# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2022070
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## Financial risk contagion and optimal control

 School of Business Administration, Hunan University, Changsha, Hunan Province 410082, China

*Corresponding author: chenxiao1992@hnu.edu.cn(Chen Xiao)

Received  September 2021 Revised  March 2022 Early access April 2022

Fund Project: This research is supported by the National Nature Science Foundation of China under Grant Nos.71790593, 72001073 and 71972066. We also would like to thank the editor Kok Lay Teo, the two very constructive referees and Lanlan Luo very much for their valuable comments and suggestions

This paper combines the epidemic model and complex network theory to study the associated credit risk contagion among financial institutions, and the basic reproduction number is obtained. When the basic reproduction number is less than 1, the risk in the financial market finally disappears, otherwise, the risk always exists. Information campaign and treatment these two measures are introduced, and we obtain the expressions of the optimal control measure pair. The theoretical model is numerically analyzed. We find that applying only information campaign measure, only treatment control measure, and both information campaign measure and treatment control measure can decrease the density of infected enterprises compared with no control measure. Moreover, we make the sensitivity analysis of two relatively important parameters infectious rate and degree. Using the cost-effectiveness analysis method, we find that treatment control measure must be included in the optimal strategy, and whether to take information campaign measure at the same time is related to the values of parameters, that is, related to risk events.

Citation: Shou Chen, Chen Xiao. Financial risk contagion and optimal control. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022070
##### References:
 [1] A. Akgul, U. Fatima, M. S. Iqbal, N. Ahmedc, A. Raza, Z. Iqbal and M. Rafiq, A fractal fractional model for computer virus dynamics, Chaos Solitons & Fractals, 147 (2021), 110947, 9 pp. doi: 10.1016/j. chaos. 2021.110947. [2] D. Alan and W. Lucy, Reputational contagion and optimal regulatory forbearance, J. Financial Economics, 110 (2013), 642-658. [3] A. Alweimine, O. Bamaarouf, A. Rachadi and H. Ez-Zahraouy, Local routing protocols performance for computer virus elimination in complex networks, Physica A: Statistical Mechanics and its Applications, 536 (2019), 120984. [4] A. Bucci, D. L. Torre, D. Liuzzi and S. Marsiglio, Financial contagion and economic development: An epidemiological approach, J. Economic Behavior & Organization, 162 (2019), 211-228. [5] N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0. [6] M. Y. Gao, H. L. Yang, Q. Z. Xiao and M. Goh, COVID-19 lockdowns and air quality: Evidence from grey spatiotemporal forecasts, Socio-Economic Planning Sciences, (2022), 101228. [7] A. Garas, P. Argyrakis, C. Rozenblat, M. Tomassini and S. Havlin, Worldwide spreading of economic crisis, New Journal of Physics, 12 (2010), 185-188. [8] E. Eryarsoy, D. Delen, B. Davazdahemami and K. Topuz, A novel diffusion-based model for estimating cases, and fatalities in epidemics: The case of COVID-19, J. Business Research, 124 (2021), 163-178. [9] K. Gkillas, A. Tsagkanos and D. I. Vortelinos, Integration and risk contagion in financial crises: Evidence from international stock markets, J. Business Research, 104 (2019), 350-365. [10] G. Jiang, S. Li and M. Li, Dynamic rumor spreading of public opinion reversal on Weibo based on a two-stage SPNR model, Physica A: Statistical Mechanics and its Applications, 558 (2020), 125005, 10 pp. doi: 10.1016/j. physa. 