Farklı İnsidans Oranlarının Etkisi Altında Bir SEIR Epidemiyolojik Modelinin Optimal Kontrolü

Bu çalışmada, farklı insidans hızı fonksiyonlarının etkisi altında bir kesirli SEIR salgın modeli için optimal kontrol problemi incelenmiştir. Bu fonksiyon, bir popülasyonun duyarlı ve enfekte bireyleri arasındaki etkileşimi ifade ederek bir salgın hastalığın en gerçekçi biçimde modellenmesinde dikkate değer bir göreve sahiptir. Bu etkileşim, hastalığın pandemiye dönüşüp dönüşmeyeceği konusunda oldukça belirleyicidir. Dolayısıyla bu fonksiyon, salgının durumuna göre farklı şekillerde tanımlanabilir. Bu çalışmada, bilineer ve doymuş insidans fonksiyonlarının etkileri tartışılmaktadır. İncelenen epidemiyolojik model, Caputo kesirli türevlidir. Temel amaç, enfekte hasta sayısını ve duyarlı önleyici bir tedbir mahiyetinde duyarlı bireylere verilen eğitim maaliyetini en aza indirmektir. Bu amaçla öncelikle ele alınan modelin optimallik koşullarına karşılık gelen Euler-Lagrange denklemleri hesaplanır. Daha sonra sağ ve sol kesirli Caputo türevli optimal sistem, kesirli Euler yöntemi ile birleştirilmiş ileri-geri süpürme yöntemi ile sayısal olarak çözülmüştür. Simülasyon sonuçlarına göre duyarlı bireylere verilen eğitimin salgını yavaşlatmada yadsınamaz derecede etkili olduğu görülmektedir.

Anahtar Kelimeler:

Optimal kontrol, Caputo kesirli türev, SEIR modeli, İnsidans oranı, kesirli Euler yöntemi.

Optimal Control for A SEIR Epidemiological Model Under the Effect of Different Incidence Rates

In this study, optimal control problem for a fractional SEIR epidemiological model under the effect of bilinear and saturate incidence rate functions is investigated. These rates play an important role in the realistic modeling of an epidemic by describing the interaction between susceptible and infected individuals of a population. This interaction is highly decisive in whether the disease will turn into a pandemic or not. Therefore, these functions can be defined in different forms depending on the course of the epidemic. The model discussed in this study is defined in terms of Caputo. Dimensional compatibility is guaranteed before posing the optimal control problem. The main objective of the proposed optimal control problem is to minimize the number of infected individuals and the cost of education given to susceptible individuals as a preventive measure. Euler-Lagrange equations corresponding to the optimality conditions of the considered model are first determined by Hamiltonian’s formalism. Afterward, the optimal system with right and left fractional Caputo derivatives are solved numerically by the forward-backward sweep method combined with the fractional Euler method. Optimal solutions are interpreted graphically for varying values of the incidence rate coefficients and the fractional parameter. According to the simulation results, it is seen that the education given to susceptible individuals is significantly effective in slowing down the epidemic.

Keywords:

Optimal control, Hamiltonian formalism, Caputo fractional derivative SEIR model, Bilinear incidence rate, Saturated incidence rate, Forward-backward sweep method, Fractional Euler method,

