1. Introduction

2. Mathematical Modelling

3. Mathematical analysis of the COVID-19 model

3.4. Backward Bifurcation Analysis of the COVID-19 Model

From condition-(c) of Theorem 2, we have the possibility of having two endemic equilibrium whenever ${\mathcal{R}}_c<1$ and hence there may exist backward bifurcation phenomenon. Now we will apply center manifold theory [37,40] to examine the existence of backward bifurcation. For this let us consider the changes of variable as $S=x_1,V=x_2,E=x_3,P=x_4,A=x_5,I=x_6,andR=x_7,$ so that in vector form the model (2) can be written as

$${\displaystyle \frac{dX}{dt}}={\left(f_1,f_2,f_3,f_4,f_5,f_6,f_7\right)}^T$$

where $X={(x_1,x_2,x_3,x_4,x_5,x_6,x_7)}^T$ and then we have

$${\displaystyle \frac{d{x}_1}{dt}}=f_1=\Lambda -\beta \left(x_6+\tau x_4\right)x_1+\sigma x_7-K_1{x}_1$$

(14a)

$${\displaystyle \frac{d{x}_2}{dt}}=f_2=\rho x_1-\left(1-\epsilon \right)\beta \left(x_6+\tau x_4\right)x_2-K_2{x}_2$$

(14b)

$${\displaystyle \frac{d{x}_3}{dt}}=f_3=\beta \left(x_6+\tau x_4\right)x_1+\left(1-\epsilon \right)\beta \left(x_6+\tau x_4\right)x_2-K_3{x}_3$$

(14c)

$${\displaystyle \frac{d{x}_4}{dt}}=f_4=\omega x_3-K_4{x}_4$$

(14d)

$${\displaystyle \frac{d{x}_5}{dt}}=f_5=\omega x_3-K_5{x}_5$$

(14e)

$${\displaystyle \frac{d{x}_6}{dt}}=f_6=\omega x_3+{\kappa }_1{x}_4-K_6{x}_6$$

(14f)

$${\displaystyle \frac{d{x}_7}{dt}}=f_7={\kappa }_2{x}_6-K_7{x}_7$$

(14g)

The Jacobian of the system (14) is given by:

where

$$J_1={\displaystyle \frac{\beta \mathsf{\Lambda }}{\rho +\mu }},J_2={\displaystyle \frac{\left(1-\epsilon \right)\rho \beta \mathsf{\Lambda }}{\mu \left(\rho +\mu \right)}},J_3={\displaystyle \frac{\beta \left\{\left(1-\epsilon \right)\rho +\mu \right\}\mathsf{\Lambda }}{\mu \left(\rho +\mu \right)}}$$

Now consider, ${\mathcal{R}}_c=1and\beta ={\beta }^{*}$ is a bifurcation parameter. Thus we get,

$$\beta ={\beta }^{*}={\displaystyle \frac{{\mu K}_3{K}_4{K}_5}{\tau \omega \mu K_5+\omega K_4^2\left\{S^{*}+\left(1-\epsilon \right)V^{*}\right\}}}$$

The Jacobian $\mathcal{J}\left({\mathcal{E}}_c\right)$ of (14) with $\beta ={\beta }^{*}$, denoted by ${\mathcal{J}}_{{\beta }^{*}}$, has a simple zero eigenvalue (with all other eigenvalues having negative real part). Hence, the center manifold theory can be used to analyze the dynamics of the model (2).

Eigenvector of ${\mathcal{J}}_{{\beta }^{\mathrm{*}}}={\left.\mathcal{J}\left({\mathcal{E}}_c\right)\right|}_{\beta ={\beta }^{\mathrm{*}}}$:

When ${\mathcal{R}}_c=1$, the Jacobian (${\mathcal{J}}_{{\beta }^{*}}$) of (14) has a right eigenvector given by $W={(w_1,w_2,w_3,w_4,w_5,w_6,w_7)}^T$ where,

$$w_1={\displaystyle \frac{J_1{w}_4+\tau J_1{w}_6-\sigma w_7}{K_1}},w_2={\displaystyle \frac{\rho w_1-J_2{w}_4+\tau {J_{2}w}_6}{K_2}},w_3=w_3,$$

$$w_4={\displaystyle \frac{\omega w_3}{K_4}},w_5={\displaystyle \frac{\omega w_3}{K_5}},w_6={\displaystyle \frac{\omega w_3+{\kappa }_1{w}_4}{K_6}},w_7={\displaystyle \frac{{\kappa }_2{w}_6}{K_7}}$$

