A new COVID-19 epidemic model with media coverage and quarantine is constructed. The model allows for the susceptibles to the unconscious and conscious susceptible compartment. First, mathematical analyses establish that the global dynamics of the spread of the COVID-19 infectious disease are completely determined by the basic reproduction number R0. If R0 ≤ 1, then the disease free equilibrium is globally asymptotically stable. If R0 > 1, the endemic equilibrium is globally asymptotically stable. Second, the unknown parameters of model are estimated by the MCMC algorithm on the basis of the total confirmed new cases from February 1, 2020 to March 23, 2020 in the UK. We also estimate that the basic reproduction number is R0 = 4.2816(95%CI: (3.8882, 4.6750)). Without the most restrictive measures, we forecast that the COVID-19 epidemic will peak on June 2 (95%CI: (May 23, June 13)) (Figure 3a) and the number of infected individuals is more than 70% of UK population. In order to determine the key parameters of the model, sensitivity analysis are also explored. Finally, our results show reducing contact is effective against the spread of the disease. We suggest that the stringent containment strategies should be adopted in the UK.
Citation: Li-Xiang Feng, Shuang-Lin Jing, Shi-Ke Hu, De-Fen Wang, Hai-Feng Huo. Modelling the effects of media coverage and quarantine on the COVID-19 infections in the UK[J]. Mathematical Biosciences and Engineering, 2020, 17(4): 3618-3636. doi: 10.3934/mbe.2020204
Related Papers:
[1]
Weike Zhou, Aili Wang, Fan Xia, Yanni Xiao, Sanyi Tang .
Effects of media reporting on mitigating spread of COVID-19 in the early phase of the outbreak. Mathematical Biosciences and Engineering, 2020, 17(3): 2693-2707.
doi: 10.3934/mbe.2020147
[2]
Fang Wang, Lianying Cao, Xiaoji Song .
Mathematical modeling of mutated COVID-19 transmission with quarantine, isolation and vaccination. Mathematical Biosciences and Engineering, 2022, 19(8): 8035-8056.
doi: 10.3934/mbe.2022376
[3]
Jiajia Zhang, Yuanhua Qiao, Yan Zhang .
Stability analysis and optimal control of COVID-19 with quarantine and media awareness. Mathematical Biosciences and Engineering, 2022, 19(5): 4911-4932.
doi: 10.3934/mbe.2022230
[4]
Jiangbo Hao, Lirong Huang, Maoxing Liu, Yangjun Ma .
Analysis of the COVID-19 model with self-protection and isolation measures affected by the environment. Mathematical Biosciences and Engineering, 2024, 21(4): 4835-4852.
doi: 10.3934/mbe.2024213
[5]
Tao Chen, Zhiming Li, Ge Zhang .
Analysis of a COVID-19 model with media coverage and limited resources. Mathematical Biosciences and Engineering, 2024, 21(4): 5283-5307.
doi: 10.3934/mbe.2024233
[6]
Hamdy M. Youssef, Najat A. Alghamdi, Magdy A. Ezzat, Alaa A. El-Bary, Ahmed M. Shawky .
A new dynamical modeling SEIR with global analysis applied to the real data of spreading COVID-19 in Saudi Arabia. Mathematical Biosciences and Engineering, 2020, 17(6): 7018-7044.
doi: 10.3934/mbe.2020362
[7]
Xiaojing Wang, Yu Liang, Jiahui Li, Maoxing Liu .
Modeling COVID-19 transmission dynamics incorporating media coverage and vaccination. Mathematical Biosciences and Engineering, 2023, 20(6): 10392-10403.
doi: 10.3934/mbe.2023456
[8]
Yujie Sheng, Jing-An Cui, Songbai Guo .
The modeling and analysis of the COVID-19 pandemic with vaccination and isolation: a case study of Italy. Mathematical Biosciences and Engineering, 2023, 20(3): 5966-5992.
doi: 10.3934/mbe.2023258
[9]
Akhil Kumar Srivastav, Pankaj Kumar Tiwari, Prashant K Srivastava, Mini Ghosh, Yun Kang .
A mathematical model for the impacts of face mask, hospitalization and quarantine on the dynamics of COVID-19 in India: deterministic vs. stochastic. Mathematical Biosciences and Engineering, 2021, 18(1): 182-213.
doi: 10.3934/mbe.2021010
[10]
Xinmiao Rong, Liu Yang, Huidi Chu, Meng Fan .
