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Modeling and analysis of Caputo-type fractional-order SEIQR epidemic model

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Abstract

In this research, a susceptible-exposed-infected-quarantine-recovered-type epidemic model containing fractional-order differential equations is suggested and examined in order to better understand the dynamical behavior of the infectious illness in the presence of vaccination and treatments. The non-negative and bounded solutions of our proposed model are examined for existence and uniqueness. We investigate the explicit formulation of a threshold \(R_0\), often known as the basic reproduction number, using the next-generation matrix technique. Depending on the value of \(R_0\), one endemic equilibrium exists and is stable for \(R_0>1\), and one disease-free equilibrium (exist for all values of \(R_0\)) is stable for \(R_0<1\). This article has also noticed the emergence of a transcritical bifurcation. The relevance of using vaccination and treatments as controls has been met by formulating a fractional-order optimal control problem. The resulting theoretical conclusions are supported by a few numerical simulations. Ultimately, a global sensitivity analysis is carried out to identify the parameters that have the greatest influence.

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Acknowledgements

The authors would like to thank Dr. Pushpendra Kumar, Guest Editor of the journal and organizer committee member of ICFCTAN-2023, for his continuous assistance regarding the preparation of this article. Also, all the authors are thankful to the anonymous reviewers and the editor for their comments and suggestions to substantially improve the revised manuscript.

Funding

The research work of Suvankar Majee is financially supported by Council of Scientific and Industrial Research (CSIR), India (File No. 08/003(0142)/2020-EMR-I, dated: March 18, 2020), the work of Snehasis Barman is financially supported by NATIONAL FELLOWSHIP FOR SCHEDULED CAST STUDENTS(UGC-NFSC, UGC-Ref.No.:-191620004584, Dated:-20/05/2020), and the work of Prof. T. K. Kar is financially supported by Science and Engineering Research Board (File No. MTR/2022/000734, dated 19/12/2022), Department of Science and Technology, Government of India.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah, West Bengal, 711103, India

    Suvankar Majee, T. K. Kar & Snehasis Barman

  2. Department of Mathematics, Ramsaday College, Amta, Howrah, West Bengal, 711401, India

    Soovoojeet Jana

  3. Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, Uttar Pradesh, 208016, India

    D. K. Das

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Appendices

Appendix A: Proof of Theorem 3.2

Let us take two points \(X=(S,E,I,Q,R)\) and \(X_1=(S_1,E_1,I_1,Q_1,R_1)\) in \(\Omega \) and the mapping \(\Upsilon :\Omega \rightarrow {\mathbb {R}}^5\) where

$$\begin{aligned} \Upsilon (X)=\left( \Upsilon _1(X),\Upsilon _2(X),\Upsilon _3(X),\Upsilon _4(X),\Upsilon _5(X)\right) \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} \Upsilon _1(X)&=A^{\alpha } (1-u_1)-\frac{m^{\alpha } S I}{a+I}-d^{\alpha } S+\rho ^{\alpha } R\\ \Upsilon _2(X)&=\frac{m^{\alpha } SI}{a+I}-(\beta _1^{\alpha } +\gamma ^{\alpha } +\sigma _1^{\alpha } +d^{\alpha } )E\\ \Upsilon _3(X)&=\gamma ^{\alpha } E-(\beta _2^{\alpha } +\mu ^{\alpha } +b^{\alpha } u_2+d^{\alpha } +\delta ^{\alpha } )I\\ \Upsilon _4(X)&=\beta _1^{\alpha } E+\beta _2^{\alpha } I-(\sigma _2^{\alpha } +d^{\alpha } )Q\\ \Upsilon _5(X)&=A^{\alpha } u_1+\sigma _1^{\alpha } E+(\mu ^{\alpha } +b^{\alpha } u_2)I+\sigma _2^{\alpha } Q\\&\quad -(d^{\alpha } +\rho ^{\alpha } )R \end{aligned} \end{aligned}$$