Further, ${\mathcal{J}}_{{\beta }^{\mathrm{*}}}$ has a left eigenvector $v=[v_1,v_2,v_3,v_4,v_5,v_6{,v}_7]$, where,

$$v_1=v_2=0,v_3={\displaystyle \frac{\omega \left(v_4+v_6\right)}{K_3}},v_4=v_4,v_5=0,v_6=v_6,v_7=0$$

Computation of $a$ and $b$ [40,41]:

After some tedious manipulations it can be shown that

$$a=2v_3\beta \left(\tau w_6+w_4\right)\left\{w_1+w_2\left(1-\epsilon \right)\right\}$$

$$b={\displaystyle \frac{\mathsf{\Lambda }{v}_3\left(\tau w_6+w_4\right)\left\{\mu +\rho \left(1-\epsilon \right)\right\}}{\mu \left(\mu +\rho \right)}}$$

Thus, it follows from Theorem 4.1 of Castillo-Chavez & Song [41] that the COVID-19 model (2) undergoes backward bifurcation at ${\mathcal{R}}_c=1$ whenever $a>0\mathrm{a}\mathrm{n}\mathrm{d}b>0.$

3.4.1. Local asymptotic stability of Endemic Equilibrium Point (EEP) for the COVID-19 model

Theorem 3. The EEP of the COVID-19 model (2) given by ${\mathcal{E}}_c$, is locally asymptotically stable (LAS) whenever ${\mathcal{R}}_c>1$.

3.6. ${\mathcal{R}}_c$ Sensitivity Analysis

In order to manage the disease, it is necessary to ensure that ${\mathcal{R}}_c$ is less than 1. This can be achieved by regulating the model parameters that influence the value of ${\mathcal{R}}_c$. If the differentiation of ${\mathcal{R}}_c$ with respect to a specific parameter yields a positive sign, it means that increasing (decreasing) the parameter value will cause ${\mathcal{R}}_c$ to increase (decrease). Conversely, if the differentiation of ${\mathcal{R}}_c$ with respect to a parameter yields a negative sign, it implies that decreasing (increasing) the parameter value will lead to an increase (decrease) in ${\mathcal{R}}_c$. It can be easily verified that

$$\begin{array}{l}{\displaystyle \frac{\partial {\mathcal{R}}_c}{\partial \tau }}={\displaystyle \frac{\left\{\beta \mathsf{\Lambda }\left(\mu +\rho -\epsilon \rho \right)\omega \right\}}{\mu \left(\mu +\rho \right)\left(\mu +3\omega \right)\left(\mu +{\kappa }_1\right)}}>0\\ {\displaystyle \frac{\partial {\mathcal{R}}_c}{\partial \omega }}={\displaystyle \frac{\beta \mathsf{\Lambda }\left(\mu +\rho -\epsilon \rho \right)\left\{\mu \tau \gamma +2\mu {\kappa }_1+{\kappa }_1^2+\mu \left(\mu +\mu \tau +\tau {\kappa }_2\right)\right\}}{\mu \left(\mu +\rho \right){\left(\mu +3\omega \right)}^2\left(\mu +{\kappa }_1\right)\left(\mu +\gamma +{\kappa }_2\right)}}>0\\ {\displaystyle \frac{\partial {\mathcal{R}}_c}{\partial \rho }}=-{\displaystyle \frac{\beta \mathsf{\Lambda }\mathsf{\omega }\mathsf{\epsilon }\left\{\mu \tau \gamma +2\mu {\kappa }_1+{\kappa }_1^2+\mu \left(\mu +\mu \tau +\tau {\kappa }_2\right)\right\}}{\mu {\left(\mu +\rho \right)}^2{\left(\mu +3\omega \right)}^2\left(\mu +{\kappa }_1\right)\left(\mu +\gamma +{\kappa }_2\right)}}<0\\ {\displaystyle \frac{\partial {\mathcal{R}}_c}{\partial \gamma }}=-{\displaystyle \frac{\beta \mathsf{\Lambda }\left(\mu +\rho -\epsilon \rho \right)\left(\mu +{\kappa }_1\right)}{{\mu }^2\left(\mu +\rho \right)\left(\mu +3\omega \right){\left(\mu +\gamma +{\kappa }_2\right)}^2}}<0\\ {\displaystyle \frac{\partial {\mathcal{R}}_c}{\partial {\kappa }_2}}=-{\displaystyle \frac{\beta \mathsf{\Lambda }\left(\mu +\rho -\epsilon \rho \right)\left(\mu +{\kappa }_1\right)}{{\mu }^2\left(\mu +\rho \right)\left(\mu +3\omega \right){\left(\mu +\gamma +{\kappa }_2\right)}^2}}<0\\ {\displaystyle \frac{\partial {\mathcal{R}}_c}{\partial \beta }}={\displaystyle \frac{\omega \mathsf{\Lambda }\left(\mu +\rho -\epsilon \rho \right)\left\{\mu \tau \gamma +2\mu {\kappa }_1+{\kappa }_1^2+\mu \left(\mu +\mu \tau +\tau {\kappa }_2\right)\right\}}{{\mu }^2\left(\mu +\rho \right)\left(\mu +{\kappa }_1\right)\left(\mu +3\omega \right){\left(\mu +\gamma +{\kappa }_2\right)}^2}}>0\end{array}$$