Effect of delay in diagnosis on transmission of COVID-19. Mathematical Biosciences and Engineering, 2020, 17(3): 2725-2740.
doi: 10.3934/mbe.2020149
Abstract
A new COVID-19 epidemic model with media coverage and quarantine is constructed. The model allows for the susceptibles to the unconscious and conscious susceptible compartment. First, mathematical analyses establish that the global dynamics of the spread of the COVID-19 infectious disease are completely determined by the basic reproduction number R0. If R0 ≤ 1, then the disease free equilibrium is globally asymptotically stable. If R0 > 1, the endemic equilibrium is globally asymptotically stable. Second, the unknown parameters of model are estimated by the MCMC algorithm on the basis of the total confirmed new cases from February 1, 2020 to March 23, 2020 in the UK. We also estimate that the basic reproduction number is R0 = 4.2816(95%CI: (3.8882, 4.6750)). Without the most restrictive measures, we forecast that the COVID-19 epidemic will peak on June 2 (95%CI: (May 23, June 13)) (Figure 3a) and the number of infected individuals is more than 70% of UK population. In order to determine the key parameters of the model, sensitivity analysis are also explored. Finally, our results show reducing contact is effective against the spread of the disease. We suggest that the stringent containment strategies should be adopted in the UK.
1.
Introduction
Since December 2019, the outbreak of the novel coronavirus pneumonia firstly occurred in Wuhan, a central and packed city of China [1,2]. The World Health Organization(WHO) has named the virus as COVID-19 On January 12, 2020. Recently, COVID-19 has spread to the vast majority of countries, as United States, France, Iran, Italy and Spain etc. The outbreak of COVID-19 has been become a globally public health concern in medical community as the virus is spreading around the world. Initially, the British government adopted a herd immunity strategy. As of March 27, cases of the COVID-19 coronavirus have been confirmed more than 11,000 mostly In the UK. The symptoms of COVID-19 most like SARS(Severe acute respiratory syndrome) and MERS(Middle East respiratory syndrome), include cough, fever, weakness and difficulty breath[3]. The period for such symptoms from mild to severe respiratory infections lasts 2–14 days. The transmission routes contain direct transmission, such as close touching and indirect transmission consist of the air by coughing and sneezing, even if contacting some contaminated things by virus particles. There are many mathematica models to discuss the dynamics of COVID-19 infection [4,5,6,7,8].
Additionally, coronaviruses can be extremely contagious and spread easily from person to person[9]. So a series of stringent control measures are necessary. For some diseases, such as influenza and tuberculosis, people often introduce the latent compartment (denoted by E), leading to an SEIR model. The latent compartment of COVID-19 is highly contagious[10,11]. Such type of models have been widely discussed in recent decades [12,13,14].
Media coverage is a key factor in the transmission process of infectious disease. People know more about the COVID-19 and enhance their self-protecting awareness by the media reporting about the COVID-19. People will change their behaviours and take correct precautions such as frequent hand-washing, wearing masks, reducing the party, keeping social distances, and even quarantining themselves at home to avoid contacting with others. Zhou et al.[15] proposed a deterministic dynamical model to examine the interaction of the disease progression and the media reports and to investigate the effectiveness of media reporting on mitigating the spread of COVID-19. The result suggested that media coverage can be considered as an effective way to mitigate the disease spreading during the initial stage of an outbreak.
Quarantine is effective for the control of infectious disease. Chinese government advises all the Chinese citizens to isolate themselves at home, and people exposed to the virus have the medical observation for 14 days. In order to get closer to the reality, many scholars have introduced quarantine compartment into epidemic model. Amador and Gomez-Corral[16] studied extreme values in an SIQS model with two different states for quarantine, termed quarantined susceptible and quarantined infective, and limited carrying capacity for the quarantine compartment. Gao and Zhuang proposed a new VEIQS worm propagation model with saturated incidence and strategies of both vaccination and quarantine[17].
Motivated by the above, we consider a new COVID-19 epidemic model with media coverage and quarantine. The model assumes that the latent stage has certain infectivity. And we also introduce the quarantine compartment into the epidemic model, and the susceptible have consciousness to checking the spread of infectious diseases in the media coverage.
The organization of this paper is as follows. In the next section, the epidemic model with media coverage and quarantine is formulated. In section 3, the basic reproduction number and the existence of equilibria are investigated. In Section 4, the global stability of the disease free and endemic equilibria are proved. In Section 5, we use the MCMC algorithm to estimate the unknown parameters and initial values of the model. The basic reproduction number R0 of the model and its confidence interval are solved by numerical methods. At the same time, we obtain the sensitivity of the unknown parameters of the model. In the last section, we give some discussions.
2.
The model formulation
2.1. System description
In this section, we introduce a COVID-19 epidemic model with media coverage and quarantine. The total population is partitioned into six compartments: the unconscious susceptible compartment (S1), the conscious susceptible compartment (S2), the latent compartment (E), the infectious compartment (I), the quarantine compartment (Q) and the recovered compartment (R). The total number of population at time t is given by
N(t)=S1(t)+S2(t)+E(t)+I(t)+Q(t)+R(t).
The parameters are described in Table 1. The population flow among those compartments is shown in the following diagram (Figure 1).
Table 1.
Parameters of the model.