2020.125005. [11] S. Jiang and H. Fan, Systemic risk in the interbank market with overlapping portfolios and cross-ownership of the subordinated debts, Physica A: Statistical Mechanics and its Applications, 562 (2021), 125355, 15 pp. doi: 10.1016/j. physa. 2020.125355. [12] T. K. Kar and S. Jana, A theoretical study on mathematical modelling of an infectious disease with application of optimal control, Bio Systems, 111 (2013), 37-50. [13] T. K. Kar, S. K. Nandi, S. Jana and M. Mandal, Stability and bifurcation analysis of an epidemic model with the effect of media, Chaos Solitons & Fractals, 120 (2019), 188-199.  doi: 10.1016/j.chaos.2019.01.025. [14] B. Kogut and G. Walker, The small world of Germany and durability of national networks, American Sociological Review, 66 (2001), 317-335. [15] O. Kostylenko, H. S. Rodrigues and D. Torres, Banking risk as an epidemiological model: An optimal control approach, Springer Proceedings in Mathematics & Statistics, 223 (2017), 165-176. [16] O. Kostylenko, H. S. Rodrigues and D. Torres, The spread of a financial virus through Europe and beyond, AIMS Math., 4 (2019), 86-98.  doi: 10.3934/Math.2019.1.86. [17] A. Lahrouz, H. E. Mahjour, A. Settati and A. Bernoussil, Dynamics and optimal control of a non-linear epidemic model with relapse and cure, Physica A: Statistical Mechanics and its Applications, 496 (2018), 299-317.  doi: 10.1016/j.physa.2018.01.007. [18] A. Lahrouz, L. Omari and D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model. Control, 16 (2011), 59-76.  doi: 10.15388/NA.16.1.14115. [19] B. Li, Y. Wang, K. Zhang and G. R. Duan, Constrained feedback control for spacecraft reorientation with an optimal gain, IEEE Transactions on Aerospace and Electronic Systems, 57 (2021), 3916-3926. [20] B. Li, J. Zhang, L. Dai, K. L. Teo and S. Wang, A hybrid offline optimization method for reconfiguration of multi-UAV formations, IEEE Transactions on Aerospace and Electronic Systems, 57 (2021), 506-520. [21] R. M. May, S. A. Levin and G. Sugihara, Complex systems: Ecology for bankers, Nature, 451 (2008), 893-895. [22] A. K. Misra, A. Sharma and J. B. Shukla, Stability analysis and optimal control of an epidemic model with awareness programs by media, Bio Systems, 13 (2015), 53-62. [23] S. K. Nandi, S. Jana, M. Mandal and T. K. Kar, Mathematical analysis of an epidemic system in the presence of optimal control and population dispersal, Biophysical Reviews and Letters, 13 (2018), 1-17. [24] D. Philippas, Y. Koutelidakis and A. Leontitsis, Insights into European interbank network contagion, Managerial Finance, 41 (2015), 754-772. [25] S. Poledna, S. Martínez-Jaramillo, F. Caccioli and S. Thurner, Quantification of systemic risk from overlapping portfolios in the financial system, J. Financial Stability, 52 (2021), 100808. [26] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience, New York, 1962. [27] Q. Qian, Y. Yang, J. Gu and H. Feng, Information authenticity, spreading willingness and credit risk contagion-A dual-layer network perspective, Physica A: Statistical Mechanics and its Applications, 536 (2019), 122519. [28] Y. F. Sun, G. Aw, B. Li, K. L. Teo and J. Sun, CVaR-based robust models for portfolio selection, J. Ind. Manag. Optim., 16 (2020), 1861-1871.  