# Developing a Novel Parameter Estimation Method for Agent-Based Model in Immune System Simulation under the Framework of History Matching: A Case Study on Influenza A Virus Infection

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## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Observation Data of Influenza A Virus (IAV)

#### 2.2. Sampling Data

#### 2.3. Non-Implausible Space

^{5}points, namely ${H}_{1}$, are drawn from a four-dimensional uniform distribution within the region $(0,2{\theta}_{0})$, and the implausibility measure is evaluated for each point of ${H}_{1}$. Since we have 10 outputs in our study, for simplicity, we use the following implausibility measure

^{5}sampling points and non-implausible sampling points for each parameter are shown in Figure 2.

#### 2.4. Fitting Experimental Data

#### 2.5. Average Relative Error

## 3. Methods

#### 3.1. Simulator: Using ABM (Agent-based Model) to Simulate the Immune System

#### 3.2. Emulator: GAM Model

#### 3.3. Reducing the Input Space by Using Implausibility Measure

#### 3.4. Parameter Estimation

**gam()**. After that, we regenerate another sampling data set ${H}_{1}$ within the region $(0,2{\theta}_{0})$ and take ${H}_{1}$ as input into ${M}_{0}$ to obtain prediction output data ${G}_{1}$. The implausibility measure (4) is employed to evaluate each point of ${H}_{1}$ and those that do not pass the implausibility test are deemed implausible, meaning that the simulator cannot match the observations given the current error specifications. The initial input space $(0,2{\theta}_{0})$ would be reduced to be a subset of non-implausible space. Next, PSO is employed to locate the optimal parameter ${\theta}^{*}$ within the non-implausible space by fitting the real experimental data, i.e., the estimated parameter ${\theta}^{*}$ is obtained by minimizing the following objective function