Parameter
Description
Λ
The birth rate of the population
βE
Transmission coefficient of the latent compartment
βI
Transmission coefficient of the infectious compartment
σ
The fraction of S2 being infected and entering E
p
The migration rate to S2 from S1, reflecting the impact of media coverage
r
The rate coefficient of transfer from the latent compartment
q
The fraction of the latent compartment E jump into the quarantine compartment Q
1−q
The fraction of the latent compartment E jump into the infectious compartment I
It is important to show positivity for the system (2.1) as they represent populations. We thus state the following lemma.
Lemma 1. If the initial values S1(0)>0, S2(0)>0, E(0)>0, I(0)>0, Q(0)>0 and R(0)>0, the solutions S1(t), S2(t), E(t), I(t), Q(t) and R(t) of system (2.1) are positive for all t>0.
Proof. Let W(t)=min{S1(t),S2(t),E(t),I(t),Q(t),R(t)}, for all t>0.
It is clear that W(0)>0. Assuming that there exists a t1>0 such that W(t1)=0 and W(t)>0, for all t∈[0,t1).
If W(t1)=S1(t1), then S2(t)≥0,E(t)≥0,I(t)≥0,Q(t)≥0,R(t)≥0 for all t∈[0,t1]. From the first equation of model (2.1), we can obtain
S1′≥−βES1E−βIS1I−(p+μ)S1,t∈[0,t1]
Thus, we have
0=S1(t1)≥S1(0)e−∫t10[βEE+βII+(p+μ)]dt>0,
which leads to a contradiction. Thus, S1(t)>0 for all t≥0.
Similarly, we can also prove that S2(t)>0, E(t)>0, I(t)>0, Q(t)>0 and R(t)>0 for all t≥0.
2.2.2. Invariant region
Lemma 2. The feasible region Ω defined by
Ω={(S1(t),S2(t),E(t),I(t),Q(t),R(t))∈R6+:N(t)≤Λμ}
with initial conditions S1(0)≥0,S2(0)≥0,E(0)≥0,I(0)≥0,Q(0)≥0,R(0)≥0 is positively invariant for system (2.1).
Proof. Adding the equations of system (2.1) we obtain
dNdt=Λ−μN−d(I+Q)≤Λ−μN.
It follows that
0≤N(t)≤Λμ+N(0)e−μt,
where N(0) represents the initial values of the total population. Thus limt→+∞supN(t)≤Λμ. It implies that the region Ω={(S1(t),S2(t),E(t),I(t),Q(t),R(t))∈R6+:N(t)≤Λμ} is a positively invariant set for system (2.1). So we consider dynamics of system (2.1) and (2.2) on the set Ω in this paper.
3.
The basic reproduction number and existence of equilibria
The model has a disease free equilibrium (S01,S02,0,0,0), where
S01=Λp+μ,S02=pΛμ(p+μ).
In the following, the basic reproduction number of system (2.2) will be obtained by the next generation matrix method formulated in [18].
Let x=(E,I,Q,S1,S2)T, then system (2.2) can be written as
Therefore, by the monotonicity of function H(I), for (3.11) there exists a unique positive root in the interval (0,Λμ) when R0>1; there is no positive root in the interval (0,Λμ) when R0≤1. We summarize this result in Theorem 3.1.
Theorem 3.1. For system (2.2), there is always the disease free equilibrium P0(S01,S02,0,0,0). When R0>1, besides the disease free equilibrium P0, system (2.2) also has a unique endemic equilibrium P∗(S∗1,S∗2,E∗,I∗,Q∗), where
We have ¯F(x,y)≤0 for x,y>0 and ¯F(x,y)=0 if and only if x=y=1. Since R0≤1, then V1′≤0. It follows from LaSalle invariance principle [19] that the disease free equilibrium P0 is globally asymptotically stable when R0≤1.
4.2. Global stability of the endemic equilibrium
For the endemic equilibrium P∗(S∗1,S∗2,E∗,I∗,Q∗), S∗1,S∗2,E∗,I∗, and Q∗ satisfies equations
Since the arithmetical mean is greater than, or equal to the geometrical mean, then, 2−x−1x≤0 for x>0 and 2−x−1x=0 if and only if x=1; 3−1x−y−xy≤0 for x,y>0 and 3−1x−y−xy=0 if and only if x=y=1; 3−1x−xuz−zu≤0 for x,z,u>0 and 3−1x−xuz−zu=0 if and only if x=1,z=u; 4−1x−xy−yuz−zu≤0 for x,y,z,u>0 and 4−1x−xy−yuz−zu=0 if and only if x=y=1,z=u. Therefore, V2′≤0 for x,y,z,u>0 and V2′=0 if and only if x=y=1,z=u, the maximum invariant set of system (2.2) on the set {(x,y,z,u):V2′=0} is the singleton (1,1,1,1). Thus, for system (2.2), the endemic equilibrium P∗ is globally asymptotically stable if R0>1 by LaSalle Invariance Principle [19].
5.
