A model for the early COVID-19 outbreak in China with case detection and behavioural change BIOMATH https://biomath.math.bas.bg/biomath/index.php/biomath B f Biomath Forum ORIGINAL ARTICLE A model for the early COVID-19 outbreak in China with case detection and behavioural change Julien Arino1, Khalid El Hail2,∗, Mohamed Khaladi3, Aziz Ouhinou2 1Department of Mathematics University of Manitoba, Winnipeg, Manitoba, Canada julien.arino@umanitoba.ca 0000-0001-6409-5027 2Department of Mathematics, Faculty of Sciences and Techniques, University of Sultan Moulay Slimane, Beni-Mellal, Morocco elhail.kha@gmail.com 0000-0001-5166-9386 a.ouhinou@usms.ma 0000-0002-0206-4935 3Laboratory of Mathematics and Population Dynamics – UMMISCO, Faculty of Sciences Semlalia, Cadi Ayyad University, Marrakech, Morocco khaladi@uca.ac.ma 0000-0002-7703-5637 Received: November 5, 2022, Accepted: December 20, 2022, Published: January 27, 2023 Abstract: We investigate a model of the early stage of the COVID-19 epidemic comprising undetected infected individuals as well as behavioural change towards the use of self-protection measures. The model is fitted to China data reported between 22 January and 29 June 2020. Using fitting results, we then consider model responses to varying screening intensities. Keywords: COVID-19, Behavioural Change, Screening Intensity, Protective Measures I. INTRODUCTION At the time of writing, the first known COVID-19 human case is one with onset on 8 December 2019, in Wuhan, China [1, 2], although there is evidence that the disease have been spreading earlier; see [3] for a timeline of early spread. On 20 January 2020, studies confirmed human-to-human transmission through res- piratory droplets [4]. There is now an unprecedentedly large body of work on the worldwide COVID-19 out- break; however, many epidemiological features such as per capita transmissibility, screening and disease- related death rates are still ambiguous and, to a large ex- tent, seem quite dependent on the location under consid- eration, with outbreak intensities varying greatly from country to country. Parameters may vary from region to region depending, for instance, on control measures taken by policymakers, availability of personal protec- tive equipment, hospitalisation, demographic pyramid, life activities and cultural aspects. Many works consider the early spread of COVID-19 in China, which at the start of the pandemic had the most data since it had the most cases; the list is far too extensive to detail here and we list just a few. In [5], the authors estimated the basic reproduction number to be up to R0 = 3.58 at the beginning of the outbreak in China. Using the official counts of confirmed cases, R0 was suggested in [6] to be on average 4.6, and, by assuming presymptomatic and mildly symptomatic infectious individuals to be twenty or forty times the reported number of infected cases, the mean of R0 was Copyright: © 2022 Julien Arino, Khalid El Hail, Mohamed Khaladi, Aziz Ouhinou. This article is distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. *Corresponding author Citation: Julien Arino, Khalid El Hail, Mohamed Khaladi, Aziz Ouhinou, A model for the early COVID-19 outbreak in China with case detection and behavioural change, Biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 1/8 https://biomath.math.bas.bg/biomath/index.php/biomath mailto:julien.arino@umanitoba.ca https://orcid.org/0000-0001-6409-5027 mailto:elhail.kha@gmail.com https://orcid.org/0000-0001-5166-9386 mailto:a.ouhinou@usms.ma https://orcid.org/0000-0002-0206-4935 mailto:khaladi@uca.ac.ma https://orcid.org/0000-0002-7703-5637 https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.55630/j.biomath.2022.12.207 Arino et al, A model for the early COVID-19 outbreak in China with case detection and behavioural change estimated to be 3.2 or 2.6, respectively; daily infection mortality and recovery rates were also estimated. In addition, in the early stage of the COVID-19 epidemic in China, [1] suggested R0 to be approximately 2.2 and the incubation period to have a mean of 5.2 days. Because infection severity differs greatly in in- fected individuals, some individuals are infectious while presymptomatic. Together with asymptomatic infec- tions, this means that some individuals may have “evaded” screening, despite contributing to the spread of the disease [6–9]. Besides isolating detected active cases and their known immediate contacts, healthcare authorities worldwide did their best to educate the pop- ulation about COVID-19 severity, its mode of transmis- sion and convince people to use all available preventive measures. Accordingly, in this work we take into consideration the population