### Formulation and Solution of an Inverse Reliability Problem to Simulate the Dynamic Behavior of COVID-19 Pandemic

F. S. Lobato, G. M. Platt, G. B. Libotte, A. J. Silva Neto

#### Abstract

Different types of mathematical models have been used to predict the dynamic behavior of the novel coronavirus (COVID-19). Many of them involve the formulation and solution of inverse problems. This kind of problem is generally carried out by considering the model, the vector of design variables, and system parameters as deterministic values. In this contribution, a methodology based on a double loop iteration process and devoted to evaluate the influence of uncertainties on inverse problem is evaluated. The inner optimization loop is used to find the solution associated with the highest probability value, and the outer loop is the regular optimization loop used to determine the vector of design variables. For this task, we use an inverse reliability approach and Differential Evolution algorithm. For illustration purposes, the proposed methodology is applied to estimate the parameters of SIRD (Susceptible-Infectious-Recovery-Dead) model associated with dynamic behavior of COVID-19 pandemic considering real data from China's epidemic and uncertainties in the basic reproduction number (R0). The obtained results demonstrate, as expected, that the increase of reliability implies the increase of the objective function value.

#### Keywords

Inverse Problem; Reliability-Based Optimization; Modeling; COVID-19

PDF

#### References

Nature Index, Coronavirus research publishing: The rise and rise of COVID-19 clinical trials, 2020 (accessed August 1, 2020). https://www.natureindex. com/news-blog/the-top-coronavirus-research-articles-by-metrics.

G. B. Libotte, F. S. Lobato, G. M. Platt, and A. J. S. Neto, “Determination

of an optimal control strategy for vaccine administration in COVID-19 pandemic treatments,” Computer Methods and Programs in Biomedicine, vol. 196, no. November 2020, p. 105664, 2020.

M. Trawicki, “Deterministic SEIRs epidemic model for modeling vital dynamics, vaccinations, and temporary immunity,” Mathematics, vol. 5, no. 1, p. 7, 2017.

J. C. Blackwood and L. M. Childs, “An introduction to compartmental modeling for the budding infectious disease modeler,” Letters in Biomathematics, vol. 5, no. 1, pp. 195–221, 2018.

T. G. Ritto, R. Sampaio, and E. Cataldo, “Timoshenko beam with uncertainty on the boundary conditions,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol. 30, no. 4, pp. 295–303, 2008.

R. Storn and K. Price, “Differential evolution: a simple and efficient adaptive scheme for global optimization over continuous spaces,” Technical Report TR-95-012, International Computer Science Institute, Berkeley, vol. 2, no. 2, pp. 1–17, 1995.

R. Storn, K. Price, and J. A. Lampinen, Differential Evolution - A Practical Approach to Global Optimization. Springer - Natural Computing Series, 2005.

F. S. Lobato, M. S. Gonçalves, B. Jahn, C. A. Ap., and S. V., “Reliability-based optimization using differential evolution and inverse reliability analysis for engineering system design,” Journal of Optimization Theory and Applications, vol. 1, no. 1, pp. 1–33, 2017.

J. M. Heffernan, R. J. Smith, and L. M. Wahl, “Perspectives on the basic reproductive ratio,” Journal of the Royal Society, Interface, vol. 2, no. 4, pp. 281–293, 2005.

J. Riou and C. L. Althaus, “Pattern of early human-to-human transmission of Wuhan 2019 novel coronavirus (2019-nCoV), december 2019 to january 2020,” Eurosurveillance, vol. 25, no. 4, pp. 1–5, 2020.

C. Anastassopoulou, L. Russo, A. Tsakris, and C. Siettos, “Data-based analysis, modelling and forecasting of the COVID-19 outbreak,” PLoS ONE, vol. 3, no. 15, pp. 1–21, 2020.

M. J. Keeling and P. Rohani, Modeling Infection Diseases in Humans and

Animals. Princeton University Press, 2005.

S. Chatterjee, A. Sarkar, S. Chatterjee, M. Karmakar, and R. Paul, “Studying the progress of COVID-19 outbreak in India using SIRD model,” Indian Journal of Physics and Proceedings of the Indian Association for the Cultivation of Science (2004), vol. Jun 23, pp. 1–17, 2020.

World Health Organization, Naming the Coronavirus Disease (COVID- 19) and the Virus that Causes it, 2020 (accessed April 12, 2020). https://www.who.int/emergencies/diseases/novel-coronavirus-2019/technical-guidance/naming-the-coronavirus-disease-(covid-2019)-and-the-virus-that-causes-it.

A. Haldar and S. Mahadevan, Probability, Reliability, and Statistical Methods in Engineering Design, vol. 1. Chichester: John Wiley & Sons, 1 ed., 2000.

E. Nikolaidis and R. Burdisso, “Reliability Based Optimization: a Safety Index Approach,” Computers & Structures, vol. 28, no. 6, pp. 781–788, 1988.

J. Tu, K. K. Choi, and Y. H. Park, “A New Study on Reliability-Based Design Optimization,” Journal of Mechanical Design, vol. 121, no. 4, pp. 557–564, 1999.

Y.-T. Wu, H. R. Millwater, and T. A. Cruse, “Advanced Probabilistic Structural Analysis Method for Implicit Performance Functions,” AIAA Journal, vol. 28, no. 9, pp. 1663–1669, 1990.

Y.-T. Wu, “Computational Methods for Efficient Structural Reliability and Reliability Sensitivity Snalysis,” AIAA Journal, vol. 32, no. 8, pp. 1717–1723, 1994.

M. Rosenblatt, “Remarks on a Multivariate Transformation,” The Annals of Mathematical Statistics, vol. 23, no. 3, pp. 470–472, 1952.

A. Ghosh, S. Das, and A. K. Das, “A simple two-phase differential evolution for improved global numerical optimization,” Soft Computing, vol. 24, pp. 6151– 6167, 2020.

G. N. Vanderplaats, Numerical Optimization Techniques for Engineering Design. Vanderplaats Research and Development, Inc; 3rd edition, ISBN-10: 0944956017, ISBN-13: 978-0944956014, 449 pages, 2001.

S. Sanche, Y. Lin-Ting, C. Xu, E. Romero-Severson, N. Hengartner, and R. Ke, “High contagiousness and rapid spread of severe acute respiratory syndrome coronavirus 2,” Emerging Infectious Diseases, vol. 26, no. 7, 2020.

H. J. Wearing, P. Rohani, and M. J. Keeling, “Appropriate models for the management of infectious diseases,” PLoS Medicine, vol. 2, no. 7, 2005.

DOI: https://doi.org/10.5540/tcam.2021.022.01.00091

### Refbacks

• There are currently no refbacks.

Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)

Indexed in:

Desenvolvido por: