = k=0

with P (B(t) = k|B(0) = 1), we denoted the conditional probability of having B(t) = k bacterial cells at time t starting from B(0) = 1. The temporal evolution of FB (s, t) is governed by the backward Kolmogorov equation in its general form [Harris (2002)]:

∂FB (s, t) = u(FB (s, t)), (11) ∂t where u(s) = a(fB (s) − s) and with the initial condition FB (s, 0) = s. In our particular case, the Kolmogorov equation can be presented as follows: ∂FB (s, t) = r(FB (s, t)2 −FB (s, t))+cb (1−FB (s, t)). (12) ∂t

IFAC BMS 2018 São Paulo, Brazil, September 3-5, 2018 Esteban A. Hernandez–Vargas et al. / IFAC PapersOnLine 51-27 (2018) 390–395

This differential equation can be solved analytically with the method of characteristics, providing the following analytical solution: cb (s − 1) − e(cb −r)t (rs − cb ) . r(s − 1) − e(cb −r)t (rs − cb )

FB (s, t) = FB (s, t)B(0) .

Single S. pneumoniae infection 1000

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In the case of a generic bacterial inoculum B(0), considering the independence property of the branching process [Harris (2002)] we obtained: 

(a)

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FB (s, t) =

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(14)

The probability of extinction (B(t) = 0) at the time point t and with the bacteria inoculum B(0), can be written as follows:  B(0)  cb − cb e(cb −r)t . (15) Pe (t) = FB (0, t) = r − cb e(cb −r)t

To evaluate the stochastic fate decision, we derived the probability of extinction by developing a stochastic Markov branching process analogous to the model (1). For the single S. pneumoniae infection, the probability of bacterial extinction at 35 hours post infection (hpi) is approximately 50% (Fig. 3a), supporting the hypothesis of cell to cell heterogeneity by Smith et al. (2013). In contrast, a stochastic fate decision with cell to cell heterogeneity would play only a minimal role in the case of coinfection (Fig. 3b).

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4.1 Stochastic fate decision explains the dichotomy in bacterial burdens.

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The bacterial load detected in the BAL and lung tissue revealed that the single S. pneumoniae infected animals divided into two groups at 18 hpi (Fig. 2): we found that 50% of the animals had cleared the bacteria while the remaining 50% were still colonized at this time point. This dichotomy was also reported by Smith et al. (2013) and was suggested to result from the heterogeneity of the biological host.

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Fig. 3. Probability of S. pneumoniae extinction dynamics. The branching process formulation governed by the Kolmogorov equation (15) provides the probability of eradication Pe (t) at time t for single S. pneumoniae infection and coinfection.

The significant death toll of about 100 million individuals in retrospective studies, performed on fatal victims of the 1918/1919 IAV pandemic, revealed a high incidence of coinfections with unrelated bacterial pathogens. The threat of newly emerging pandemic IAV strains together with the increasing prevalence of antibiotic-resistant bacterial pathogens underline the need for a complete understanding of the mechanisms for secondary pneumonia. Independently both IAV and S. pneumoniae are responsible for a large proportion of respiratory infection-related deaths worldwide, expectedly being more lethal during the concurrent occurrence.

a pathogenic action of cytokines rather than their protective role. For example, Tumor Necrosis Factor-α (TNFα) and Interleukin-6 (IL-6) are drastically elevated in coinfected lungs compared to the single infection. Interestingly, lung lesions are proportional to the TNF-α levels while IL-6 is an established marker for sepsis and also indicating meningitis in most cases [Scheller et al. (2011)]. Furthermore, neutralization of TNF-α ameliorated pulmonary immunopathology and improved survival of coinfected animals. This was a resultant effect of retraction in TNF-α driven cytotoxicity and immune cells infiltration. Furthermore, Sun and Metzger (2008) showed that Interferon-γ (IFN-γ) suppresses macrophage phagocytosis and increases oxidative radicals by downregulating the expression of scavenger receptors on alveolar macrophages, indicating a bipolar mode of action of inflammatory mediators.