Now for \(X,X_1\in \Omega \), we can write

$$\begin{aligned}&||\Upsilon (X)-\Upsilon (X_1)||\\&\quad =\left| -\frac{m^{\alpha } S I}{a+I}+\frac{m^{\alpha } S_1 I_1}{a+I_1}-d^{\alpha } (S-S_1)+\rho ^{\alpha } (R-R_1)\right| \\&\qquad +\left| \frac{m^{\alpha } SI}{a+I}\!-\!\frac{m^{\alpha } S_1 I_1}{a+I_1}\!-\!(\beta _1^{\alpha } \!+\!\gamma ^{\alpha } +\sigma _1^{\alpha } +d^{\alpha } )(E-E_1)\right| \\&\qquad +|\gamma ^{\alpha } (E-E_1)\!-\!(\beta _2^{\alpha } \!+\!\mu ^{\alpha } \!+\!b^{\alpha } u_2\!+\!d^{\alpha } +\delta ^{\alpha } )(I-I_1)|\\&\qquad +|\beta _1^{\alpha } (E-E_1)+\beta _2^{\alpha } (I-I_1)-(\sigma _2^{\alpha } +d^{\alpha } )(Q-Q_1)|\\&\qquad +|\sigma _1^{\alpha } (E-E_1)+(\mu ^{\alpha } +b^{\alpha } u_2)(I-I_1)\\&\qquad +\sigma _2^{\alpha } ( Q-Q_1)-(d^{\alpha } +\rho ^{\alpha } )(R-R_1)| \\&\quad \le \frac{|m^{\alpha } SI(a+I_1)-m^{\alpha } S_1I_1(a+I)|}{|(a+I)(a+I_1)|}\\&\qquad +d^{\alpha } |S-S_1|+\rho ^{\alpha } |R-R_1|\\&\qquad +\frac{|m^{\alpha } SI(a+I_1)-m^{\alpha } S_1I_1(a+I)|}{|(a+I)(a+I_1)|}\\&\qquad +(\beta _1^{\alpha } +\gamma ^{\alpha } +\sigma _1^{\alpha } +d^{\alpha } )|E-E_1|\\&\qquad +\gamma ^{\alpha } |E-E_1|+(\beta _2^{\alpha } +\mu ^{\alpha } +b^{\alpha } u_2+d^{\alpha }\\&\qquad +\delta ^{\alpha } )|I-I_1|+\beta _1^{\alpha } |E-E_1|+\beta _2^{\alpha } |I-I_1|\\&\qquad +(\sigma _2^{\alpha } +d^{\alpha } )|Q-Q_1|+\sigma _1^{\alpha } |E-E_1|\\&\qquad +(\mu ^{\alpha } +b^{\alpha } u_2)|I-I_1|+\sigma _2^{\alpha } |Q-Q_1|\\&\qquad +(d^{\alpha } +\rho ^{\alpha } )|R-R_1| \\&\quad = 2\frac{a m^{\alpha } |I||S\!-\!S_1|\!+\!a m^{\alpha } |S_1||I\!-\!I_1| \!+\!m^{\alpha } |I_1||I||S\!-\!S_1|}{|(a\!+\!I)(a\!+\!I_1)|}\\&\qquad +d^{\alpha } |S-S_1|+(2\beta _1^{\alpha } +2\gamma ^{\alpha } +2\sigma _1^{\alpha } +d^{\alpha } )|E-E_1|\\&\qquad +(2\beta _2^{\alpha } +2\mu ^{\alpha } +2b^{\alpha } u_2+d^{\alpha } +\delta ^{\alpha } )|I-I_1|\\&\qquad +(2\sigma _2^{\alpha } +d^{\alpha } )|Q-Q_1|+(d^{\alpha } +2\rho ^{\alpha } )|R-R_1| \\&\quad \le 2k\{a m^{\alpha } p_2|S-S_1|+a m^{\alpha } p_2|I-I_1|\\&\qquad +m^{\alpha } p_2^2|S-S_1|\}+d^{\alpha } |S-S_1|\\&\qquad +(2\beta _1^{\alpha } +2\gamma ^{\alpha } +2\sigma _1^{\alpha } +d^{\alpha } )|E-E_1|\\&\qquad +(2\beta _2^{\alpha } +2\mu ^{\alpha } +2b^{\alpha } u_2+d^{\alpha } +\delta ^{\alpha } )|I-I_1|\\&\qquad +(2\sigma _2^{\alpha } +d^{\alpha } )|Q-Q_1|+(d^{\alpha } +2\rho ^{\alpha } )|R-R_1|\\&\quad \text { where }|(a+I)(a+I_1)|\ge \frac{1}{k} \\&\quad \le (2akp_2m^{\alpha } +2kp_2^2 m^{\alpha } + d^{\alpha } ) |S-S_1|+(2\beta _1^{\alpha }\\&\qquad +2\gamma ^{\alpha } +2\sigma _1^{\alpha } +d^{\alpha } )|E-E_1|\\&\qquad +(2akp_2 m^{\alpha } +2\beta _2^{\alpha } +2\mu ^{\alpha } +2b^{\alpha } u_2+d^{\alpha } \\&\qquad +\delta ^{\alpha } )|I-I_1|+(2\sigma _2^{\alpha } +d^{\alpha } )|Q-Q_1|\\&\qquad +(d^{\alpha } +2\rho ^{\alpha } )|R-R_1|\\&\quad \le l(|S-S_1|+|E-E_1|+|I-I_1|+|Q-Q_1|\\&\qquad +|R-R_1|)= l||X-X_1|| \end{aligned}$$