doi: 10.3934/jimo.2019032. [29] Y. F. Sun, G. Aw, R. Loxton and K. L. Teo, An optimal machine maintenance problem with probabilistic state constraints, Inform. Sci., 281 (2014), 386-398.  doi: 10.1016/j.ins.2014.05.051. [30] Y. F. Sun, G. Aw, R. Loxton and K. L. Teo, Chance-constrained optimization for pension fund portfolios in the presence of default risk, European J. Oper. Res., 256 (2017), 205-214.  doi: 10.1016/j.ejor.2016.06.019. [31] I. Takaidza, O. D. Makinde and O. K. Okosun, Computational modelling and optimal control of Ebola virus disease with non-linear incidence rate, J. Physics Conference, 818 (2017), 012003. [32] K. L. Teo, B. Li, C. Yu and V. Rehbock, Applied and Computational Optimal Control: A Control Parametrization Approach, Springer, 2021. doi: 10.1007/978-3-030-69913-0. [33] G. T. Tilahun, O. D. Makinde and D. Malonza, Co-dynamics of pneumonia and typhoid fever diseases with cost effective optimal control analysis, Appl. Math. Comput., 316 (2018), 438-459.  doi: 10.1016/j.amc.2017.07.063. [34] M. Tirado, Complex network for a crisis contagion on an interbank system, International J. Modern Physic C, 23 (2012), 1-20. [35] M. Toivanen, Contagion in the interbank network: An epidemiological approach, Research Discussion Papers, (2013). [36] L. Wang, S. Li and T. Chen, Investor behavior, information disclosure strategy and counterparty credit risk contagion, Chaos Solitons & Fractals, 119 (2019), 37-49.  doi: 10.1016/j.chaos.2018.12.007. [37] P. Wang and L. Zong, Contagion effects and risk transmission channels in the housing, stock, interest rate and currency markets: An empirical study in China and the US, The North American Journal of Economics and Finance, 54 (2020), 101113. [38] D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442. [39] P. Xu and X. Yu, Research on the application of risk contagion model of mutual guarantee financing for SMEs cluster, Accounting Research, 1 (2018), 82-88. [40] F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings, VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems, J. Ind. Manag. Optim., 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781. [41] G. F. Yu, D. F. Li, D. C. Liang and G. X. Li, An intuitionistic fuzzy multi-objective goal programming approach to portfolio selection, International J. Information Technology and Decision Making, 20 (2021), 1477-1497. [42] W. Zhang, G. Zhang and J. Helwege, Cross country linkages and transmission of sovereign risk: Evidence from China's credit default swaps, J. Financial Stability, 58 (2022), 100838. [43] Q. Zhou, S. Sun and Q. Liu, The capital flow of stock market studies based on epidemic model with double delays, Physica A: Statistical Mechanics and its Applications, 526 (2019), 120733, 18 pp. doi: 10.1016/j. physa. 2019.03.098.

show all references

##### References:
 [1] A. Akgul, U. Fatima, M. S. Iqbal, N. Ahmedc, A. Raza, Z. Iqbal and M. Rafiq, A fractal fractional model for computer virus dynamics, Chaos Solitons & Fractals, 147 (2021), 110947, 9 pp. doi: 10.1016/j. chaos. 2021.110947. [2] D. Alan and W. Lucy, Reputational contagion and optimal regulatory forbearance, J. Financial Economics, 110 (2013), 642-658. [3] A. Alweimine, O. Bamaarouf, A. Rachadi and H. Ez-Zahraouy, Local routing protocols performance for computer virus elimination in complex networks, Physica A: Statistical Mechanics and its Applications, 536 (2019), 120984. [4] A. Bucci, D. L. Torre, D. Liuzzi and S. Marsiglio, Financial contagion and economic development: An epidemiological approach, J. Economic Behavior & Organization, 162 (2019), 211-228. [5] N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0. [6] M. Y. Gao, H. L. Yang, Q. Z. Xiao and M. Goh, COVID-19 lockdowns and air quality: Evidence from grey spatiotemporal forecasts, Socio-Economic Planning Sciences, (2022), 101228. [7] A. Garas, P. Argyrakis, C. Rozenblat, M. Tomassini and S. Havlin, Worldwide spreading of economic crisis, New Journal of Physics, 12 (2010), 185-188. [8] E. Eryarsoy, D. Delen, B. Davazdahemami and K. Topuz, A novel diffusion-based model for estimating cases, and fatalities in epidemics: The case of COVID-19, J. Business Research, 124 (2021), 163-178. [9] K. Gkillas, A. Tsagkanos and D. I. Vortelinos, Integration and risk contagion in financial crises: Evidence from international stock markets, J. Business Research, 104 (2019), 350-365. [10] G. Jiang, S. Li and M. Li, Dynamic rumor spreading of public opinion reversal on Weibo based on a two-stage SPNR model, Physica A: Statistical Mechanics and its Applications, 558 (2020), 125005, 10 pp. doi: 10.1016/j. physa. 2020.125005. [11] S. Jiang and H. Fan, Systemic risk in the interbank market with overlapping portfolios and cross-ownership of the subordinated debts, Physica A: Statistical Mechanics and its Applications, 562 (2021), 125355, 15 pp. doi: 10.1016/j. physa. 2020.125355. [12] T. K. Kar and S. Jana, A theoretical study on mathematical modelling of an infectious disease with application of optimal control, Bio Systems, 111 (2013), 37-50. [13] T. K. Kar, S. K. Nandi, S. Jana and M. Mandal, Stability and bifurcation analysis of an epidemic model with the effect of media, Chaos Solitons & Fractals, 120 (2019), 188-199.  doi: 10.1016/j.chaos.2019.01.025. [14] B. Kogut and G. Walker, The small world of Germany and durability of national networks, American Sociological Review, 66 (2001), 317-335. [15] O. Kostylenko, H. S. Rodrigues and D. Torres, Banking risk as an epidemiological model: An optimal control approach, Springer Proceedings in Mathematics & Statistics, 223 (2017), 165-176. [16] O. Kostylenko, H. S. Rodrigues and D. Torres, The spread of a financial virus through Europe and beyond, AIMS Math., 4 (2019), 86-98.  doi: 10.3934/Math.2019.1.86. [17] A. Lahrouz, H. E. Mahjour, A. Settati and A. Bernoussil, Dynamics and optimal control of a non-linear epidemic model with relapse and cure, Physica A: Statistical Mechanics and its Applications, 496 (2018), 299-317.  doi: 10.1016/j.physa.2018.01.007. [18] A. Lahrouz, L. Omari and D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model. Control, 16 (2011), 59-76.  doi: 10.15388/NA.16.1.14115. [19] B. Li, Y. Wang, K. Zhang and G. R. Duan, Constrained feedback control for spacecraft reorientation with an optimal gain, IEEE Transactions on Aerospace and Electronic Systems, 57 (2021), 3916-3926. [20] B. Li, J. Zhang, L. Dai, K. L. Teo and S. Wang, A hybrid offline optimization method for reconfiguration of multi-UAV formations, IEEE Transactions on Aerospace and Electronic Systems, 57 (2021), 506-520. [21] R. M. May, S. A. Levin and G. Sugihara, Complex systems: Ecology for bankers, Nature, 451 (2008), 893-895. [22] A. K. Misra, A. Sharma and J. B. Shukla, Stability analysis and optimal control of an epidemic model with awareness programs by media, Bio Systems, 13 (2015), 53-62. [23] S. K. Nandi, S. Jana, M. Mandal and T. K. Kar, Mathematical analysis of an epidemic system in the presence of optimal control and population dispersal, Biophysical Reviews and Letters, 13 (2018), 1-17. [24] D. Philippas, Y. Koutelidakis and A. Leontitsis, Insights into European interbank network