A case study
In this section, we estimate the unknown parameters of model (2.2) on the basis of the total confirmed new cases in the UK from February 1, 2020 to March 23, 2020 by using MCMC algorithm. By estimating the unknown parameters, we estimate the mean and confidence interval of the basic reproduction number R0.
5.1. Parameter estimation and model fitting
The total confirmed cases can be expressed as follows
dCdt=qrE+εI,
where C(t) indicates the total confirmed cases.
As for the total confirmed new cases, it can be expressed as following
NC=C(t)−C(t−1),
(5.1)
where NC represents the total confirmed new cases.
We use the MCMC method [20,21,22] for 20000 iterations with a burn-in of 5000 iterations to fit the Eq (5.1) and estimate the parameters and the initial conditions of variables (see Table 2). Figure 2 shows a good fitting between the model solution and real data, well suggesting the epidemic trend in the United Kingdom. According to the estimated parameter values and initial conditions as given in Table 2, we estimate the mean value of the reproduction number R0=4.2816 (95%CI:(3.8882,4.6750)).
Table 2.
The parameters values and initial values of the model (2.2).
Figure 2.
The fitting results of the total confirmed new cases from February 1, 2020 to March 23, 2020. The blue line is the simulated curve of model (2.2). The red dots represent the actual data. The light blue area is the 95% confidence interval (CI) for all 5000 simulations.
Applying the estimated parameter values, without the most restrictive measures in UK, we forecast that the peak size is 1.2902×106 (95%CI:(1.1429×106,1.4374×106)), the peak time is June 2 (95%CI:(May23,June13)) (Figure 3a), and the final size is 4.9437×107 (95%CI:(4.7199×107,5.1675×107)) in the UK (Figure 3b).
Figure 3.
(a) Forecasting trends of total confirmed new cases. (b) Forecasting trends of total confirmed cases. The blue line is the simulated curve of model (2.2). The red dots represent the actual data. The light blue area is the 95% confidence interval (CI) for all 5000 simulations.
In this section, we do the sensitivity analysis for four vital model parameters σ, p, q and ε, which reflect the intensity of contact, media coverage and isolation, respectively.
Figure 4 and Table 3 show that reducing the fraction σ of the conscious susceptible S2 contacting with the latent compartment (E) and the infectious compartment (I) delays the peak arrival time, decreases the peak size of confirmed cases and decreases the final size. Reducing the fraction σ is in favor of controlling COVID-19 transmission.
Figure 4.
(a) Effects of different Parameter σ on The numbers of total confirmed new cases. (b) Effects of different Parameter σ on The numbers of total confirmed cases.
Figure 5 and Table 3 show that reducing migration rate p to S2 from S1, reflecting the impact of media coverage advances the peak arrival time, increase the peak size of confirmed cases and the final size.
Figure 5.
(a) Effects of different Parameter p on The numbers of total confirmed new cases. (b) Effects of different Parameter p on The numbers of total confirmed cases.
Figures 6, 7 and Table 3 show that, with increase the fraction q (Individuals in the latent compartment E jump into the quarantine compartment Q) and the transition rate ε (the infectious compartment I jump into the quarantine compartment Q), the peak time delays, the the peak size and the final size decrease. This show that Increasing the intensity of detection and isolation may affect the spread of COVID-19.
Figure 6.
(a) Effects of different Parameter q on The numbers of total confirmed new cases. (b) Effects of different Parameter q on The numbers of total confirmed cases.
Figure 7.
(a) Effects of different Parameter ε on The numbers of total confirmed new cases. (b) Effects of different Parameter ε on The numbers of total confirmed cases.
Then the test set data is used to verify the short-term prediction effect of the model (Figure 8). we fit the model with the total confirmed new cases from February 1, 2020 to March 23, 2020, and verify the fitting results with the new cases from March 24, 2020 to April 12, 2020. The model has a good fit to the trajectory of the coronavirus prevalence for a short time in the UK.
Figure 8.
The test set data is used to verify the short-term prediction effect of the model.
We have formulated the COVID-19 epidemic model with media coverage and quarantine and investigated their dynamical behaviors. By means of the next generation matrix, we obtained their basic reproduction number, R0, which play a crucial role in controlling the spread of COVID-19. By constructing Lyapunov function, we proved the global stability of their equilibria: when the basic reproduction number is less than or equal to one, all solutions converge to the disease free equilibrium, that is, the disease dies out eventually; when the basic reproduction number exceeds one, the unique endemic equilibrium is globally stable, that is, the disease will persist in the population and the number of infected individuals tends to a positive constant. We use the MCMC algorithm to estimate the unknown parameters and initial values of the model (2.2) on the basis of the total confirmed new cases in the UK. The sensitivity of all parameters are evaluated.
Through the mean and confidence intervals of the parameters in Table 2, we obtain the basic reproduction number R0=4.2816(95%CI:(3.8882,4.6750)), which means that the novel coronavirus pneumonia is still pandemic in the crowd. The sensitivity of the parameters provides a possible intervention to reduce COVID-19 infection. People should wear masks, avoid contact or reduce their outings, take isolation measure to reduce the spread of virus during COVID-19 outbreaks.
