# Modeling Influenza Virus Infection: A Roadmap for Influenza Research

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## Abstract

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## 1. Introduction

#### IAV Pathogenesis

**Figure 1.**Influenza A virus (IAV) infection and dynamics. (

**a**) Description of the main phases of IAV infection within a host. After entering the respiratory tract, each virion binds to a target cell. Then, virions enter the eclipse phase (5–12 hpi), before starting to replicate and infecting other cells; (

**b**) IAV and immune response (IR) dynamics. The innate IR is mainly represented by interferon (IFN)-I and by natural killer (NK) cells, whereas the adaptive IR is mainly driven by cytotoxic CD8${}^{+}$ T cells (CTLs) and antibodies (Abs). Days post infection is abbreviated with dpi.

## 2. Mathematical Models of IAV Infections

#### 2.1. In Vivo Systems

**Figure 2.**Target cell model. (

**Left**) IAV (V) infects susceptible cells (U) with rate β. Infected cells are cleared with rate δ. Once cells are productively infected (I), they release virus at rate p and virus particles are cleared at rate c. The symbol ϕ represents clearance; (

**Right**) Computational simulations of the target cell model. Parameter values used for model simulation are taken from [26]. The susceptible cells (red line) are rapidly infected while the virus (black line) and infected cells (blue line) peak at day one approximately. The viral growth is limited by the number of susceptible cells, decreasing the viral load and the number of infected cells to undetectable levels.

#### 2.2. Mathematical Models Including the Immune Response

#### 2.3. In Vitro Systems

#### 2.4. Data for Modeling: Scarce and Diverse

References | In Vitro | In Vivo | Host | Coinfection | Aging | |
---|---|---|---|---|---|---|

Innate | Adaptive | |||||

Antia et al. [48] | √ | |||||

Baccam et al. [26] | √ | |||||

Beauchemin et al. [31] | √ | |||||

Bocharov and Romanyukha [38] | √ | |||||

Canini and Carrat [45] | √ | |||||

Cao et al. [43] | √ | |||||

Chen et al. [60] | √ | |||||

Dobrovolny et al. [35] | √ | Various | ||||

Hancioglu et al. [39] | √ | |||||

Handel et al. [33] | √ | √ | ||||

Handel and Antia [49] | √ | |||||

[61] | √ | |||||

Hernandez-Vargas et al. [42] | √ | √ | √ | |||

Holder et al. [57] | √ | |||||

Holder and Beauchemin [32] | √ | |||||

Le et al. [50] | √ | |||||

Lee et al. [52] | √ | √ | ||||

Miao et al. [25] | √ | √ | ||||

Mitchell et al. [62] | √ | |||||

Moehler et al. [55] | √ | |||||

Paradis et al. [58] | √ | |||||

Pawelek et al. [40] | √ | |||||

Petrie et al. [36] | √ | |||||

Pinilla et al. [21] | √ | |||||

Price et al. [51] | √ | √ | ||||

Reperant et al. [63] | √ | √ | ||||

Saenz et al. [41] | √ | |||||

Schulze-Horsel et al. [56] | √ | |||||

Smith et al. [64] | √ | √ | ||||

Tridane and Kuang [54] | √ |

#### 2.5. Parameter Estimation: A Continuous Challenge

**step 1**). Note that multiple models can provide the same fit with observed experimental data. Thus, it becomes necessary to choose between different models. The standard approach to model selection is first estimate all model parameters from the data, then select the model with the best-fit error and some penalty on model complexity (Akaike Information Criterion, Bayesian Information Criterion [65]).