contagion, Managerial Finance, 41 (2015), 754-772. [25] S. Poledna, S. Martínez-Jaramillo, F. Caccioli and S. Thurner, Quantification of systemic risk from overlapping portfolios in the financial system, J. Financial Stability, 52 (2021), 100808. [26] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience, New York, 1962. [27] Q. Qian, Y. Yang, J. Gu and H. Feng, Information authenticity, spreading willingness and credit risk contagion-A dual-layer network perspective, Physica A: Statistical Mechanics and its Applications, 536 (2019), 122519. [28] Y. F. Sun, G. Aw, B. Li, K. L. Teo and J. Sun, CVaR-based robust models for portfolio selection, J. Ind. Manag. Optim., 16 (2020), 1861-1871.  doi: 10.3934/jimo.2019032. [29] Y. F. Sun, G. Aw, R. Loxton and K. L. Teo, An optimal machine maintenance problem with probabilistic state constraints, Inform. Sci., 281 (2014), 386-398.  doi: 10.1016/j.ins.2014.05.051. [30] Y. F. Sun, G. Aw, R. Loxton and K. L. Teo, Chance-constrained optimization for pension fund portfolios in the presence of default risk, European J. Oper. Res., 256 (2017), 205-214.  doi: 10.1016/j.ejor.2016.06.019. [31] I. Takaidza, O. D. Makinde and O. K. Okosun, Computational modelling and optimal control of Ebola virus disease with non-linear incidence rate, J. Physics Conference, 818 (2017), 012003. [32] K. L. Teo, B. Li, C. Yu and V. Rehbock, Applied and Computational Optimal Control: A Control Parametrization Approach, Springer, 2021. doi: 10.1007/978-3-030-69913-0. [33] G. T. Tilahun, O. D. Makinde and D. Malonza, Co-dynamics of pneumonia and typhoid fever diseases with cost effective optimal control analysis, Appl. Math. Comput., 316 (2018), 438-459.  doi: 10.1016/j.amc.2017.07.063. [34] M. Tirado, Complex network for a crisis contagion on an interbank system, International J. Modern Physic C, 23 (2012), 1-20. [35] M. Toivanen, Contagion in the interbank network: An epidemiological approach, Research Discussion Papers, (2013). [36] L. Wang, S. Li and T. Chen, Investor behavior, information disclosure strategy and counterparty credit risk contagion, Chaos Solitons & Fractals, 119 (2019), 37-49.  doi: 10.1016/j.chaos.2018.12.007. [37] P. Wang and L. Zong, Contagion effects and risk transmission channels in the housing, stock, interest rate and currency markets: An empirical study in China and the US, The North American Journal of Economics and Finance, 54 (2020), 101113. [38] D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442. [39] P. Xu and X. Yu, Research on the application of risk contagion model of mutual guarantee financing for SMEs cluster, Accounting Research, 1 (2018), 82-88. [40] F. Yang, K. L. Teo, R. Loxton, V. Rehbock, B. Li, C. Yu and L. Jennings, VISUAL MISER: An efficient user-friendly visual program for solving optimal control problems, J. Ind. Manag. Optim., 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781. [41] G. F. Yu, D. F. Li, D. C. Liang and G. X. Li, An intuitionistic fuzzy multi-objective goal programming approach to portfolio selection, International J. Information Technology and Decision Making, 20 (2021), 1477-1497. [42] W. Zhang, G. Zhang and J. Helwege, Cross country linkages and transmission of sovereign risk: Evidence from China's credit default swaps, J. Financial Stability, 58 (2022), 100838. [43] Q. Zhou, S. Sun and Q. Liu, The capital flow of stock market studies based on epidemic model with double delays, Physica A: Statistical Mechanics and its Applications, 526 (2019), 120733, 18 pp. doi: 10.1016/j. physa. 2019.03.098.