(15)

Table 1. Estimated parameters for model (2).

Table 1. Estimated parameters for model (2).

Parameter	Baseline values	Units	References
$\mathsf{\Lambda }$	10,000	Day-1	Assumed
$\beta$	0.0005	Day-1	[43]
$\omega$	0.05	-	[44]
$\rho$	0.80	Day-1	[45]
$\epsilon$	0.85	-	Assumed
$\tau$	0.8	-	Assumed
$\sigma$	0.0001	Day-1	[43]
$\gamma$	0.0095	Day-1	[44]
${\kappa }_1,{\kappa }_2$	0.15, 0.2	Day-1	[43]
$\mu$	0.0001	Day-1	[44]

4. Numerical Simulations

5. Conclusion

References

1. Barlow, N.S.; Weinstein, S.J. Accurate closed-form solution of the SIR epidemic model. Physica. D 2020, 408, 132540–132540. [Google Scholar] [CrossRef] [PubMed]
2. Tomchin, D.A.; Fradkov, A.L. Prediction of the COVID-19 spread in Russia based on SIR and SEIR models of epidemics. IFAC PapersOnLine 2020, 53, 833–838. [Google Scholar] [CrossRef] [PubMed]
3. Marinov, T.T.; Marinova, R.S. Dynamics of COVID-19 using inverse problem for coefficient identification in SIR epidemic models. Chaos, Solitons & Fractals: X 2020, 5, 100041–100041. [Google Scholar] [CrossRef]
4. Neves, A.G.M.; Guerrero, G. Predicting the evolution of the COVID-19 epidemic with the A-SIR model: Lombardy, Italy and São Paulo state, Brazil. Physica. D 2020, 413, 132693–132693. [Google Scholar] [CrossRef]
5. Alenezi, M.N.; Al-Anzi, F.S.; Alabdulrazzaq, H. Building a sensible SIR estimation model for COVID-19 outspread in Kuwait. Alexandria engineering journal 2021, 60, 3161–3175. [Google Scholar] [CrossRef]
6. Din, R.u.; Algehyne, E.A. Mathematical analysis of COVID-19 by using SIR model with convex incidence rate. Results in physics 2021, 23, 103970–103970. [Google Scholar] [CrossRef]
7. Malavika, B.; Marimuthu, S.; Joy, M.; Nadaraj, A.; Asirvatham, E.S.; Jeyaseelan, L. Forecasting COVID-19 epidemic in India and high incidence states using SIR and logistic growth models. Clinical epidemiology and global health 2021, 9, 26–33. [Google Scholar] [CrossRef]
8. Kozyreff, G. Hospitalization dynamics during the first COVID-19 pandemic wave: SIR modelling compared to Belgium, France, Italy, Switzerland and New York City data. Infectious disease modelling 2021, 6, 398–404. [Google Scholar] [CrossRef]
9. Campos, E.L.; Cysne, R.P.; Madureira, A.L.; Mendes, G.L.Q. Multi-generational SIR modeling: Determination of parameters, epidemiological forecasting and age-dependent vaccination policies. Infectious disease modelling 2021, 6, 751–765. [Google Scholar] [CrossRef]
10. Kudryashov, N.A.; Chmykhov, M.A.; Vigdorowitsch, M. Analytical features of the SIR model and their applications to COVID-19. Applied Mathematical Modelling 2021, 90, 466–473. [Google Scholar] [CrossRef]
11. Cooper, I.; Mondal, A.; Antonopoulos, C.G. A SIR model assumption for the spread of COVID-19 in different communities. Chaos, solitons and fractals 2020, 139, 110057–110057. [Google Scholar] [CrossRef] [PubMed]
12. Nesteruk, I. Statistics-Based Procedure of Parameter Identification for the Classical SIR Model; Springer: Singapore, 2021; pp. 33–36. [Google Scholar] [CrossRef]
13. Nesteruk, I. Applications of the General SIR Model for Calculations of the COVID-19 Epidemic Waves in Ukraine; Springer: Singapore, 2021; pp. 141–146. [Google Scholar] [CrossRef]
14. Turkyilmazoglu, M. Explicit formulae for the peak time of an epidemic from the SIR model. Physica. D 2021, 422, 132902–132902. [Google Scholar] [CrossRef] [PubMed]