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Miao, H.Y.; Xia, X.H.; Perelson, A.S.; Wu, H.L. On identifiability of nonlinear ode models and applications in viral dynamics. SIAM Rev.
**2011**, 53, 3–39. [Google Scholar] [CrossRef] [PubMed] - Miao, H.Y.; Hollenbaugh, J.A.; Zand, M.S.; Holden-Wiltse, J.; Mosmann, T.R.; Perelson, A.S.; Wu, H.; Topham, D.J. Quantifying the early immune response and adaptive immune response kinetics in mice infected with influenza A virus. J. Virol.
**2010**, 84, 6687–6698. [Google Scholar] [CrossRef] [PubMed] - Miao, H.Y.; Dykes, C.; Demeter, L.M.; Wu, H.L. Differential equation modeling of hiv viral fitness experiments: Model identification, model selection, and multimodel inference. Biometrics
**2009**, 65, 292–300. [Google Scholar] [CrossRef] [PubMed] - Øksendal, B. Stochastic Differential Equations; Springer: Berlin, Germany, 2003. [Google Scholar]
- Jones, D.S.; Plank, M.J.; Sleeman, B.D. Differential Equations and Mathematical Biology; Food and Agriculture Organization: Rome, Italy, 2010. [Google Scholar]
- Ho, W.H.; Chan, L.F. Hybrid Taguchi-differential evolution algorithm for parameter estimation of differential equation models with application to HIV dynamics. Math. Probl. Eng.
**2011**, 2011, 514756. [Google Scholar] [CrossRef] - Miao, H.; Wu, H.; Xue, H. Generalized ordinary differential equation models. J. Am. Stat. Assoc.
**2014**, 109, 1672. [Google Scholar] [CrossRef] [PubMed] - Jiang, W.; Sullivan, A.M.; Su, C.L.; Zhao, X.P. An agent-based model for the transmission dynamics of Toxoplasma gondii. J. Theor. Biol.
**2012**, 293, 15–26. [Google Scholar] [CrossRef] [PubMed] - Folcik, V.A.; An, G.C.; Orosz, C.G. The Basic Immune Simulator: An agent-based model to study the interactions between innate and adaptive immunity. Theor. Biol. Med. Model.
**2007**, 4, 39. [Google Scholar] [CrossRef] [PubMed] - Segovia-Juarez, J.L.; Ganguli, S.; Kirschner, D. Identifying control mechanisms of granuloma formation during M-tuberculosis infection using an agent-based model. J. Theor. Biol.
**2004**, 231, 357–376. [Google Scholar] [CrossRef] [PubMed] - Jacob, C.; Litorco, J.; Lee, L. Immunity through swarms: Agent-based simulations of the human immune system. In Proceedings of the International Conference on Artificial Immune Systems, Sicily, Italy, 13–16 September 2014; pp. 400–412. [Google Scholar]
- Wang, Z.H.; Butner, J.D.; Kerketta, R.; Cristini, V.; Deisboeck, T.S. Simulating cancer growth with multiscale agent-based modeling. Semin. Cancer Biol.
**2015**, 30, 70–78. [Google Scholar] [CrossRef] [PubMed] - Chiacchio, F.; Pennisi, M.; Russo, G.; Motta, S.; Pappalardo, F. Agent-based modeling of the immune system: Netlogo, a promising framework. BioMed Res. Int.
**2014**, 2014, 907171. [Google Scholar] [CrossRef] [PubMed] - Zhang, L.; Jiang, B.; Wu, Y.; Strouthos, C.; Sun, P.Z.; Su, J.; Zhou, X.B. Developing a multiscale, multi-resolution agent-based brain tumor model by graphics processing units. Theor. Biol. Med. Model.
**2011**, 8, 46. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhang, L.; Qiao, M.; Gao, H.; Hu, B.; Tan, H.; Zhou, X.; Li, C.M. Investigation of mechanism of bone regeneration in a porous biodegradable calcium phosphate (CaP) scaffold by a combination of a multi-scale agent-based model and experimental optimization/validation. Nanoscale
**2016**, 8, 14877. [Google Scholar] [CrossRef] [PubMed] - Zhang, L.; Xue, Y.; Jiang, B.N.; Strouthos, C.; Duan, Z.F.; Wu, Y.K.; Su, J.; Zhou, X.B. Multiscale agent-based modelling of ovarian cancer progression under the stimulation of the STAT 3 pathway. Int. J. Data Min. Bioinform.
**2014**, 9, 235–253. [Google Scholar] [CrossRef] [PubMed] - Moedomo, R.L.; Pancoro, A.; Ibrahim, J.; Ahmad, A.S.; Mardiyanto, M.S.; Belatiff, M.B.; Tasman, H. Simulation of influenza pandemic based on genetic algorithm and agent-based modeling: A multi-objective optimization problem solving. J. Matematika Sains
**2010**, 15, 47–59. [Google Scholar] - Zhang, L.; Zhang, S. Using game theory to investigate the epigenetic control mechanisms of embryo development: Comment on: “Epigenetic game theory: How to compute the epigenetic control of maternal-to-zygotic transition” by Qian Wang et al. Phys. Life Rev.
**2017**, 20, 140. [Google Scholar] [CrossRef] [PubMed] - Zhang, L.; Wang, Z.; Sagotsky, J.; Deisboeck, T. Multiscale agent-based cancer modeling. J. Math. Biol.
**2009**, 58, 545. [Google Scholar] [CrossRef] [PubMed] - Tong, X.M.; Chen, J.H.; Miao, H.Y.; Li, T.T.; Zhang, L. Development of an Agent-Based Model (ABM) to simulate the immune system and integration of a regression method to estimate the key abm parameters by fitting the experimental data. PLoS ONE
**2015**, 10, e0141295. [Google Scholar] [CrossRef] [PubMed] - Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the IEEE International Conference on Neural Networks, Honolulu, HI, USA, 12–17 May 2002; Volume 4, pp. 1942–1948. [Google Scholar]
- Kennedy, J.; Eberhart, R.C.; Shi, Y. Swarm Intelligence; Morgan Kaufmann Publishers Inc.: San Francisco, CA, USA, 2001. [Google Scholar]
- Poli, R. An Analysis of Publications on Particle Swarm Optimization Applications; Department of Computer Science, University of Essex: Essex, UK, 2007. [Google Scholar]
- Poli, R. Analysis of the publications on the applications of particle swarm optimisation. J. Artif. Evol. Appl.
**2008**, 2008, 685175. [Google Scholar] [CrossRef] - Clerc, M. Standard Particle Swarm Optimisation. 2012. Available online: https://hal.archives-ouvertes.fr/hal-00764996/ (accessed on 13 December, 2012).
- Pedersen, M.E.H.; Chipperfield, A.J. Simplifying particle swarm optimization. Appl. Soft Comput.
**2010**, 10, 618–628. [Google Scholar] [CrossRef] - Fan, J.; Gijbels, R. Local Polynomial Modelling and Its Applications; Chapman & Hall: London, UK, 1996. [Google Scholar]
- Walsh, W.A.; Kleiber, P. Generalized additive model and regression tree analyses of blue shark (Prionace glauca) catch rates by the Hawaii-based commercial longline fishery. Fish. Res.
**2001**, 53, 115–131. [Google Scholar] [CrossRef] - Hastie, T.; Tibshirani, R. Generalized Additive Models; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2017. [Google Scholar]
- Trevor, H.; Tibshirani, R. Generalized additive models. Stat. Sci.
**1986**, 1, 297–310. [Google Scholar] - Andrianakis, I.; Vernon, I.R.; McCreesh, N.; McKinley, T.J.; Oakley, J.E.; Nsubuga, R.N.; Goldstein, M.; White, R.G. Bayesian history matching of complex infectious disease models using emulation: A tutorial and a case study on HIV in Uganda. PLoS Comput. Biol.
**2015**, 11, e1003968. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mckay, M.D.; Beckman, R.J.; Conover, W.J. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics
**1979**, 21, 239–245. [Google Scholar] - Pukelsheim, F. The three sigma rule. Am. Stat.
**1994**, 48, 88–91. [Google Scholar] - López-Moreno, J.I.; Nogués-Bravo, D. A generalized additive model for the spatial distribution of snowpack in the Spanish Pyrenees. Hydrol. Process.
**2005**, 19, 3167–3176. [Google Scholar] [CrossRef] - Ramsay, T.; Burnett, R.; Krewski, D. Exploring bias in a generalized additive model for spatial air pollution data. Environ. Health Perspect.
**2003**, 111, 1283–1288. [Google Scholar] [CrossRef] [PubMed] - Murase, H.; Nagashima, H.; Yonezaki, S.; Matsukura, R.; Kitakado, T. Application of a generalized additive model (GAM) to reveal relationships between environmental factors and distributions of pelagic fish and krill: A case study in Sendai Bay, Japan. ICES J. Mar. Sci.
**2009**, 66, 1417–1424. [Google Scholar] [CrossRef] - Jiang, B.N.; Struthers, A.; Sun, Z.; Feng, Z.; Zhao, X.Q.; Zhao, K.Y.; Dai, W.Z.; Zhou, X.B.; Berens, M.E.; Zhang, L. Employing graphics processing unit technology, alternating direction implicit method and domain decomposition to speed up the numerical diffusion solver for the biomedical engineering research. Int. J. Numer. Methods Biomed. Eng.
**2011**, 27, 1829–1849. [Google Scholar] [CrossRef] - Jamshed, S. Graphics Processing Unit Technology; Elsevier: Amsterdam, The Netherlands, 2015. [Google Scholar]