response to education campaigns. We use a system of differential equations to model the COVID-19 epidemic with, including burden-dependent behavioural change. We calibrate some of the parame- ters so that the model fits the COVID-19 data in China from 22 January until 29 June 2020. This article is organised as follows. In Section II, the mathematical model of COVID-19 transmission is derived. Section III presents numerical results, which are then discussed further in Section IV. II. MATHEMATICAL MODELLING The mathematical model considered in this paper comprises seven epidemiological compartments, S, E, IS, IE, JD, JT and R, as well as an auxiliary variable, A, used to account for awareness of the disease. The flow diagram is shown in Figure 1; let us elaborate on this structure. S are susceptible individuals in the classical sense, while E denotes educated susceptible individuals using self-protective measures against the infection. Infected individuals are divided into four compartments. IS and IE are, respectively, non-educated and educated undetected infectious individuals; individuals in both of these compartments who get screened move, upon detection, into the isolation compartment JD, where they wait for recovery or a potential hospitalisation [10]. If their infection goes undetected, upon recovery or death, they progress directly to the removed compart- ment. The fourth infected compartment, JT , contains infected cases who are under treatment in hospital. Both JD and JT are isolated and as a consequence, they are not infectious to others. The compartment R is for Table I: State variables. Variable Definition S Susceptible individuals E Educated susceptible individuals IS Undetected infectious non-educated individuals IE Undetected infectious educated individuals JD Isolated infected individuals JT Hospitalised infected individuals R Removed individuals A Disease awareness (auxiliary variable) removals due to recovery or death. Table I summarises the definition of all state variables. We use the auxiliary variable A to represent aware- ness of the disease. Awareness is based on available information: known (detected) cases, hospitalisations and deaths from the disease. In the case of COVID-19, many people used personal protection equipment and practiced social distancing when they became aware of the presence of the disease. This was further reinforced by stringent social distancing and confinement measures imposed or recommended by authorities. However, even if they are aware of the presence of the disease and with strong or even coercive governmental policies, not all individuals follow public health recommendations or orders. We therefore assume that susceptible individuals become educated (and therefore follow public health recommendations) at the rate e(A) = e0A 2 A∗2 + A2 , giving a Hill functional form [11] and described in [12, 13]. Here A∗, is the awareness level producing half of the maximum education response e0 to campaign efforts; see [11, 14] for more details. We model individuals flow between different com- partments using the following system of differential equations S′ = −λS −e(A)S (1a) E′ = e(A)S − (1−ε)λE (1b) I′S = λS − (α + e(A) + δ)IS (1c) I′E = (1−ε)λE + e(A)IS − (α + δ)IE (1d) J′D = (1−θ)α(IS + IE)− (γ1 + w)JD (1e) J′T = θα(IS + IE)−γ2JT + wJD (1f) R′ = δ(IE + IS) + γ1JD + γ2JT (1g) A = JD + (1 + pγ2)JT , (1h) with initial conditions (S0,E0,IS0,IE0,JD0,JT 0,R0) ∈ R7+. Biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 2/8 https://doi.org/10.55630/j.biomath.2022.12.207 Arino et al, A model for the early COVID-19 outbreak in China with case detection and behavioural change S E IS IE JD JT RA e (A )S e (A )I S λS (1−ε)λE δIE (1−θ)αIS θαI S (1 −θ )α IE θαIE δIS γ 1 J D w J D γ 2 J T Fig. 1: Flow diagram of the model. Dark red compartments are infectious, blue represents awareness. Plain arcs are flows of individuals between compartments, dashed lines indicate the influence of compartments on A, dotted lines show the flows on which A acts. The force of infection takes the form λ = β IS + (1−ε)IE N , where β is the per capita transmission rate per unit time, ε ∈ [0,1] is the efficacy of self-protective mea- sures, α is the detection rate and θ is the treatment rate. Table II summarises the parameters used. Table II: Definition of parameters. Param. Definition β Transmission rate α Detection rate δ Natural recovery rate θ Proportion of individuals needing hospitalisation after detection γ1 Recovery rate for individuals in self-isolation γ2 Removal rate for individuals under treatment w Hospitalisation rate for self-isolating individuals p Proportion of deaths among removed individuals A∗ Burden level producing