Acknowledgments
We are grateful to the anonymous referees and the editors for their valuable comments and suggestions which improved the quality of the paper. This work is supported by the National Natural Science Foundation of China (11861044 and 11661050), and the HongLiu first-class disciplines Development Program of Lanzhou University of Technology.
Conflict of interest
The authors declare there is no conflict of interest.
References
[1]
P. Zhou, X. L. Yang, X. G. Wang, B. Hu, L. Zhang, W. Zhang, et al., A pneumonia outbreak associated with a new coronavirus of probable bat origin, Nature, 579 (2020), 270-273. doi: 10.1038/s41586-020-2012-7
[2]
Q. Li, X. Guan, P. Wu, X. Wang, L. Zhou, Y. Tong, et al., Early transmission dynamics in Wuhan, China, of Noval Coronavirus-infected Pneumonia, N. Engl. J. Med., 382 (2020), 1199-1207. doi: 10.1056/NEJMoa2001316
[3]
P. Wu, X. Hao, E. H. Y. Lau, J. Y. Wong, K. S. M. Leung, J. T. Wu, et al., Real-time tentative assessment of the epidemiological characteristics of novel coronavirus infections in Wuhan, China, as at 22 January 2020, Euro. Surveill., 25 (2020), 2000044.
[4]
T. Chen, J. Rui, Q. Wang, Z. Zhao, J. Cui, L. Yin, A mathematical model for simulating the transmission of Wuhan novel Coronavirus, Infect. Dis. Poverty, 9 (2020), 1-8. doi: 10.1186/s40249-019-0617-6
[5]
M. W. Shen, Z. H. Peng, Y. N. Xiao, L. Zhang, Modelling the epidemic trend of the 2019 novel coronavirus outbreak in China, bioRxiv, (2020), 2020.01.23.916726.
[6]
Q. Y. Lin, S. Zhao, D. Z. Gao, Y. J. Lou, S. Yang, S. S. Musa, et al., A conceptual model for the outbreak of Coronavirus disease 2019 (COVID-19) in Wuhan, China with individual reaction and governmental action, Int. J. Infect. Dis., 93 (2020), 211-216. doi: 10.1016/j.ijid.2020.02.058
[7]
L. L. Wang, Y. W. Zhou, J. He, B. Zhu, F. Wang, L. Tang, et al., An epidemiological forecast model and software assessing interventions on COVID-19 epidemic in China, medRxiv, (2020).
[8]
S. Y. Tang, B. Tang, N. L. Bragazzi, F. Xia, T. J. Li, S. He, et al., Stochastic discrete epidemic modeling of COVID-19 transmission in the Province of Shaanxi incorporating public health intervention and case importation, medRxiv, (2020).
[9]
X. F. Luo, S. S. Feng, J. Y. Yang, X. L. Peng, X. C. Cao, J. P. Zhang, et al., Analysis of potential risk of COVID-19 infections in China based on a pairwise epidemic model, (2020). doi: 10.20944/preprints202002.0398.v1.
[10]
H. Nishiura, N. M. Linton, A. R. Akhmetzhanov, Serial interval of novel coronavirus (2019-nCoV) infections, medRxiv, (2020). doi: 10.1101/2020.02.03.20019497.
[11]
Z. W. Du, X. K. Xu, Y. Wu, L. Wang, B. J. Cowling, L. A. Meyers, The serial interval of COVID-19 among publicly reported confirmed cases, medRxiv, (2020). doi: 10.1101/2020.02.19.20025452.
[12]
H. F. Huo, S. J. Dang, Y. N. Li, Stability of a Two-Strain Tuberculosis Model with General Contact Rate, Abstr. Appl. Anal., 2010 (2010), 1-31.
[13]
H. F. Huo, L. X. Feng, Global stability for an HIV/AIDS epidemic Model with different latent stages and treatment, Appl. Math. Model., 37 (2013), 1480-1489. doi: 10.1016/j.apm.2012.04.013
[14]
P. Shao, Y. G. Shan, Beware of asymptomatic transmission: Study on 2019-nCov preventtion and control measures based on SEIR model, BioRxiv, (2020). doi: 10.1101/2020.01.28.923169.
[15]
W. K. Zhou, A. L. Wang, F. Xia, Y. N. Xiao, S. Y. Tang, Effects of media reporting on mitigating spread of COVID-19 in the early phase of the outbreak, Math. Biosci. Eng., 17 (2020), 2693-2707. doi: 10.3934/mbe.2020147
[16]
J. Amador, A. Gomez-Corral, A stochastic epidemic model with two quarantine states and limited carrying capacity for quarantine, Physica A, 544 (2020), 121899. doi: 10.1016/j.physa.2019.121899
[17]
Q. W. Gao, J. Zhuang, Stability analysis and control strategies for worm attack in mobile networks via a VEIQS propagation model, Appl. Math. Comput., 368 (2020), 124584.