**Figure 3.**Fitting accuracy of the proposed method and ordinary differential equation (ODE) method [2].

**Figure 4.**Average relative error (ARE) of IABMR [20] and our proposed method for each parameter with three different level of random noises.

Time Points (Day^{−1}) | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Samples | 0 | 0.125 | 0.25 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 | 3.5 | 4 | 4.5 | 5 |

1 | 4.25 | 2.5 | 3.5 | 4.25 | 5.5 | 6.5 | 6.33 | 6.75 | 6.5 | 6.5 | 6.5 | 7 | 6.33 |

2 | 3.75 | 2.5 | 4.75 | 3.25 | 6.75 | 6.75 | 7.5 | 3.5 | 7.33 | 7.25 | 6.25 | 6.5 | 5.5 |

3 | 4.25 | 3.5 | 4.75 | 5.25 | 6.5 | 7.75 | 7.75 | 7.5 | 7.33 | 7.25 | 6.5 | 6.25 | 5.75 |

4 | 3.75 | 3.5 | 4.13 | 5.75 | 7.25 | NA | 7.25 | 6.5 | 6.25 | 5.5 | NA | NA | NA |

5 | 4.55 | 2.75 | 2.5 | 5.75 | NA | NA | NA | 7.5 | 6.75 | 6.5 | NA | NA | NA |

6 | 4.25 | NA | 4.75 | 5.5 | NA | NA | NA | NA | 7.25 | 5.75 | NA | NA | NA |

**Table 2.**Sampling data set as inputs into the simulator ABM, where ${\theta}_{k}$, $k=1,2,3,4$ represent proliferation rate, infection rate and death rate per hour for epithelial cells, infected epithelial cells, and virus, separately.