half maximum of educa- tion response e0 Maximum of education response ε Efficacy of self-protective means Let us briefly comment on some characteristics of the model. In a standard way as in [11], one can show that system (1) is well-posed and has a unique positive solution whenever the initial condition is positive. By construction, the total population N = S + E + IS + IE + JD + JT + R is constant. The disease-free equilibrium (DFE) is x∗ = (N,0,0,0,0,0,0) and at x∗ awareness is A = 0 and thus e = 0. To apply the next generation matrix method [15], we focus on the infected compartments. Although still infected, individuals in JD and JT no longer contribute to the infection and can be considered as having been removed. As a consequence, the infected compartments considered for the computation are IS and IE. We get F = ( λS (1−ε)λE ) and V = ( (α + e + δ)IS −eIS + (α + δ)IE ) . Therefore, the next generation matrix near x∗ is FV −1, where F = ( β (1−ε)β 0 0 ) and V = ( α + δ 0 0 α + δ ) . Hence, the basic reproduction number for (1) is given by R0 = ρ ( FV −1 ) = β α + δ , where ρ(·) is the spectral radius. Using the method in [16], it is also possible to derive a final size relation. Biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 3/8 https://doi.org/10.55630/j.biomath.2022.12.207 Arino et al, A model for the early COVID-19 outbreak in China with case detection and behavioural change III. PARAMETER ESTIMATION AND NUMERICAL SIMULATIONS A. Parameter estimation We now estimate the parameters that are used in numerical simulations. First, let us establish the initial conditions used. We take the initial time to be 8 December 2019, when the first known patient developed symptoms of COVID-19 [2]. The initial susceptible population is 1.438 billion, the estimated total popula- tion of China at the time [17]. At this very early stage of the pandemic, people did not yet know about the disease and had thus not changed their behaviour to use adequate self-protection measures. Thus, initially, the educated compartments E0 and IE0 are empty. Also, there were no reported recoveries or deaths of the new disease [2]. Table III summarises the initial conditions considered for (1) in simulations. Table III: Initial conditions on 8 December 2019 (B stands for billion). Compart- S0 E0 IS0 IE0 JD0 JT 0 R0 ment Value 1.438B 0 1 0 1 0 0 Sequencing of the SARS-CoV-2 genome was accom- plished early on, in January 2020, and as a consequence, PCR tests followed in the same month. However, be- cause of limitations in test processing capacities, many jurisdictions, including China, have, at least at times, imposed criteria that individuals had to satisfy in order to be tested. During the period considered, in China and Wuhan in particular, two different sets of criteria were used [18–21]. Most of the time, a restrictive set of criteria was in effect, requiring individuals to show many symptoms in order to be considered as suspected cases and therefore be eligible for testing, leading to what we also refer to as normal screening. During this period, we use the detection rate αr. Then, between 12 and 19 February 2020, the criteria for screening were temporarily changed from the restrictive set to a milder set requiring less symptoms, implying that far more tests were carried out. During that short time period of intense screening, we use the detection rate αm ≥ αr. Almost all parameters in the model are fitted. How- ever, for the proportion of deaths among removed individuals, we use the estimation in [22], namely, 0.04. To estimate parameters, we use data on cases in China between 22 January 2020 and 29 June 2020 as reported in [22]. We use the Python Optimize Module to fit our model to cumulative and active cases (see Figure 2) and calibrate the parameters. Note that for active cases, this means we fit JD(t) +JT (t). Table IV presents the parameter values found by that process. Table IV: Parameter values found by fitting. Parameter Value Remark/Source β 0.347 Fitted (αr,αm) (0.1,0.274) Fitted as a step function δ 0.058 Fitted θ 0.023 Fitted γ2 0.092 Fitted γ1 9.16 × 10−6 Fitted p 0.04 [22] ε 0.662 Fitted w 0.099 Fitted e0 0.389 Fitted A∗ 36432.82 Fitted The transmission rate obtained is 0.347 day−1, which is close to the mean value estimated in [23]. The time between infection and detection is calibrated to be α−1r = 10 days, while the value α −1 m = 3.649 days is found during the intense screening period between 12 and 19 February 2020. The present study suggests a hospitalisation rate of w = 0.099 for 98% of detected individuals, while the remaining detected individuals go directly to hospitals, which is consistent with the results in [1], where the authors