Abstract
In this study, we have proposed a mathematical model to describe the dynamics of the spread of Middle East Respiratory Syndrome disease. The model consists of six-coupled ordinary differential equations. The existence of the corona-free equilibrium and endemic equilibrium points has been proved. The threshold condition for which the disease will die out or becomes permanent has been computed. That is the corona-free equilibrium point is locally asymptotically stable whenever the reproduction number is less than unity, and it is globally asymptotically stable (GAS) whenever the reproduction number is greater than unity. Moreover, we have proved that the endemic equilibrium point is GAS whenever the reproduction number is greater than unity. The results of the model analysis have been illustrated by numerical simulations.
1 Introduction
In recent years, the threat of emerging and re-emerging viruses on public health and economics has increased drastically. In the past few decades, human beings have suffered from several coronaviruses such as severe acute respiratory syndrome (SARS), Middle East Respiratory Syndrome (MERS-Cov), and recently the coronavirus disease (COVID-19) pandemic. Coronaviruses are single RNA viruses that cause respiratory tract infections. Those viruses were discovered in the 1960s [1,2].
The MERS-Cov is a respiratory disease caused by a coronavirus that attacks the respiratory tract. The main symptoms of MERS-Cov are fever, shortness of breath, and cough from mild to severe with an incubation period of 2–14 days. In some cases, it may cause diarrhea, nausea, vomiting, and myalgia [2].
The first case was recorded in Saudi Arabia in 2012 as it was reported by the World Health Organization (WHO). Moreover, the number of confirmed cases since 2012 is 2,464 with a death rate of 34% in 27 countries. 2073 confirmed cases were recorded in Saudi Arabia with a death rate of 37% [3].
In 2015, South Korea has suffered from the MERS-Cov outbreak. The reported number of confirmed cases was 168 cases. The latter was the largest in the region other than Middle East [4].
Mathematical models have been used to explore the dynamical spread of disease, such as hepatitis, cholera, HIV, SARS, Ebola, yellow fever, smallpox, and MERS [5,6, 7,8,9, 10,11,12, 13,14,15, 16,17,18, 19,20,21, 22,23,24, 25,26]. Tang et al. [27] have studied the spread of MERS-Cov and the expression of dipeptidyl peptidase 4 (DPP4). They have discussed the global stability of the equilibrium points of the model, and they have shown that the spread of MERS-Cov can be controlled by decreasing the rate of DPP4. Berrai et al. [28] have used the semi-group theory and optimal control to show the existence of a solution for the proposed model and to prove the existence of optimal control. Manaqib et al. [29] have studied the effect of the medical mask usage and vaccination on the spread of MERS-Cov. They have found that if the reproduction number is less than one, then the disease will die out. On the other hand, if the reproduction number is more than unity, the disease will be permanent. DarAssi et al. [9] have discussed a discrete-time MERS-Cov model. They have studied the global stability of the equilibrium points, and they have shown that the disease will die out if the reproduction number is less than unity.
In this paper, we have proposed a mathematical model to describe the spread of MERS-Cov. coronavirus. This model consists of six compartments: susceptible, exposed, infected with symptoms, infected without symptoms, hospitalized, and recovered or removed individuals. This paper is organized as follows: In Section 2, we formulate the model. In Section 3, the local and global stability of corona-free equilibrium is investigated. In Section 4, we have studied the existence of the endemic equilibrium point, and we have discussed the local stability at this point. The numerical simulation and the conclusion have been discussed in Section 5.
2 Mathematical model and basic properties
The SEAIHR epidemic model for MERS-Cov has been proposed. We have divided the total population
The susceptible individuals
where
The aforementioned model is subject to the following initial conditions:
Let
Figure 1 illustrates the flow diagram of the model (1).
2.1 Basic properties
All the parameters of the model (1) are non negative because it is a human population model. Moreover, the following result holds.
Lemma 1
The variables of the model (1) are non negative for all time t.
Proof
Let
Thus,
so that,
In a similar manner, we can prove that
Moreover, the method in Appendix A of [30] can be used to obtain the same results mentioned earlier. The following result can be claimed.
Lemma 2
The closed set
is positively invariant.
Proof
By adding all the equations in (1), we have,
If
It is sufficient to study the dynamics of the flow generated by the model (1) in
3 Stability of corona-free equilibrium
3.1 Local stability of disease free equilibrium (DFE)
The model (1) has DFE given by
The next-generation method [33,34] will be used to investigate the local stability of the corona-free equilibrium point (CFE). Upon using the same notations in [34], the matrix of the new infection terms,
and
It is known that the basic reproduction number [35,36] is denoted by
Using Theorem 2 in [34], the following result is established.