[18]
P. Van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6
[19]
J. P. LaSalle, The stability of dynamical systems, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976.
[20]
H. Haario, E. Saksman, J. Tamminen, An adaptive metropolis algorithm, Bernoulli, 7 (2001), 223-242. doi: 10.2307/3318737
[21]
H. Haario, M. Laine, A. Mira, E. Saksman, DRAM: efficient adaptive MCMC, Stat. Comput., 16 (2006), 339-354. doi: 10.1007/s11222-006-9438-0
[22]
D. Gamerman, H. F. Lopes, Markov chain Monte Carlo: stochastic simulation for Bayesian inference, Technometrics, 50 (2008), 97.
[23]
W. J. Guan, Z. Y. Ni, Y. Hu, W. H. Liang, C. Q. Ou, J. X. He, et al., Clinical Characteristics of Coronavirus Disease 2019 in China, New Engl. J. Med., 382 (2020), 1708-1720. doi: 10.1056/NEJMoa2002032
[24]
B. Tang, X. Wang, Q. Li, N. L. Bragazzi, S. Y. Tang, Y. N. Xiao, et al., Estimation of the transmission risk of 2019-nCov and its implication for public health interventions, J. Clin. Med., 9 (2020), 462. doi: 10.3390/jcm9020462
Marouane Mahrouf, Adnane Boukhouima, Houssine Zine, El Mehdi Lotfi, Delfim F. M. Torres, Noura Yousfi,
Modeling and Forecasting of COVID-19 Spreading by Delayed Stochastic Differential Equations,
2021,
10,
2075-1680,
18,
10.3390/axioms10010018
2.
M A A Pratama, A K Supriatna, N Anggriani,
A mathematical model to study the effect of travel between two regions on the COVID-19 infections,
2021,
1722,
1742-6588,
012044,
10.1088/1742-6596/1722/1/012044
3.
Guo-Rong Xing, Ming-Tao Li, Li Li, Gui-Quan Sun,
The Impact of Population Migration on the Spread of COVID-19: A Case Study of Guangdong Province and Hunan Province in China,
2020,
8,
2296-424X,
10.3389/fphy.2020.587483
4.
Chandan Maji, Mayer Humi,
Impact of Media-Induced Fear on the Control of COVID-19 Outbreak: A Mathematical Study,
2021,
2021,
1687-9651,
1,
10.1155/2021/2129490
5.
Cheng-Cheng Zhu, Jiang Zhu,
Dynamic analysis of a delayed COVID-19 epidemic with home quarantine in temporal-spatial heterogeneous via global exponential attractor method,
2021,
143,
09600779,
110546,
10.1016/j.chaos.2020.110546
6.
Ahmed A Mohsen, Hassan Fadhil AL-Husseiny, Xueyong Zhou, Khalid Hattaf,
Global stability of COVID-19 model involving the quarantine strategy and media coverage effects,
2020,
7,
2327-8994,
587,
10.3934/publichealth.2020047
7.
A. Babaei, M. Ahmadi, H. Jafari, A. Liya,
A mathematical model to examine the effect of quarantine on the spread of coronavirus,
2021,
142,
09600779,
110418,
10.1016/j.chaos.2020.110418
8.
Tanvir Ahammed, Aniqua Anjum, Mohammad Meshbahur Rahman, Najmul Haider, Richard Kock, Md Jamal Uddin,
Estimation of novel coronavirus (
COVID
‐19) reproduction number and case fatality rate: A systematic review and meta‐analysis
,
2021,
4,
2398-8835,
10.1002/hsr2.274
9.
Ahmed A. Mohsen, Hassan F. AL-Husseiny, Raid Kamel Naji,
The dynamics of Coronavirus pandemic disease model in the existence of a curfew strategy,
2022,
25,
0972-0502,
1777,
10.1080/09720502.2021.2001139
10.
Abdelhamid Ajbar, Rubayyi T. Alqahtani, Mourad Boumaza,
Dynamics of an SIR-Based COVID-19 Model With Linear Incidence Rate, Nonlinear Removal Rate, and Public Awareness,
2021,
9,
2296-424X,
10.3389/fphy.2021.634251
11.
Ousmane Koutou, Abou Bakari Diabaté, Boureima Sangaré,
Mathematical analysis of the impact of the media coverage in mitigating the outbreak of COVID-19,
2023,
205,
03784754,
600,
10.1016/j.matcom.2022.10.017
12.
Folashade B. Agusto, Eric Numfor, Karthik Srinivasan, Enahoro A. Iboi, Alexander Fulk, Jarron M. Saint Onge, A. Townsend Peterson,
Impact of public sentiments on the transmission of COVID-19 across a geographical gradient,
2023,
11,
2167-8359,
e14736,
10.7717/peerj.14736
13.