conclude that most patients were hospitalised after at least 5 days and that this delay can go up to 14 days. On the other hand, [10] reports an average recovery time after symptoms onset of 24.7 days and the mean time to death to be 17.8 days, while our fitting suggests a mean period between detection and hospital discharge by recovery or death of γ1 (γ1 + w)2 + w γ1 + w ( 1 γ1 + w + 1 γ2 ) = 20.832 days for 98% of detected individuals and 1/γ2 = 10.86 days for 2% of detected individuals, finally obtaining a mean time from detection to removal of 20.632 days. Using parameter values in Table IV, the basic reproduction number is estimated to be R0 = 2.118, close to the value 2.2 obtained in [1]. B. Numerical simulations Figure 2 shows the evolution of the cumulative number and number of active cases reported by the Chinese government from 22 January to June 29 2020, as well as the result of fitting model (1) to this data, displaying good agreement with the real data. Overall, our simulation results are in accordance with both real data and published findings. We obtained that Biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 4/8 https://doi.org/10.55630/j.biomath.2022.12.207 Arino et al, A model for the early COVID-19 outbreak in China with case detection and behavioural change Jan -18 Feb -02 Feb -17 Mar -03 Mar -18 Apr -02 Apr -17 May -02 May -17 Jun -01 Jun -16 Jul- 01 Date 0 20000 40000 60000 80000 Cu m ul at iv e ca se c ou nt Real data Fitted model (a) Cumulative cases Jan -18 Feb -02 Feb -17 Mar -03 Mar -18 Apr -02 Apr -17 May -02 May -17 Jun -01 Jun -16 Jul- 01 Date 0 10000 20000 30000 40000 50000 60000 Nu m be r o f a ct iv e ca se s Real data Fitted model (b) Active cases Fig. 2: Actual data and fitted solution from 22 January to 29 June 2020. Dots are the real data and dotted lines are obtained from simulations. (a) cumulative number of reported cases; (b) active reported cases. the per capita per day transmissibility rate is about β = 0.346 days−1, giving a basic reproduction number of R0 = 2.11. The obtained values are fairly consistent with the approximations in [1]. Figure 3 shows the percentage of undetected infected individuals (including asymptomatic, mild and symp- tomatic individuals) among the total number of COVID- 19 cases. We observe, in the beginning, an increase of the percentage of undeclared infected individuals, reaching 66% during the outbreak and remaining above 50% until 3 February 2020. This significant percentage ensured that infection continued despite the isolation of detected cases, explaining in part the persistence of transmission of COVID-19 during the early stages of the epidemic. To get more insight into the impact of the screening protocol change, we investigate the effect of the timing De c-0 4 De c-1 1 De c-1 8 De c-2 5 Jan -01 Jan -08 Jan -15 Jan -22 Jan -29 Fe b-0 5 Fe b-1 2 Fe b-1 9 Fe b-2 6 Ma r-0 4 Ma r-1 1 Ma r-1 8 Ma r-2 5 Ap r-0 1 Ap r-0 8 Ap r-1 5 Ap r-2 2 Ap r-2 9 Date 0% 20% 40% 60% 80% 100% Pe rc en ta ge Undetected cases Detected cases Fig. 3: Percentage repartition of undetected and detected infected individuals among the total number of infected individuals. of the modification of that protocol on the intensity of the COVID-19 outbreak as shown in Figure 4. We use the parameter values in Table IV, but use αr until the day of the modification and αm afterwards. We observe that the sooner we adopt a less stringent set of symptoms needed to trigger testing, the lower the percentage of undetected infected individuals, leading to a reduction of outbreak intensity. Figure 4a shows that the timing of screening criteria change strongly affects the burden of the disease. A change on 22 January 2020, for instance, leads to a burden equal to about a third of the burden that is observed when no change in screening criteria occur. This fact is explained by Figure 4b, where, with policy change on 22 January, the percentage of hidden infected individuals declines immediately and exponentially, while it does not with criteria modification on 12 February. This implicitly confirms that the presymptomatic period contains hid- den infectious individuals who contributed to the persis- tent transmission in the early stages of the COVID-19 epidemic. We furthermore deduce that increasing the detection rate α early substantially helps to control the COVID-19 epidemic. On the contrary, we observe that a late screening intensity increase after 12 February does not have remarkable effects in dampening the disease intensity. This might be due to behavioural changes of individuals or effectiveness of preventive measures. Figures 5, 6 and 7 address the sensitivity of the dynamics of the COVID-19 outbreak to the