Animal and human studies have shown that preceding IAV infection enhances all aspects of S. pneumoniae pathogenesis from colonization to IPD, indicating a strong predisposition to lethal secondary bacterial infection in IAV infected patients [Taubenberger and Morens (2008); McCullers and Rehg (2002); McCullers (2006)]. In a state of coinfection, there is accumulating evidence supporting

Although abundant proinflammatory responses are reported in acute IAV infection, its role in facilitating bacterial colonization remain unclear. In the study developed by Duvigneau et al. (2016), model selection procedures highlighted IFN-γ kinetics as a decisive candidate responsible to impair bacterial clearance promoting the bacterial burden and systemic dissemination observed 18 hpsi

5. DISCUSSION

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with S. pneumoniae during acute IAV infection. Moreover, Duvigneau et al. (2016) found a contribution but not conclusive of IL-6 to impaired bacterial clearance. Duvigneau et al. (2016) results challenge current beliefs that TNF-α response, increase nutrient source and impairment in macrophages dynamics modulated by IAV infection contribute to the fate decision of a secondary bacterial infection. Here, using the experimental data from Duvigneau et al. (2016), we were focused more to evaluate nutrients availability hypothesis and the cell to cell heterogeneity. Mathematical analysis showed that more nutrients for bacterial growth would impact only the level that the bacteria would reach, but not the bacterial decision to colonize a host. In contrast, experimental evidence by Siegel et al. (2014) revealed that influenza infection accelerates not only bacterial replication in vivo but also sialic acid, which stimulates pneumococcal proliferation. Our findings suggest that more nutrients could imply the increase of pneumococcal proliferation but it is not fully conclusive to invade a host. Single pneumococcal infection experiments by Smith et al. (2013) and Duvigneau et al. (2016) revealed that mice could be divided into a group that sustains bacterial titers while the other one cleared the bacteria. Smith et al. (2013) attributed this dichotomy as a result of biological heterogeneity. Here, stochastic simulations confirmed that cell to cell heterogeneity in single bacterial infection is relevant while in the case of coinfection is minor. REFERENCES Baccam, P., Beauchemin, C., Macken, C.a., Hayden, F.G., and Perelson, A.S. (2006). Kinetics of influenza A virus infection in humans. Journal of virology, 80(15), 7590–9. doi:10.1128/JVI.01623-05. Ballinger, M.N. and Standiford, T.J. (2010). Postinfluenza bacterial pneumonia: host defenses gone awry. Journal of interferon and cytokine research, 30(9), 643–52. Beauchemin, C.a.a. and Handel, A. (2011). A review of mathematical models of influenza A infections within a host or cell culture: lessons learned and challenges ahead. BMC public health, 11 Suppl 1(Suppl 1), S7. doi:10.1186/1471-2458-11-S1-S7. Boianelli, A., Nguyen, V.K., Ebensen, T., Schulze, K., Wilk, E., Sharma, N., Stegemann-Koniszewski, S., Bruder, D., Toapanta, F.R., Guzm´ an, C., MeyerHermann, M., and Hernandez-Vargas, E.A. (2015). Modeling Influenza Virus Infection: A Roadmap for Influenza Research. Viruses, 7(10), 5274–5304. Boianelli, A., Sharma-Chawla, N., Bruder, D., and Hernandez-Vargas, E.A. (2016). Oseltamivir PK/PD Modeling and Simulation to Evaluate Treatment Strategies against Influenza-Pneumococcus Coinfection. Frontiers in Cellular and Infection Microbiology, 6. Dobrovolny, H.M., Reddy, M.B., Kamal, M.a., Rayner, C.R., and Beauchemin, C.a.a. (2013). Assessing Mathematical Models of Influenza Infections Using Features of the Immune Response. PLoS ONE, 8(2), e57088. doi: 10.1371/journal.pone.0057088. Duvigneau, S., Sharma-Chawla, N., Boianelli, A., Stegemann-Koniszewski, S., Nguyen, V.K., Bruder, D., and Hernandez-Vargas, E.A. (2016). Hierarchical effects 394

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