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[1] H. W. Hethcote, “The mathematics of infectious diseases,” SIAM review, vol. 42, no. 4, pp. 599-653, 2000.
[2] J. D. Murray, Mathematical biology I. An introduction, 3rd ed., New York, USA: Springer, 2002.
[3] L. J. Allen, F. Brauer, P. Van den Driessche, and J. Wu, Mathematical epidemiology, vol. 1945, Berlin, Germany: Springer, 2008.
[4] J. C. Frauenthal, Mathematical modeling in epidemiology, New York, USA: Springer, 2012.
[5] J. Mishra, R. Agarwal, and A. Atangana (Eds.), Mathematical Modeling and Soft Computing in Epidemiology, New York, USA: CRC Press, 2020.
[6] V. S. Erturk and P. Kumar, “Solution of a COVID-19 model via new generalized Caputo-type fractional derivatives,” Chaos, Solitons & Fractals, vol. 139, Article ID 110280, 2020.
[7] M. A. Dokuyucu and H. Dutta, “A fractional order model for Ebola Virus with the new Caputo fractional derivative without singular kernel,” Chaos, Solitons & Fractals, vol. 134, Article ID 109717, 2020.
[8] P. A. Naik, M. Yavuz, S. Qureshi, J. Zu, and S. Townley, “Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan,” The European Physical Journal Plus, vol. 135, no. 10, pp. 1-42, 2020.
[9] A. Akgül, N. Ahmed, A. Raza, Z. Iqbal, M. Rafiq, D. Baleanu, and M. A. U. Rehman, “New applications related to Covid-19,” Results in physics, vol. 20, Article ID 103663, 2021.
[10] P. Veeresha, E. Ilhan, D. G. Prakasha, H. M. Baskonus, and W. Gao, “A new numerical investigation of fractional order susceptible-infected-recovered epidemic model of childhood disease,” Alexandria Engineering Journal, vol. 61, no. 2, pp. 1747-1756, 2022.
[11] N. Sene, “SIR epidemic model with Mittag–Leffler fractional derivative. Chaos, Solitons & Fractals,” vol. 137, Article ID 109833, 2020.
[12] B. S. T. Alkahtani and I. Koca, “Fractional stochastic sır model,” Results in Physics, vol. 24, Article ID 104124, 2021.
[13] B. Daşbaşı, “Stability analysis of an incommensurate fractional-order SIR model,” Mathematical Modelling and Numerical Simulation with Applications, vol. 1, no. 1, pp. 44-55, 2021.
[14] A. Kaddar, A. Abta H. T. Alaoui, “A comparison of delayed SIR and SEIR epidemic models,” Nonlinear Analysis: Modelling and Control, vol. 16, no. 2, pp. 181-190, 2011.
[15] E. Demirci and A. Unal, “A fractional order SEIR model with density dependent death rate,” Hacettepe Journal of Mathematics and Statistics, vol. 40, no. 2, pp. 287-295, 2011.
[16] S. He, Y. Peng, and K. Sun, “SEIR modeling of the COVID-19 and its dynamics,” Nonlinear dynamics, vol. 101, no. 3, pp. 1667-1680, 2020.
[17] D. J. Gerberry and F. A. Milner, “An SEIQR model for childhood diseases,” Journal of Mathematical Biology, vol. 59, no. 4, pp. 535-561, 2009.
[18] M. Sinan, A. Ali, K. Shah, T. A. Assiri, and T. A. Nofal, “Stability analysis and optimal control of Covid-19 pandemic SEIQR fractional mathematical model with harmonic mean type incidence rate and treatment,” Results in Physics, vol. 22, Article ID 103873, 2021.
[19] H. M. Youssef, N. Alghamdi, M. A. Ezzat, A. A. El-Bary, and A. M. Shawky, “A proposed modified SEIQR epidemic model to analyze the COVID-19 spreading in Saudi Arabia,” Alexandria Engineering Journal, vol. 61, no. 3, pp. 2456-2470, 2022.
[20] S. Lenhart and J. T. Workman, Optimal control applied to biological models, London, UK: Chapman and Hall/CRC, 2007.
[21] D. S. Naidu, Optimal control systems, London, UK: CRC press, 2002.
[22] J. J. Wang, J. Z. Zhang, and Z. Jin, “Analysis of an SIR model with bilinear incidence rate,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 2390-2402, 2010.
[23] R. M. Anderson and R. M. May, Infectious diseases of humans: dynamics and control, Oxford, UK: Oxford university press, 1992.
[24] J. R. Beddington, “Mutual interference between parasites or predators and its effect on searching efficiency,” J. Anim. Ecol., vol. 44, pp. 331–340, 1975.