rate α of de- tection and the efficiency ε of self-protective measures. We assume no change in the screening strategy; other parameter values are taken from Table IV. In Figure 5, Biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 5/8 https://doi.org/10.55630/j.biomath.2022.12.207 Arino et al, A model for the early COVID-19 outbreak in China with case detection and behavioural change Jan -18 Jan -28 Feb -07 Feb -17 Feb -27 Mar -08 Mar -18 Mar -28 Apr -07 Apr -17 Apr -27 May -07 May -17 May -27 Jun -06 Jun -16 Jun -26 Jul- 06 Date 0 10000 20000 30000 40000 50000 Nu m be r o f a ct iv e ca se s Criteria modification on Jan-24 Criteria modification on Feb-02 Criteria modification on Feb-12 Criteria modification on Feb-22 No criteria modification (a) Active cases Jan -18 Jan -28 Feb -07 Feb -17 Feb -27 Mar -08 Mar -18 Mar -28 Apr -07 Apr -17 Apr -27 May -07 May -17 May -27 Jun -06 Jun -16 Jun -26Jul- 06 Date 0% 10% 20% 30% 40% 50% 60% Pe rc en ta ge o f u nd et ec te d ca se s Criteria modification on Jan-24 Criteria modification on Feb-02 Criteria modification on Feb-12 Criteria modification on Feb-22 No criteria modification (b) Percentage undetected cases Fig. 4: Effect of the timing of the relaxation of the criteria for screening, i.e., of the change from αr to αm. All dates in 2020. (a) Number of detected active cases. (b) Percentage of undetected cases among infected individuals. we observe that the outbreak peak is very sensitive to parameters α and ε. Figure 6a and 7a show how serious the epidemic would be with a low detection rate (see near the ε axis). Figure 6b is a zoom of Figure 6a, around the region in the (ε,α)-space close to the fitted parameters for China. This confirms that the parameters found lie in a region where solutions are quite sensitive to parameter variations, confirming the significant sensitivity to parameter α observed in Figure 5. IV. DISCUSSION In this work, we present a simple model for the spread of COVID-19 taking into account undetected cases, the isolation of detected cases and education favouring the use of protective measures. We fitted this model to Chinese data corresponding to the period from Jan -18 Fe b-0 7 Fe b-2 7 Ma r-1 8 Ap r-0 7 Ap r-2 7 Ma y-1 7 Jun -06 Jun -26 Date 0 10000 20000 30000 40000 50000 Nu m be r o f a ct iv e ca se s = 0.56 = 0.63 = 0.7 = 0.76 (a) Sensitivity to ε Jan -18 Fe b-0 7 Fe b-2 7 Ma r-1 8 Ap r-0 7 Ap r-2 7 Ma y-1 7 Jun -06 Jun -26 Date 0 20000 40000 60000 80000 100000 120000 140000 Nu m be r o f a ct iv e ca se s = 0.02 = 0.08 = 0.14 = 0.21 (b) Sensitivity to α Fig. 5: Number of active cases when (a) the efficiency ε of protective measures and (b) detection rates α vary, with all other parameters as in Table IV. All dates in 2020. the start of the epidemic to the end of June 2020. The model does a good job of fitting that data, as can be seen in Figure 2. In order to obtain this fit, though, we introduced two different values of the intensity α of screening: αm for a period of intense screening cor- responding to a loose definition of symptoms required for screening and αr for a period with more restrictive set of symptoms leading to lower testing rates. The calibrated value αm = 0.438 is consistent with the results in [18–21] reporting the enhancement of the detection process on 12 February. Taking the calibrated values, we then explore in more detail the effect of changing the intensity of screening. We saw in Figure 3 that stringent criteria for screening giving a detection parameter αr = 0.17 led to an extended time period during which over 54% of the infected individuals evaded detection. These Biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 6/8 https://doi.org/10.55630/j.biomath.2022.12.207 Arino et al, A model for the early COVID-19 outbreak in China with case detection and behavioural change 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Nu m be r o f c as es a t t he p ea k 1e8 (a) Sensitivity to ε and α 0.62 0.64 0.66 0.68 0.70 0.06 0.08 0.10 0.12 0.14 ( , ) for China o 20000 30000 40000 50000 60000 70000 80000 90000 Nu m be r o f c as es a t t he p ea k (b) Zoom around parameters found for China Fig. 6: Sensitivity of the number of active infected cases at the peak to the detection rate α and efficacy ε of protective measures. undetected infectious individuals may not know about their infection and keep interacting with the population causing new cases even among loved ones [8]. Figure 4 strengthens the findings of Figure 3 and emphasises the effect of detection strategy change. Thus, the require- ment that individuals show a large number of symptoms in order to be tested might have contributed to a longer persistence of the outbreak in