**Figure 3.**Mathematical modeling approach. From an experimental data set available, a mathematical model is developed/applied (

**step 1**); Then, the identifiability analysis (

**step 2**) should be carried out; Then, parameter uncertainty (

**step 3**) is evaluated providing parameter confidence intervals. In this phase scatter plots can inform on the parameters relation and their influence on the mathematical model; Once reasonable parameter values are obtained, model prediction (

**step 4**) can be performed generating new knowledge on the biological process and testing different scenarios.

**step 2**) of model parameters. A mathematical model is said to be identifiable when the parameter set can be uniquely determined. This can be achieved from the mathematical model structure (structural) and from the experimental data (practical). The structural and practical identifiability are necessary in mathematical models to reach significant predictions [66,67,68,69,70,71]. A very reliable method to test both the structural and practical identifiability is the profile likelihood method proposed by Raue et al. [72]. The idea behind this approach is to explore the parameter values, requiring for each parameter the optimization procedure of the cost function with respect to all other parameters. In particular, for each parameter, a range of values centered at the optimized value is generated in an adaptive manner. Re-optimization of the cost function with respect to the other parameters is performed for each value of the parameters. The aim of this approach is to detect directions where the likelihood flattens out [72].

**step 3**) is necessary to assess the large variability usually encountered in the biological data. The most frequent approach for parameter estimation is the bootstrap method. Bootstrapping is a statistic method for assigning measures of accuracy to estimates [73]. The nonparametric bootstrap considers data to be independent and identically distributed, whereas the parametric bootstrap requires imposing on the data a distribution assumption which is usually unknown. Bootstrap methods are frequently used as a conventional tool to take into account the uncertainty of the estimated parameter by calculating the confidence interval from bootstrap samples [10,25,42,57,74]. The bootstrap methods can be affected by the large variation of a few measurements or by the imposed distribution assumption that is usually unknown. As a result, the parameters confidence intervals can span a broad range [9]. Improvements of the bootstrap method in mathematical modeling have been proposed in [75,76,77]. Alternatively, the Bayesian approach could deal more efficiently with the parameters uncertainty, as well as the model prediction [78]. At the moment, there were only a few applications of the Bayesian methodology in mathematical modeling literature [79].

**step 4**), generate new knowledge or hypotheses of the biological process of interest and guide the design of new experiments.

#### 2.6. Case Study: Identification of a Mathematical Model of IAV Infection Including the Immune Response

**Figure 4.**Viral infection model with CTLs response. IAV (V) induces CTLs (E) clonal expansion with a rate r which inhibits the viral replication through the clearance of the infected cell, this effect can be included in ${\text{c}}_{\text{v}}$. CTLs are replenished with rate ${\text{s}}_{\text{E}}$ and die with rate ${\text{c}}_{\text{e}}$.

#### Step 1: Mathematical Modeling

#### Step 2: Identifiability Analysis

#### Step 3: Parameter Uncertainty

**Figure 5.**Profile likelihood for the model parameters. (

**a**) p is the viral replication rate; (

**b**) ${\text{c}}_{\text{v}}$ represents the viral clearance; (

**c**) ${\text{k}}_{\text{e}}$ is the CTLs half saturation constant; (

**d**) r represents the CTLs proliferation rate.

**Figure 6.**Viral infection model fitting: (

**a**) The model fitting is shown with a blue line, viral load data is presented in red squares for mice infected with the IAV (H1N1) (PR8) strain; (

**b**) the model fitting is shown in a blue line, the CTLs data is presented in red squares.

**Figure 7.**Nonparametric bootstrap results. The distributions of the nonparametric bootstrap obtained with 1000 samples for the model parameters (

**a**) p; (

**b**) ${\text{c}}_{\text{v}}$; (

**c**) ${\text{k}}_{\text{e}}$; (

**d**) r.

**Table 2.**Model parameter estimates (median), 95% confidence intervals and constraints used in the optimization algorithm [84].