Khalid Hattaf, Ahmed A. Mohsen, Jamal Harraq, Naceur Achtaich,
Modeling the dynamics of COVID-19 with carrier effect and environmental contamination,
2021,
12,
1793-9623,
2150048,
10.1142/S1793962321500483
14.
Rubayyi T. Alqahtani, Abdelhamid Ajbar,
Study of Dynamics of a COVID-19 Model for Saudi Arabia with Vaccination Rate, Saturated Treatment Function and Saturated Incidence Rate,
2021,
9,
2227-7390,
3134,
10.3390/math9233134
15.
Tridip Sardar, Sk Shahid Nadim, Sourav Rana,
Detection of multiple waves for COVID-19 and its optimal control through media awareness and vaccination: study based on some Indian states,
2023,
111,
0924-090X,
1903,
10.1007/s11071-022-07887-5
16.
M. El Sayed, M. A. El Safty, M. K. El-Bably,
Topological approach for decision-making of COVID-19 infection via a nano-topology model,
2021,
6,
2473-6988,
7872,
10.3934/math.2021457
17.
Azhar Iqbal Kashif Butt, Muhammad Imran, Saira Batool, Muneerah AL Nuwairan,
Theoretical Analysis of a COVID-19 CF-Fractional Model to Optimally Control the Spread of Pandemic,
2023,
15,
2073-8994,
380,
10.3390/sym15020380
18.
S. A. Alblowi, M. El Sayed, M. A. El Safty,
Decision Making Based on Fuzzy Soft Sets and Its Application in COVID-19,
2021,
30,
1079-8587,
961,
10.32604/iasc.2021.018242
19.
M. A. El Safty, S. A. Alblowi, Yahya Almalki, M. El Sayed,
Coronavirus Decision-Making Based on a Locally -Generalized Closed Set,
2022,
32,
1079-8587,
483,
10.32604/iasc.2022.021581
20.
Yan Wang, Feng Qing, Haozhan Li, Xuteng Wang,
Timely and effective media coverage's role in the spread of Corona Virus Disease 2019,
2022,
0170-4214,
10.1002/mma.8732
21.
Jinxing Guan, Yang Zhao, Yongyue Wei, Sipeng Shen, Dongfang You, Ruyang Zhang, Theis Lange, Feng Chen,
Transmission dynamics model and the coronavirus disease 2019 epidemic: applications and challenges,
2022,
2,
2749-9642,
89,
10.1515/mr-2021-0022
22.
A.I.K. Butt, W. Ahmad, M. Rafiq, D. Baleanu,
Numerical analysis of Atangana-Baleanu fractional model to understand the propagation of a novel corona virus pandemic,
2022,
61,
11100168,
7007,
10.1016/j.aej.2021.12.042
23.
Jiraporn Lamwong, Puntani Pongsumpun, I-Ming Tang, Napasool Wongvanich,
Vaccination’s Role in Combating the Omicron Variant Outbreak in Thailand: An Optimal Control Approach,
2022,
10,
2227-7390,
3899,
10.3390/math10203899
24.
Abdelhamid Ajbar, Rubayyi T. Alqahtani, Mourad Boumaza,
Dynamics of a COVID-19 Model with a Nonlinear Incidence Rate, Quarantine, Media Effects, and Number of Hospital Beds,
2021,
13,
2073-8994,
947,
10.3390/sym13060947
25.
Asma Hanif, Azhar Iqbal Kashif Butt, Waheed Ahmad,
Numerical approach to solve Caputo‐Fabrizio‐fractional model of corona pandemic with optimal control design and analysis,
2023,
0170-4214,
10.1002/mma.9085
26.
M. L. Diagne, H. Rwezaura, S. Y. Tchoumi, J. M. Tchuenche, Jan Rychtar,
A Mathematical Model of COVID-19 with Vaccination and Treatment,
2021,
2021,
1748-6718,
1,
10.1155/2021/1250129
27.
Cheng-Cheng Zhu, Jiang Zhu, Jie Shao,
Epidemiological Investigation: Important Measures for the Prevention and Control of COVID-19 Epidemic in China,
2023,
11,
2227-7390,
3027,
10.3390/math11133027
28.
Abdisa Shiferaw Melese,
Vaccination Model and Optimal Control Analysis of Novel Corona Virus Transmission Dynamics,
2023,
271,
1072-3374,
76,
10.1007/s10958-023-06277-5
29.
Andrea Miconi, Simona Pezzano, Elisabetta Risi,
Framing pandemic news. Empirical research on Covid-19 representation in the Italian TV news,
2023,
35,
0862397X,
65,
10.58193/ilu.1752
30.
Rubayyi T. Alqahtani, Abdelhamid Ajbar, Nadiyah Hussain Alharthi,
Dynamics of a Model of Coronavirus Disease with Fear Effect, Treatment Function, and Variable Recovery Rate,
2024,
12,
2227-7390,
1678,
10.3390/math12111678
31.