15. Anand, N.; Sabarinath, A.; Geetha, S.; Somanath, S. Predicting the Spread of COVID-19 Using SIR Model Augmented to Incorporate Quarantine and Testing. Transactions of the Indian National Academy of Engineering (Online) 2020, 5, 141–148. [Google Scholar] [CrossRef]
16. Lounis, M.; Bagal, D.K. Estimation of SIR model’s parameters of COVID-19 in Algeria. Bulletin of the National Research Centre 2020, 44, 180–180. [Google Scholar] [CrossRef]
17. Ramos, A.M.; Ferrández, M.R.; Vela-Pérez, M.; Kubik, A.B.; Ivorra, B. A simple but complex enough θ-SIR type model to be used with COVID-19 real data. Application to the case of Italy. Physica. D 2021, 421, 132839–132839. [Google Scholar] [CrossRef] [PubMed]
18. Shahrear, P.; Rahman, S.M.S.; Nahid, M.M.H. Prediction and mathematical analysis of the outbreak of coronavirus (COVID-19) in Bangladesh. Results in Applied Mathematics 2021, 10, 100145–100145. [Google Scholar] [CrossRef] [PubMed]
19. Uddin, M.B.; Hasan, M.; Harun-Al-Rashid, A.; Ahsan, M.I.; Imran, M.A.S.; Ahmed, S.S.U. Ancestral origin, antigenic resemblance and epidemiological insights of novel coronavirus (SARS-CoV-2): Global burden and Bangladesh perspective. Infection, genetics and evolution 2020, 84, 104440–104440. [Google Scholar] [CrossRef]
20. Ahmed, J.; Jaman, M.H.; Saha, G.; Ghosh, P. Effect of environmental and socio-economic factors on the spreading of COVID-19 at 70 cities/provinces. Heliyon 2021, 7, e06979–e06979. [Google Scholar] [CrossRef]
21. Alrabaiah, H.; Arfan, M.; Shah, K.; Mahariq, I.; Ullah, A. A comparative study of spreading of novel corona virus disease by ussing fractional order modified SEIR model. Alexandria Engineering Journal 2021, 60, 573–585. [Google Scholar] [CrossRef]
22. López, L.; Rodó, X. A modified SEIR model to predict the COVID-19 outbreak in Spain and Italy: Simulating control scenarios and multi-scale epidemics. Results in physics 2021, 21, 103746–103746. [Google Scholar] [CrossRef]
23. Hussain, T.; Ozair, M.; Ali, F.; Rehman, S.u.; Assiri, T.A.; Mahmoud, E.E. Sensitivity analysis and optimal control of COVID-19 dynamics based on SEIQR model. Results in physics 2021, 22, 103956–103956. [Google Scholar] [CrossRef] [PubMed]
24. Li, J.; Zhong, J.; Ji, Y.-M.; Yang, F. A new SEIAR model on small-world networks to assess the intervention measures in the COVID-19 pandemics. Results in physics 2021, 25, 104283–104283. [Google Scholar] [CrossRef]
25. Youssef, H.; Alghamdi, N.; Ezzat, M.A.; El-Bary, A.A.; Shawky, A.M. Study on the SEIQR model and applying the epidemiological rates of COVID-19 epidemic spread in Saudi Arabia. Infectious disease modelling 2021, 6, 678–692. [Google Scholar] [CrossRef]
26. Yu, Z.; Arif, R.; Fahmy, M.A.; Sohail, A. Self organizing maps for the parametric analysis of COVID-19 SEIRS delayed model. Chaos, solitons and fractals 2021, 150, 111202–111202. [Google Scholar] [CrossRef]
27. Adhikary, A.; Pal, A. A six compartments with time-delay model SHIQRD for the COVID-19 pandemic in India: During lockdown and beyond. Alexandria engineering journal 2022, 61, 1403–1412. [Google Scholar] [CrossRef]
28. Akindeinde, S.O.; Okyere, E.; Adewumi, A.O.; Lebelo, R.S.; Fabelurin, O.O.; Moore, S.E. Caputo fractional-order SEIRP model for COVID-19 Pandemic. Alexandria engineering journal 2022, 61, 829–845. [Google Scholar] [CrossRef]
29. Basnarkov, L. SEAIR Epidemic spreading model of COVID-19. Chaos, solitons and fractals 2021, 142, 110394–110394. [Google Scholar] [CrossRef] [PubMed]