Parameter | Median | Confidence Interval (95%) | Constraints for Optimization Algorithm |
---|---|---|---|

$\text{p}\left[{\text{d}}^{-1}\right]$ | 4.4 | [3.43 ; 6.08] | [1; 8] |

${\text{c}}_{\text{v}}\left[{\text{d}}^{-1}\text{cel}{\text{l}}^{-1}\right]$ | $1.24\times {10}^{-6}$ | [$6.1\times {10}^{-7}$ ; $2.73\times {10}^{-6}$] | [$5\times {10}^{-8}$; ${10}^{-5}$ ] |

$\text{r}\left[{\text{d}}^{-1}\right]$ | 0.33 | [0.20 ; 0.42] | [0.01; 1] |

${\text{k}}_{\text{e}}[\text{PFU}/\text{ml}]$ | $2.7\times 10{}^{3}$ | [$5.10\times 10{}^{2}$ ; $1.06\times {10}^{4}$ ] | [$4\times {10}^{2}$; $3\times {10}^{4}$] |

**Figure 8.**Scatter plot results. The scatter plots of (

**a**) p-${\text{c}}_{\text{v}}$; (

**b**) ${\text{k}}_{\text{e}}$-r; (

**c**) ${\text{k}}_{\text{e}}$-p; (

**d**) r-p; (

**e**) r-${\text{c}}_{\text{v}}$, (

**f**) ${\text{k}}_{\text{e}}$-${\text{c}}_{\text{v}}$. The numerical values are obtained from the nonparametric bootstrap distributions in Figure 7. The plots show dependencies between parameters p-${\text{c}}_{\text{v}}$, p-r, ${\text{c}}_{\text{v}}$-r.

## 3. Discussion and Future Perspectives

**Figure 9.**Main challenges in IAV infection. IAV infections facilitate secondary bacterial infections impairing the IR deputed to the bacterial clearance. The IAV infection is controlled by IR, which in turn is shaped by host genetic factors, previous infections, vaccination, and aging.

#### 3.1. Bacterial Coinfection

#### 3.2. Aging of the Immune System and the Role in IAV Infections

#### 3.3. Challenges for Influenza Vaccination

**Figure 10.**Emerging vaccination strategies. Novel vaccination strategies can for example (i) enhance mucosal IR against IAV reducing horizontal transmission and virus spread; (ii) display improved efficacy in poor responders such as elderly. Novel technologies will enable rapid production of emerging virus strains (e.g., synthetic mRNA or RNA replicon based vaccines) or universal vaccines covering major clades (e.g., designed hemagglutinins triggering broad neutralizing antibodies or vaccines triggering cross-protective CTL responses).

#### 3.4. Host and IAV Genetic Factors

**Table 3.**Host Genetic factors. Host factors identified having a crucial role to determine severity of IAV infection in different hosts.

Host Factors | Role |
---|---|

IFITM3 | Restrict morbidity and mortality of IAV infection [161,162,163] |

CPT2 | Related complication as influenza-associated encephalopathy [164] |

TMPRSS2 | Resistance to IAV infection [165,166,167] |

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Boianelli, A.; Nguyen, V.K.; Ebensen, T.; Schulze, K.; Wilk, E.; Sharma, N.; Stegemann-Koniszewski, S.; Bruder, D.; Toapanta, F.R.; Guzmán, C.A.;
et al. Modeling Influenza Virus Infection: A Roadmap for Influenza Research. *Viruses* **2015**, *7*, 5274-5304.
https://doi.org/10.3390/v7102875

**AMA Style**

Boianelli A, Nguyen VK, Ebensen T, Schulze K, Wilk E, Sharma N, Stegemann-Koniszewski S, Bruder D, Toapanta FR, Guzmán CA,
et al. Modeling Influenza Virus Infection: A Roadmap for Influenza Research. *Viruses*. 2015; 7(10):5274-5304.
https://doi.org/10.3390/v7102875

**Chicago/Turabian Style**

Boianelli, Alessandro, Van Kinh Nguyen, Thomas Ebensen, Kai Schulze, Esther Wilk, Niharika Sharma, Sabine Stegemann-Koniszewski, Dunja Bruder, Franklin R. Toapanta, Carlos A. Guzmán,
and et al. 2015. "Modeling Influenza Virus Infection: A Roadmap for Influenza Research" *Viruses* 7, no. 10: 5274-5304.
https://doi.org/10.3390/v7102875