Lemma 3
The CFE of the model (1) is locally asymptotically stable if
The value of
3.2 Global stability of CFE
Theorem 1
The CFE of the model (1), given by (5), is globally asymptotically stable (GAS) in
Proof
The proof of the aforementioned theorem is established using a suitable Lyapunov function [39] and LaSalle’s invariance principle [39,40] by defining the following Lyapunov function:
where
and the derivative of
4 Existence and stability of endemic equilibria
In this section, the possibility of the existence and stability of endemic (positive) equilibria of the model (1) will be investigated (i.e., equilibria where at least one of the infected components of the model is nonzero).
4.1 Existence of endemic equilibrium point
The endemic equilibrium point (EEP) of model (1) is denoted by
Lemma 4
The model (1) has a unique endemic equilibrium point
Proof
Consider the following steady-state equations of model (1)
On solving the equations (6), we obtain
Substituting equations (7) in the expression of
where
Equation (8) has two solutions, namely,
4.2 Local stability of endemic equilibrium point
Theorem 2
The associated unique endemic equilibrium of the model (1) is GAS whenever
To show the global stability of the endemic equilibrium
where
The derivative
Now, we assume that
Thus, the function
Next, construct the function set
There are five groups associated with
Define the function:
where
According to the aforementioned group, we define the function H as follows:
To find appropriate parameters
This system has the following solution set:
It is clear that
Upon substituting the values of
The equating of the coefficients for the same terms between
The completion of the preceding steps indicates that the function
According to the property that the arithmetic mean is greater than or equal to the geometric mean, we have
Since the singleton
5 Discussion and conclusion
In this paper, we have proposed a mathematical model to study the dynamics of the MERS-Cov coronavirus. The population has been divided into six compartments which are susceptible, exposed, asymptomatic, symptomatic, hospitalized, and recovered. Figure 1 shows the proposed model. The latter has been analyzed and two equilibrium points were found. The existence of the corona-free equilibrium point (CFP) and the endemic equilibrium point (EEP) have been discussed. Both locally and globally stability analyses of the CFP and EEP have been conducted. Figure 2 shows the behavior of the population compartments when
Parameters | Values (per day) | Sources |
---|---|---|
|
[0.05, 0.1] | [41] |
|
0.0000351 | [42] |
|
0.03521 | [43] |
|
0.042553 | [43] |
|
0.0535 | [44] |
|
0.04227 | [45] |
|
0.027855 | [43] |
|
0.20619 | [43] |
|
136 | [41] |
|
0.1 | [41] |
|
0.5 | Assumed |
|
(0, 1] | Variable |
Figure 3 depicts the effect of the hospitalization (isolation) on the virus spread curve in both cases when
Figure 4 illustrates the contour plots of the reproduction number
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
-
Conflict of interest: The authors state no coflict of interest.
-
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
References
[1] J. S. Khan and K. Mclntosh, History and recent advances in coronavirus discovery, Pediatr. Infect. Dis. J. 24 (2005), 16378050. 10.1097/01.inf.0000188166.17324.60Search in Google Scholar
[2] World Health Organization, Middle East respiratory syndrome coronavirus (MERS-Cov), 2019, November 1. 10.1007/978-3-319-74365-3_49-1Search in Google Scholar
[3] G. Dudas, L. Carvalho, and A. Rambaut, MERS-Cov Spillover at the camel-human interface, eLive 7 (2018), 1–23. 10.7554/eLife.31257Search in Google Scholar
[4] Y. Kim, H. Ryu, and S. Lee, Effectiveness of intervention strategies on MERS-CoV transmission dynamics in South Korea, 2015: Simulations on the network based on the real-world contact data, Int. J. Environ. Res. Public Health 18 (2021), 3530. 10.3390/ijerph18073530Search in Google Scholar
[5] H. Alrabaiah, M. A. Safi, M. H. DarAssi, B. Al-Hdaibat, S. Ullah, M. A. Khan, et al., Optimal control analysis of hepatitis B virus with treatment and vaccination, Results Phys. 19 (2020), 103599. 10.1016/j.rinp.2020.103599Search in Google Scholar