Samples | ${\mathit{\theta}}_{1}$ | ${\mathit{\theta}}_{2}$ | ${\mathit{\theta}}_{3}$ | ${\mathit{\theta}}_{4}$ |
---|---|---|---|---|

1 | 3.466758 × 10^{−9} | 2.288938 × 10^{−7} | 2.460326 × 10^{−2} | 8.616152 × 10^{−2} |

2 | 8.001264 × 10^{−9} | 4.300130 × 10^{−7} | 8.741329 × 10^{−2} | 3.955367 × 10^{−1} |

3 | 1.081166 × 10^{−8} | 1.932323 × 10^{−7} | 1.004010 × 10^{−1} | 3.100995 × 10^{−1} |

4 | 1.090549 × 10^{−8} | 2.812863 × 10^{−7} | 8.654013 × 10^{−2} | 3.202220 × 10^{−1} |

5 | 9.102252 × 10^{−9} | 4.513295 × 10^{−7} | 4.608862 × 10^{−2} | 1.196989 × 10^{−1} |

6 | 3.405003 × 10^{−9} | 3.370440 × 10^{−8} | 1.130993 × 10^{−1} | 6.104288 × 10^{−1} |

7 | 8.092254 × 10^{−9} | 4.017315 × 10^{−8} | 2.174145 × 10^{−2} | 2.430601 × 10^{−1} |

8 | 2.010234 × 10^{−9} | 1.745676 × 10^{−8} | 6.418247 × 10^{−2} | 1.722317 × 10^{−1} |

9 | 1.691198 × 10^{−9} | 3.527068 × 10^{−7} | 1.158533 × 10^{−1} | 1.302177 × 10^{−2} |

10 | 2.912003 × 10^{−9} | 2.957414 × 10^{−7} | 2.715800 × 10^{−2} | 2.602361 × 10^{−1} |

11 | 2.554265 × 10^{−9} | 9.798854 × 10^{−8} | 1.866300 × 10^{−2} | 5.577698 × 10^{−1} |

12 | 6.864842 × 10^{−9} | 4.184238 × 10^{−7} | 1.079778 × 10^{−1} | 7.730243 × 10^{−1} |

13 | 1.121311 × 10^{−9} | 3.102666 × 10^{−7} | 2.784293 × 10^{−3} | 5.343298 × 10^{−2} |

14 | 9.583759 × 10^{−9} | 6.668325 × 10^{−8} | 3.832125 × 10^{−2} | 7.836137 × 10^{−1} |

15 | 8.762499 × 10^{−9} | 5.740286 × 10^{−8} | 6.656615 × 10^{−2} | 1.462840 × 10^{−1} |

16 | 1.167708 × 10^{−8} | 1.339571 × 10^{−7} | 1.286682 × 10^{−2} | 7.554816 × 10^{−1} |

17 | 5.319678 × 10^{−9} | 4.057522 × 10^{−7} | 7.242679 × 10^{−2} | 6.884958 × 10^{−1} |

18 | 7.634766 × 10^{−9} | 8.162650 × 10^{−8} | 9.417931 × 10^{−2} | 8.124229 × 10^{−1} |

19 | 9.973253 × 10^{−9} | 1.641823 × 10^{−7} | 5.553776 × 10^{−2} | 1.506380 × 10^{−1} |

20 | 6.455279 × 10^{−9} | 1.729994 × 10^{−7} | 7.591240 × 10^{−2} | 4.765285 × 10^{−1} |

21 | 3.989849 × 10^{−9} | 9.193181 × 10^{−8} | 9.013124 × 10^{−2} | 4.137486 × 10^{−1} |

22 | 1.212724 × 10^{−8} | 4.454997 × 10^{−7} | 1.593432 × 10^{−2} | 1.957182 × 10^{−1} |

23 | 9.163350 × 10^{−10} | 3.647394 × 10^{−7} | 7.019446 × 10^{−2} | 5.823037 × 10^{−1} |

24 | 4.437117 × 10^{−9} | 3.801312 × 10^{−7} | 8.076542 × 10^{−3} | 6.162935 × 10^{−1} |

25 | 1.186253 × 10^{−8} | 3.221797 × 10^{−7} | 6.068513 × 10^{−2} | 2.227833 × 10^{−1} |

26 | 5.134477 × 10^{−9} | 2.635175 × 10^{−7} | 9.752898 × 10^{−2} | 5.182151 × 10^{−1} |

27 | 4.222250 × 10^{−9} | 2.151502 × 10^{−7} | 1.059426 × 10^{−1} | 8.356908 × 10^{−1} |

28 | 1.246306 × 10^{−9} | 3.445810 × 10^{−7} | 5.116530 × 10^{−2} | 4.604467 × 10^{−1} |