V. R. Saiprasad, V. Vikram, R. Gopal, D. V. Senthilkumar, V. K. Chandrasekar,
Effect of vaccination rate in multi-wave compartmental model,
2023,
138,
2190-5444,
10.1140/epjp/s13360-023-04634-6
32.
Adesoye Idowu Abioye, Olumuyiwa James Peter, Hammed Abiodun Ogunseye, Festus Abiodun Oguntolu, Tawakalt Abosede Ayoola, Asimiyu Olalekan Oladapo,
A fractional-order mathematical model for malaria and COVID-19 co-infection dynamics,
2023,
4,
27724425,
100210,
10.1016/j.health.2023.100210
33.
Abou Bakari Diabaté, Boureima Sangaré, Ousmane Koutou,
Optimal control analysis of a COVID-19 and Tuberculosis (TB) co-infection model with an imperfect vaccine for COVID-19,
2024,
81,
2254-3902,
429,
10.1007/s40324-023-00330-8
34.
Lili Han, Mingfeng He, Xiao He, Qiuhui Pan,
Synergistic effects of vaccination and virus testing on the transmission of an infectious disease,
2023,
20,
1551-0018,
16114,
10.3934/mbe.2023719
35.
A.I.K. Butt, W. Ahmad, M. Rafiq, N. Ahmad, M. Imran,
Optimally analyzed fractional Coronavirus model with Atangana–Baleanu derivative,
2023,
53,
22113797,
106929,
10.1016/j.rinp.2023.106929
36.
Pan Tang, Ning Wang, Tong Zhang, Longxing Qi,
Modeling the effect of health education and individual participation on the increase of sports population and optimal design,
2023,
20,
1551-0018,
12990,
10.3934/mbe.2023579
37.
Zhong-Ning Li, Yong-Lu Tang, Zong Wang,
Optimal control of a COVID-19 dynamics based on SEIQR model,
2025,
2025,
2731-4235,
10.1186/s13662-025-03869-0
38.
Azhar Iqbal Kashif Butt, Waheed Ahmad, Hafiz Ghulam Rabbani, Muhammad Rafiq, Shehbaz Ahmad, Naeed Ahmad, Saira Malik,
Exploring optimal control strategies in a nonlinear fractional bi-susceptible model for Covid-19 dynamics using Atangana-Baleanu derivative,
2024,
14,
2045-2322,
10.1038/s41598-024-80218-3
39.
Fahad Al Basir, Kottakkaran Sooppy Nisar, Ibraheem M. Alsulami, Amar Nath Chatterjee,
Dynamics and optimal control of an extended SIQR model with protected human class and public awareness,
2025,
140,
2190-5444,
10.1140/epjp/s13360-025-06108-3
40.
Gabriel McCarthy, Hana M. Dobrovolny,
Determining the best mathematical model for implementation of non-pharmaceutical interventions,
2025,
22,
1551-0018,
700,
10.3934/mbe.2025026
Li-Xiang Feng, Shuang-Lin Jing, Shi-Ke Hu, De-Fen Wang, Hai-Feng Huo. Modelling the effects of media coverage and quarantine on the COVID-19 infections in the UK[J]. Mathematical Biosciences and Engineering, 2020, 17(4): 3618-3636. doi: 10.3934/mbe.2020204
Li-Xiang Feng, Shuang-Lin Jing, Shi-Ke Hu, De-Fen Wang, Hai-Feng Huo. Modelling the effects of media coverage and quarantine on the COVID-19 infections in the UK[J]. Mathematical Biosciences and Engineering, 2020, 17(4): 3618-3636. doi: 10.3934/mbe.2020204
Figure 1. The transfer diagram for the model (2.1)
Figure 2. The fitting results of the total confirmed new cases from February 1, 2020 to March 23, 2020. The blue line is the simulated curve of model (2.2). The red dots represent the actual data. The light blue area is the 95% confidence interval (CI) for all 5000 simulations
Figure 3. (a) Forecasting trends of total confirmed new cases. (b) Forecasting trends of total confirmed cases. The blue line is the simulated curve of model (2.2). The red dots represent the actual data. The light blue area is the 95% confidence interval (CI) for all 5000 simulations
Figure 4. (a) Effects of different Parameter σ on The numbers of total confirmed new cases. (b) Effects of different Parameter σ on The numbers of total confirmed cases
Figure 5. (a) Effects of different Parameter p on The numbers of total confirmed new cases. (b) Effects of different Parameter p on The numbers of total confirmed cases
Figure 6. (a) Effects of different Parameter q on The numbers of total confirmed new cases. (b) Effects of different Parameter q on The numbers of total confirmed cases
Figure 7. (a) Effects of different Parameter ε on The numbers of total confirmed new cases. (b) Effects of different Parameter ε on The numbers of total confirmed cases
Figure 8. The test set data is used to verify the short-term prediction effect of the model