[6] C. T. Bauch, J. O. Lloyd-Smith, and M. P. Coffee, Dynamically modeling SARS and other newly emerging respiratory illnesses: past, present, and future, Epidemiology 16 (2005), 791–801. 10.1097/01.ede.0000181633.80269.4cSearch in Google Scholar
[7] Y. Bechah, C. Capo, J. L. Mege, and D. Raoult, Epidemic typhus, Lancet Infect. Dis. 8 (2008), 417–426. 10.1016/S1473-3099(08)70150-6Search in Google Scholar
[8] M. H. DarAssi, M. A. Safi, and B. Al-Hdaibat, A delayed SEIR epidemic model with pulse vaccination and treatment, Nonlinear Studies 25 (2018), no. 3, 1–16. Search in Google Scholar
[9] M. H. DarAssi, M. A. Safi, and M. Ahmad, Global dynamics of a discrete-time MERS-Cov model, Mathematics 9 (2021), no. 5, 563. 10.3390/math9050563Search in Google Scholar
[10] M. H. DarAssi and M. A. Safi, Analysis of an SIRS epidemic model for a disease geographic spread, Nonlinear Dynam. Syst. Theory 21 (2021), 1, 56–67. Search in Google Scholar
[11] M. H. DarAssi, M. A. Safi, M. A. Khan, A Beigi, A. A. Aly, and M. Y. Alshahrani, A mathematical model for SARS-CoV-2 in variable-order fractional derivative, Eur. Phys. J. Spec. Top. (2022), https://doi.org/10.1140/epjs/s11734-022-00458-0. Search in Google Scholar PubMed PubMed Central
[12] P. Daszak, A. A. Cunningham, and A. D. Hyat, Emerging infectious diseases of wildlife – threats to biodiversity and human health, Science 287 (2000), no. 5452, 443–449. 10.1126/science.287.5452.443Search in Google Scholar PubMed
[13] O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases, Chisteter: John Wiley & Son, 2000. Search in Google Scholar
[14] S. Funk, M. Salathé, and V. A. A. Jansen, Modeling the influence of human behavior on the spread of infectious diseases: a review, J. R. Soc. 50 (2010), 1247–1256. 10.1098/rsif.2010.0142Search in Google Scholar PubMed PubMed Central
[15] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, New Jersey, USA: Princeton University Press, 2008. 10.1515/9781400841035Search in Google Scholar
[16] M. A. Khan, K. Khan, M. A. Safi, and M. H. DarAssi, A discrete model of TB dynamics in Khyber Pakhtunkhwa-Pakistan, CMES – Comput. Model. Eng. Sci. 123 (2020), no. 2, 777–795. 10.32604/cmes.2020.08208Search in Google Scholar
[17] J. D. Murray, Mathematical Biology, Berlin: Springer-Verlag, 1989. 10.1007/978-3-662-08539-4Search in Google Scholar
[18] H. Sato, H. Nakada, R. Yamaguchi, S. Imoto, S. Miyano, and M. Kami, When should we intervene to control the 2009 influenza A(H1N1) pandemic? Euro Surveill. 15 (2010), no. 1, 19455.10.2807/ese.15.01.19455-enSearch in Google Scholar PubMed
[19] M. A. Safi, A. B. Gumel, and E. H. Elbasha, Qualitative analysis of an age-structured SEIR epidemic model with treatment, Appl. Math. Comput. 219 (2013), 10627–10642. 10.1016/j.amc.2013.03.126Search in Google Scholar
[20] M. A. Safi and M. H. DarAssi, Mathematical analysis of a model for ectoparasite-borne diseases, J. Comput. Methods Sci. Eng. 41 (2018), no. 17, 8248–8257. 10.1002/mma.5287Search in Google Scholar
[21] M. A. Safi and M. H. DarAssi, Mathematical analysis of an age-structured HSV-2 model, J. Comput. Methods Sci. Eng. 19 (2019), no. 3, 841–856. 10.3233/JCM-181111Search in Google Scholar
[22] M. A. Safi, B. Al-Hdaibat, M. H. DarAssi, and M. A. Khan, Global dynamics for a discrete quarantine/isolation model, Results Phys. 21 (2021), 103788. 10.1016/j.rinp.2020.103788Search in Google Scholar
[23] Z. Y. He, A. Abbes, H. Jahanshahi, N. D. Alotaibi, and Y. Wang, Fractional-order discrete-time SIR epidemic model with vaccination: chaos and complexity, Mathematics 10 (2022), 165, https://doi.org/10.3390/math10020165. Search in Google Scholar
[24] T. H. Zha, O. Castillo, H. Jahanshahi, A. Yusuf, M. O. Alassafi, F. E. Alsaadi, et al., A fuzzy-based strategy to suppress the novel coronavirus (2019-NCOV) massive outbreak, Appl. Comput. Math. 20 (2021), no. 1, 160–176. Search in Google Scholar
[25] M. A. Khan, K. Ali, E. Bonyah, K. O. Okosun, S. Islam, and A. Khan, Mathematical modeling and stability analysis of Pine Wilt disease with optimal control, Sci. Rep. 7 (2017), 3115, https://doi.org/10.1038/s41598-017-03179-w. Search in Google Scholar PubMed PubMed Central