29 | 1.134631 × 10^{−8} | 1.157556 × 10^{−7} | 3.036958 × 10^{−2} | 6.538406 × 10^{−1} |

30 | 2.343542 × 10^{−9} | 1.483302 × 10^{−7} | 7.804259 × 10^{−2} | 4.427248 × 10^{−1} |

31 | 4.911390 × 10^{−9} | 1.124756 × 10^{−8} | 1.018424 × 10^{−1} | 2.351053 × 10^{−2} |

32 | 1.043893 × 10^{−8} | 4.616473 × 10^{−7} | 5.736185 × 10^{−2} | 2.911685 × 10^{−1} |

33 | 5.792641 × 10^{−9} | 4.757734 × 10^{−7} | 1.195959 × 10^{−1} | 7.008772 × 10^{−1} |

34 | 7.162705 × 10^{−9} | 1.314465 × 10^{−7} | 1.195076 × 10^{−2} | 3.444968 × 10^{−1} |

35 | 6.743423 × 10^{−9} | 2.437133 × 10^{−7} | 5.718468 × 10^{−3} | 6.614001 × 10^{−2} |

36 | 8.583690 × 10^{−9} | 3.363820 × 10^{−7} | 3.510043 × 10^{−2} | 4.877613 × 10^{−1} |

37 | 1.832510 × 10^{−10} | 1.983512 × 10^{−7} | 4.478183 × 10^{−2} | 3.777456 × 10^{−1} |

38 | 5.980517 × 10^{−9} | 3.927292 × 10^{−7} | 4.956199 × 10^{−2} | 6.684690 × 10^{−1} |

39 | 5.754720 × 10^{−10} | 2.763359 × 10^{−7} | 4.084022 × 10^{−2} | 5.291611 × 10^{−1} |

40 | 9.649890 × 10^{−9} | 2.419316 × 10^{−7} | 8.210135 × 10^{−2} | 7.266126 × 10^{−1} |

Parameters | Initial Interval | Non-Implausible Interval |
---|---|---|

${\theta}_{1}$ | [0, 1.240000 × 10−8] | [3.8139 × 10−14, 1.2400 × 10−8] |

${\theta}_{2}$ | [0, 4.840000 × 10−7] | [2.5844 × 10−14, 4.8400 × 10−7] |

${\theta}_{3}$ | [0, 1.196000 × 10−1] | [8.7906 × 10−7, 1.1960 × 10−1] |

${\theta}_{4}$ | [0, 8.460000 × 10−1] | [6.1473 × 10−6, 8.4600 × 10−1] |

**Table 4.**Initial parameters and the means with standard errors in brackets of the 50 parameter estimates.

Parameters | ||||
---|---|---|---|---|

Model | ${\theta}_{1}$ | ${\theta}_{2}$ | ${\theta}_{3}$ | ${\theta}_{4}$ |

Initial Parameters | 6.2000 × 10^{−9} | 2.4200 × 10^{−7} | 5.9800 × 10^{−2} | 4.2300 × 10^{−1} |

Our Estimates | 6.5656 × 10^{−9} | 7.2467 × 10^{−9} | 2.7739 × 10^{−2} | 1.2595 × 10^{−1} |

(4.2290 × 10^{−9}) | (6.4759 × 10^{−11}) | (2.8178 × 10^{−7}) | (3.1538 × 10^{−6}) |

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## Share and Cite

**MDPI and ACS Style**

Li, T.; Cheng, Z.; Zhang, L.
Developing a Novel Parameter Estimation Method for Agent-Based Model in Immune System Simulation under the Framework of History Matching: A Case Study on Influenza A Virus Infection. *Int. J. Mol. Sci.* **2017**, *18*, 2592.
https://doi.org/10.3390/ijms18122592

**AMA Style**

Li T, Cheng Z, Zhang L.
Developing a Novel Parameter Estimation Method for Agent-Based Model in Immune System Simulation under the Framework of History Matching: A Case Study on Influenza A Virus Infection. *International Journal of Molecular Sciences*. 2017; 18(12):2592.
https://doi.org/10.3390/ijms18122592

**Chicago/Turabian Style**

Li, Tingting, Zhengguo Cheng, and Le Zhang.
2017. "Developing a Novel Parameter Estimation Method for Agent-Based Model in Immune System Simulation under the Framework of History Matching: A Case Study on Influenza A Virus Infection" *International Journal of Molecular Sciences* 18, no. 12: 2592.
https://doi.org/10.3390/ijms18122592