[25] D. L. DeAngelis, R. A. Goldsten, and R.V. O’Neill, “A model for trophic interaction,” Ecology, vol. 56, pp. 881–892, 1975.
[26] G. H. Li, Y. X. Zhang, “Dynamic behaviors of a modified SIR model in epidemic diseases using nonlinear incidence and recovery rates,” PLoS One, vol. 12, no. 4, Article ID e0175789, 2017.
[27] U. D. Purwati, F. Riyudha, and H. Tasman, “Optimal control of a discrete age-structured model for tuberculosis transmission,” Heliyon, vol. 6, no. 1, Article ID e03030, pp. 1-10, 2020.
[28] M. A. Khan, S. Ullah, Y. Khan, M. Farhan, “Modeling and scientific computing for the transmission dynamics of Avian Influenza with Half-Saturated Incidence,” International Journal of Modeling, Simulation, and Scientific Computing, vol. 11, no. 04, Article ID 2050035, 2020.
[29] M. A. Khan, S. Ullah, S. Ullah, and M. Farhan, “Fractional order SEIR model with generalized incidence rate,” AIMS Math., vol. 5, no. 4, pp. 2843-2857, 2020.
[30] R. Shi, J. Ren, and C. Wang, “Analysis of a fractional order mathematical model for tuberculosis with optimal control,” J. Nonlinear Funct. Anal., vol. 2020, Article ID 15, pp. 1-18, 2020.
[31] I. A. Baba, R. A. Abdulkadir, and P. Esmaili, “Analysis of tuberculosis model with saturated incidence rate and optimal control,” Physica A, vol. 540, Article ID 123237, 2020.
[32] R. Zarin, I. Ahmed, P. Kuman, A. Zeb, and A. Din, “Fractional modeling and optimal control analysis of rabies virus under the convex incidence rate,” Results in Physics, vol. 28, Article ID 104665, 2021.
[33] A. Khan, R. Zarin, A. Akgül, A. Saeed, and T. Gul, “Fractional optimal control of COVID-19 pandemic model with generalized Mittag-Leffler function,” Advances in Difference Equations, vol. 2021, no. 1, pp. 1-22, 2021.
[34] A. Boukhouima, E. M. Lotfi, M. Mahrouf, S. Rosa, D. F. Torres, and N. Yousfi, “Stability analysis and optimal control of a fractional HIV-AIDS epidemic model with memory and general incidence rate,” The European Physical Journal Plus, vol. 136, Article ID 103, pp. 1-20, 2021.
[35] I. Podlubny, Fractional Differential Equations, New York, USA: Academic Press, 1999.
[36] O. P. Agrawal, “A general formulation and solution scheme for fractional optimal control problems,” Nonlinear Dynamics, vol. 38, no. 1, pp. 323-337, 2004.
[37] O. P. Agrawal, “A formulation and numerical scheme for fractional optimal control problems,” Journal of Vibration and Control, vol. 1, no. 9-10, pp. 1291-1299, 2008.
[38] H. Kheiri and M. Jafari, “Optimal control of a fractional-order model for the HIV/AIDS epidemic,” International Journal of Biomathematics, vol. 11, no. 07, 1850086.
[39] I. Ameen, D. Baleanu, and H. M. Ali, “An efficient algorithm for solving the fractional optimal control of SIRV epidemic model with a combination of vaccination and treatment,” Chaos, Solitons & Fractals, vol. 137, Article ID 109892, 2020.
[40] A. Lotfi, M. Dehghan, and S. A. Yousefi, “A numerical technique for solving fractional optimal control problems,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1055-1067, 2011.
[41] X. Liu and L. Yang, “Stability analysis of an SEIQV epidemic model with saturated incidence rate,” Nonlinear Anal-Real, vol. 13, pp. 2671–2679, 2012.
[42] P. H. Crowley and E. K. Martin, “Functional responses and interference within and between year classes of a dragonfly population,” J. North. Am. Benthol. Soc., vol. 8, pp. 211–221, 1989.
[43] K. Hattaf, M. Mahrouf, J. Adnani, and N. Yousfi, “Qualitative analysis of a stochastic epidemic model with specific functional response and temporary immunity,” Physica A: Statistical Mechanics and its Applications, vol. 490, pp. 591-600, 2018.
[44] A. Rachah, “Analysis, simulation and optimal control of a SEIR model for Ebola virus with demographic effects,” Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 67, no. 1, pp. 179-197, 2018.