The process of risk contagion among associated enterprises
Comparison of the density of infected enterprises under Strategies A and B
Variation of the adjoint variables with Strategy B
Variation of the control parameter as time evolves with Strategy B
Comparison of the density of infected enterprises under Strategies A and C
Variation of the adjoint variables with Strategy C
Variation of the control parameter as time evolves with Strategy C
Comparison of the density of infected enterprises under Strategies A and D
Variation of the adjoint variables with Strategy D
Variation of the control parameter as time evolves with Strategy D
Variation of the density of infected enterprises for different control strategies
Control strategies in order of increasing the amount of total infection averted
 Strategies Total infection averted Total cost Strategy A 0 0 Strategy C 53.34 45.94 Strategy B 62.21 65.87 Strategy D 64.28 46.31
 Strategies Total infection averted Total cost Strategy A 0 0 Strategy C 53.34 45.94 Strategy B 62.21 65.87 Strategy D 64.28 46.31
The total infection averted and the total cost with different $\alpha$ under Strategy B
 $\alpha$ Total infection averted Total cost 0.1 27.62 24.44 0.2 48.72 46.16 0.3 56.28 55.75 0.4 60.13 61.11 0.5 62.47 64.70 0.6 64.03 67.40 0.7 65.14 69.27 0.8 65.98 71.85 0.9 66.64 73.93 1.0 67.17 75.23
 $\alpha$ Total infection averted Total cost 0.1 27.62 24.44 0.2 48.72 46.16 0.3 56.28 55.75 0.4 60.13 61.11 0.5 62.47 64.70 0.6 64.03 67.40 0.7 65.14 69.27 0.8 65.98 71.85 0.9 66.64 73.93 1.0 67.17 75.23
The total infection averted and the total cost with different $\alpha$ under Strategy C
 $\alpha$ Total infection averted Total cost 0.1 30.39 4.07 0.2 50.62 15.02 0.3 57.39 34.66 0.4 57.62 48.01 0.5 54.35 46.59 0.6 52.01 45.07 0.7 50.30 43.99 0.8 49.01 43.09 0.9 48.02 42.44 1.0 47.21 41.94
 $\alpha$ Total infection averted Total cost 0.1 30.39 4.07 0.2 50.62 15.02 0.3 57.39 34.66 0.4 57.62 48.01 0.5 54.35 46.59 0.6 52.01 45.07 0.7 50.30 43.99 0.8 49.01 43.09 0.9 48.02 42.44 1.0 47.21 41.94
The total infection averted and the total cost with different $\alpha$ under Strategy D
 $\alpha$ Total infection averted Total cost 0.1 30.42 3.91 0.2 50.79 12.49 0.3 58.04 23.67 0.4 61.76 34.33 0.5 64.02 43.55 0.6 65.54 50.84 0.7 66.63 56.65 0.8 67.44 61.51 0.9 68.08 65.43 1.0 68.60 68.74
 $\alpha$ Total infection averted Total cost 0.1 30.42 3.91 0.2 50.79 12.49 0.3 58.04 23.67 0.4 61.76 34.33 0.5 64.02 43.55 0.6 65.54 50.84 0.7 66.63 56.65 0.8 67.44 61.51 0.9 68.08 65.43 1.0 68.60 68.74
The total infection averted and the total cost with different $k$ under Strategy B
 $k$ Total infection averted Total cost 2 59.94 61.19 3 63.89 66.90 4 65.89 71.61 5 67.09 75.14 6 67.89 76.63 7 68.46 77.40 8 67.92 77.80
 $k$ Total infection averted Total cost 2 59.94 61.19 3 63.89 66.90 4 65.89 71.61 5 67.09 75.14 6 67.89 76.63 7 68.46 77.40 8 67.92 77.80
The total infection averted and the total cost with different $k$ under Strategy C
 $k$ Total infection averted Total cost 2 57.86 48.02 3 52.20 45.34 4 49.17 43.21 5 47.32 42.11 6 46.11 41.11 7 45.25 40.49 8 44.35 39.90
 $k$ Total infection averted Total cost 2 57.86 48.02 3 52.20 45.34 4 49.17 43.21 5 47.32 42.11 6 46.11 41.11 7 45.25 40.49 8 44.35 39.90
The total infection averted and the total cost with different $k$ under Strategy D
 $k$ Total infection averted Total cost 2 61.57 33.74 3 65.40 50.20 4 67.35 61.12 5 68.52 68.19 6 69.30 73.11 7 69.86 77.83 8 69.85 80.73
 $k$ Total infection averted Total cost 2 61.57 33.74 3 65.40 50.20 4 67.35 61.12 5 68.52 68.19 6 69.30 73.11 7 69.86 77.83 8 69.85 80.73
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