30. Khedher, N.b.; Kolsi, L.; Alsaif, H. A multi-stage SEIR model to predict the potential of a new COVID-19 wave in KSA after lifting all travel restrictions. Alexandria engineering journal 2021, 60, 3965–3974. [Google Scholar] [CrossRef]
31. Pekpe, K.M.; Zitouni, D.; Gasso, G.; Dhifli, W.; Guinhouya, B.C. From SIR to SEAIRD: A novel data-driven modeling approach based on the Grey-box System Theory to predict the dynamics of COVID-19. Applied intelligence (Dordrecht, Netherlands) 2022, 52, 71–80. [Google Scholar] [CrossRef]
32. Zhu, W.-j.; Shen, S.-f. An improved SIR model describing the epidemic dynamics of the COVID-19 in China. Results in physics 2021, 25, 104289–104289. [Google Scholar] [CrossRef]
33. Chen, M.; Li, M.; Hao, Y.; Liu, Z.; Hu, L.; Wang, L. The introduction of population migration to SEIAR for COVID-19 epidemic modeling with an efficient intervention strategy. Information fusion 2020, 64, 252–258. [Google Scholar] [CrossRef] [PubMed]
34. Contreras, S.; Villavicencio, H.A.; Medina-Ortiz, D.; Biron-Lattes, J.P.; Olivera-Nappa, Á. A multi-group SEIRA model for the spread of COVID-19 among heterogeneous populations. Chaos, solitons and fractals 2020, 136, 109925–109925. [Google Scholar] [CrossRef]
35. Avila-Ponce de León, U.; Pérez, Á.G.C.; Avila-Vales, E. An SEIARD epidemic model for COVID-19 in Mexico: Mathematical analysis and state-level forecast. Chaos, solitons and fractals 2020, 140, 110165–110165. [Google Scholar] [CrossRef]
36. Diekmann, O.; Heesterbeek, H.; Metz, J.A.J. On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. Journal of mathematical biology 1990, 28, 365–382. [Google Scholar] [CrossRef] [PubMed]
37. van den Driessche, P.; Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences 2002, 180, 29–48. [Google Scholar] [CrossRef]
38. Chavez, C.C.; Feng, Z.; Huang, W. On the computation of R0 and its role on global stability. Mathematical approaches for emerging and re-emerging infection diseases: an introduction 2002, 125, 31–65. [Google Scholar]
39. Hethcote, H.W. The Mathematics of Infectious Diseases. SIAM review 2000, 42, 599–653. [Google Scholar] [CrossRef]
40. Carr, J. Applications of Centre Manifold Theory; Applied Mathematical Sciences; Springer, 2012; Volume 35. [Google Scholar]
41. Castillo-Chavez, C.; Song, B. Dynamical models of tuberculosis and their applications. Math. Biosci. Eng 2004, 1, 361–404. [Google Scholar] [CrossRef] [PubMed]
42. La Salle, J.P., The stability of dynamical systems. 1976: SIAM.
43. Ngonghala, C.N.; Iboi, E.; Eikenberry, S.; Scotch, M.; MacIntyre, C.R.; Bonds, M.H.; Gumel, A.B. Mathematical assessment of the impact of non-pharmaceutical interventions on curtailing the 2019 novel Coronavirus. Mathematical biosciences 2020, 325, 108364–108364. [Google Scholar] [CrossRef]
44. Saha, A.K.; Podder, C.N.; Niger, A.M. Dynamics of novel COVID-19 in the presence of Co-morbidity. Infectious disease modelling 2022, 7, 138–160. [Google Scholar] [CrossRef]
45. Mancuso, M.; Eikenberry, S.E.; Gumel, A.B. Will vaccine-derived protective immunity curtail COVID-19 variants in the US? Infectious disease modelling 2021, 6, 1110–1134. [Google Scholar] [CrossRef] [PubMed]
46. Burden and Faires, Numerical Analysis. 2015: Cengage Learning.
47. Weinstein, S.J.; Holland, M.S.; Rogers, K.E.; Barlow, N.S. Analytic solution of the SEIR epidemic model via asymptotic approximant. Physica. D 2020, 411, 132633–132633. [Google Scholar] [CrossRef] [PubMed]
48. Odagaki, T. Exact properties of SIQR model for COVID-19. Physica A 2021, 564, 125564–125564. [Google Scholar] [CrossRef] [PubMed]