where \(l=\max \{(2akp_2m^{\alpha } +2kp_2^2\,m^{\alpha } + d^{\alpha } ),(2\beta _1^{\alpha } +2\gamma ^{\alpha } +2\sigma _1^{\alpha } +d^{\alpha } ),(2akp_2 m^{\alpha } +2\beta _2^{\alpha } +2\mu ^{\alpha } +2b^{\alpha } u_2+d^{\alpha } +\delta ^{\alpha } ),(2\sigma _2^{\alpha } +d^{\alpha } ),(d^{\alpha } +2\rho ^{\alpha } )\}\). Therefore, we get \(||\Upsilon (X)-\Upsilon (X_1)||\le l||X-X_1||\), so \(\Upsilon \) satisfies the Lipschitz’s condition. Now lemma 3.1 implies that fractional-order system (3) has a unique solution \(X\in \Omega \) with respect to the starting condition \(X_{t_0}=(S_{t_0},E_{t_0},I_{t_0},Q_{t_0},R_{t_0})\in \Omega \). Hence the theorem.

Appendix B: Proof of Theorem 5.2

From Theorem 5.1, we see that the endemic equilibrium \(P_1\) exists when \(R_0>1\). Now, the Jacobian matrix of system (3) at the endemic equilibrium point \(P_1(S^*,E^*,I^*,Q^*,R^*)\) is as follows:

$$\begin{aligned} J(P_1)=\begin{pmatrix} -d^{\alpha }-\frac{m^{\alpha } I^*}{a+I^*}&{}0&{}\frac{m^{\alpha } S^* I^* }{(a+I^*)^2}-\frac{m^{\alpha } S^*}{a+I^*}&{}0&{}\rho ^{\alpha }\\ \frac{m^{\alpha } I^*}{a+I^*}&{}-(\beta _1^{\alpha }+\gamma ^{\alpha }+\sigma _1^{\alpha }+d^{\alpha })&{}-\frac{m^{\alpha } S^* I^* }{(a+I^*)^2}+\frac{m^{\alpha } S^*}{a+I^*}&{}0&{}0\\ 0&{}\gamma ^{\alpha }&{}-(\beta _2^{\alpha }+\mu ^{\alpha }+b^{\alpha } u_2+d^{\alpha }+\delta ^{\alpha })&{}0&{}0\\ 0&{}\beta _1^{\alpha }&{}\beta _2^{\alpha }&{}-(\sigma _2^{\alpha }+d^{\alpha })&{}0\\ 0&{}\sigma _1^{\alpha }&{}\mu ^{\alpha }+b^{\alpha } u_2&{}\sigma _2^{\alpha }&{}-(\rho ^{\alpha }+d^{\alpha }) \end{pmatrix} \end{aligned}$$

Then the characteristic equation is \(\lambda ^5+c_1\lambda ^4+c_2\lambda ^3+c_3\lambda ^2+c_4\lambda +c_5=0\) where \(c_i(i=1,2,3,4,5)\) are the coefficient of the characteristic equation, which can be obtained from \(det(J(P_1)-\lambda I)=0.\) It is clear that all \(c_i~(i=1,2,3,4,5)\) are positive. So, by Proposition 1 of Ref. [36], we can say that the endemic equilibrium \(P_1(S^*,E^*,I^*,Q^*,R^*)\) is locally asymptotically stable.

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Majee, S., Jana, S., Kar, T.K. et al. Modeling and analysis of Caputo-type fractional-order SEIQR epidemic model. Int. J. Dynam. Control 12, 148–166 (2024). https://doi.org/10.1007/s40435-023-01348-6

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