[26] N. Trebi, Emerging and neglected infectious diseases: insights, advances, and challenges, BioMed. Res. Int. 2017 (2017), 5245021, https://doi.org/10.1155/2017/5245021. Search in Google Scholar PubMed PubMed Central
[27] S. Tang, W. Ma, and P. Bai, A novel dynamic model describing the spread of the MERS-CoV and the expression of dipeptidyl peptidase 4, Comput. Math. Methods. Med. 2017 (2017), 5285810. 10.1155/2017/5285810Search in Google Scholar
[28] I. Berrai, K. Adnaoui, and J. Bouyaghroumni, Mathematical study of Coronavirus (MERS-Cov), Commun. Math. Biol. Neurosci. 2020 (2020), 44. Search in Google Scholar
[29] M. Manaqib, I. Fauziah, and M. Mujiyanti, Mathematical model for Mers-CoV disease transmission with medical mask usage and vaccination, Indones. J. Pure Appl. Math. 1 (2019), 97–109. 10.15408/inprime.v1i2.13553Search in Google Scholar
[30] H. R. Thieme, Mathematics in Population Biology, New Jersey, USA: Princeton University Press, 2003. 10.1515/9780691187655Search in Google Scholar
[31] V. Lakshmikantham, S. Leela, and A. A. Matynyuk, Stability Analysis of Nonlinear Systems, New York and Basel: Marcel Dekker Inc., 1989. Search in Google Scholar
[32] H. W. Hethcote and H. R. Thieme, Stability of the endemic equilibrium in epidemic models with subpopulations, Math. Biosci. 75 (1985), 205–227. 10.1016/0025-5564(85)90038-0Search in Google Scholar
[33] O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, On the definition and computation of the basic reproduction ratio R0 in models for infectious disease in heterogeneous population, J. Math. Biol. 28 (1990), 365–382. 10.1007/BF00178324Search in Google Scholar
[34] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29–48. 10.1016/S0025-5564(02)00108-6Search in Google Scholar
[35] R. M. Anderson and R. M. May, Population Biology of Infectious Diseases, Berlin, Heidelrberg, New York: Springer-Verlag; 1982. 10.1007/978-3-642-68635-1Search in Google Scholar
[36] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000), 599–653. 10.1137/S0036144500371907Search in Google Scholar
[37] J. Li, Y. Xiao, F. Zhang, and Y. Yang, An algebraic approach to proving the global stability of a class of epidemic models, Nonlinear Anal. Real World Appl. 28 (2012), 2006–2016. 10.1016/j.nonrwa.2011.12.022Search in Google Scholar
[38] J. Li, Y. Yang, and Y. Zhou, Global stability of an epidemic model with latent stage and vaccination, Nonlinear Anal. Real World Appl. 12 (2011), 2163–2173. 10.1016/j.nonrwa.2010.12.030Search in Google Scholar
[39] J. K. Hale, Ordinary Differential Equations, New York: John Wiley and Sons, 1969. Search in Google Scholar
[40] J. P. LaSalle, The stability of dynamical systems, CBMS-NSF Regional Conference Series in Applied Mathematics, Philadelphia: SIAM, 1976. Search in Google Scholar
[41] A. B. Gumel, S. Ruan, T. Day, J. Watmough, F. Brauer, P. Van den Driessche, et al., Modeling strategies for controlling SARS outbreaks, Proc. Biol. Sci. 271 (2004), 2223–2232. 10.1098/rspb.2004.2800Search in Google Scholar
[42] Kong in Figures 2006 Edition, Census and Statistics Department Hong Kong Special Administrative Region. Search in Google Scholar
[43] G. Chowell, C. Castillo-Chavez, P. Fenimore, C. Kribs-Zaleta, L. Arriola, and J. Hyman, Model parameters and outbreak control for SARS, EID 10 (2004), 1258–1263. 10.3201/eid1007.030647Search in Google Scholar
[44] C. Donnelly, A. Ghani, G. Leung, A. J. Hedley, C. Fraser, S. Riley, et al., Epidemiological determinants of spread of casual agent of severe acute respiratory syndrome in Hong Kong, Lancet. 361 (2003), 1761–1766. 10.1016/S0140-6736(03)13410-1Search in Google Scholar
[45] G. Leung, A. Hedley, L. Ho, P. Chau, I. O. Wong, T. Q. Thach, et al., The epidemiology of severe acute respiratory syndrome in the 2003 Hong Kong epidemic: an analysis of all 1755 patients, Ann. Intern. Med. 9 (2004), 662–673. 10.7326/0003-4819-141-9-200411020-00006Search in Google Scholar PubMed
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