China. Interestingly, modification of screening intensity after 12 February 2020 does not have much effect (Figure 4). This may be because, as the epidemic was well estab- lished at the time, public awareness of the crisis had increased concomitantly with an expansion of the set of public protection measures, leading to an increase in 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.2 0.4 0.6 0.8 1.0 100 101 102 103 104 105 106 107 108 109 Nu m be r o f c as es a t t he p ea k (a) Sensitivity to ε and α (Logarithmic scale) 0.62 0.64 0.66 0.68 0.70 0.06 0.08 0.10 0.12 0.14 The case of China o 103 104 105 106 Nu m be r o f c as es a t t he p ea k (b) Zoom around parameters found for China Fig. 7: Sensitivity of the number of active infected cases at the peak to the detection rate α and efficacy ε of protective measures (Logarithmic scale). uptake of a wider variety of measures. Besides detecting infected individuals before illness onset and isolating them, thereby reducing the chance of transmission of the disease to susceptible individuals, reporting the real number of infected individuals alerts the population about the actual danger presented by the disease. This means that more individuals, including undetected infected individuals, change their behaviour and consider all possible actions to protect themselves or others from the infection. Figure 5 considers the sensitivity of the COVID-19 outbreak dynamics to the efficacy of self-protective measures and detection rates, when these parameters are near the parameter values found for China. It shows that the outbreak is Biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 7/8 https://doi.org/10.55630/j.biomath.2022.12.207 Arino et al, A model for the early COVID-19 outbreak in China with case detection and behavioural change sensitive to both parameters, with a particularly marked sensitivity to α, the rate of detection. The contour plot in Figure 6 confirms this: movement along the (self- protective measures) ε axis induces less variation than movement along the (detection) α axis. Altogether, this highlights that good detection, for in- stance by deploying more tests in highly affected areas and using strategies favouring the tracing of infected individuals, has a significant effect on early spread. According to figure 6, this provides more capacity to control spread than behavioural changes and efficacy of protective measures whose use is made obligatory when detection rates are low. It would be interesting to study an optimal control problem considering the combination of the different types of interventions used here. REFERENCES [1] Q. Li, X. Guan, P. Wu, X. Wang, L. Zhou, Y. Tong, et al., “Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus-Infected Pneumonia”, New England Journal of Medicine, 382(13):1199-1207, 2020. [2] Z. Wu, J. M. McGoogan, “Characteristics of and Important Lessons From the Coronavirus Disease 2019 (COVID-19) Out- break in China: Summary of a Report of 72 314 Cases From the Chinese Center for Disease Control and Prevention”, JAMA, 323(13):1239-1242, 2020. [3] J. Arino, “Describing, Modelling and Forecasting the Spatial and Temporal Spread of COVID-19: A Short Review”, Math- ematics of Public Health, Fields Institute Communications, 85:25-51, Springer, Cham, 2022. [4] 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 epidemiolog- ical characteristics of novel coronavirus infections in Wuhan, China, as at 22 January 2020”, Eurosurveillance, 25(3), 2020. [5] S. Zhao, Q. Lin, J. Ran, S. S. Musa, G. Yang, W. Wang, et al., “Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020: A data-driven analysis in the early phase of the outbreak”, International Journal of Infectious Diseases, 92:214-217, 2020. [6] C. Anastassopoulou, L. Russo, A. Tsakris, C. Siettos, “Data- based analysis, modelling and forecasting of the COVID-19 outbreak”, PLoS ONE, 15(3):e0230405, 2020. [7] Y. Bai, L. Yao, T. Wei, F. Tian, D.-Y. Jin, L. Chen, M. Wang, “Presumed Asymptomatic Carrier Transmission of COVID-19”, JAMA, 323(14):1406-1407, 2020. [8] P. Li, J.-B. Fu, K.-F. Li, J.-N. Liu, H.-L. Wang, L.-J. Liu, et al., “Transmission of COVID-19 in the terminal stages of the incubation period: A familial cluster”, International Journal of Infectious Diseases, 96:452-453, 2020. [9] K. El Hail, M. Khaladi, A. Ouhinou, “Early-confinement strat- egy to tackling COVID-19 in Morocco; a mathematical mod- elling study”, RAIRO-Operations Research, 56(6):4023-4033, 2022. [10] R. Verity, L. C. Okell, I. Dorigatti, P. Winskill, C. Whittaker, N. Imai, et al., “Estimates of the severity of coronavirus disease 2019: a model-based analysis”, The Lancet Infectious Diseases, 20(6):669-677, 2020. [11] S. M. Kassa, A. Ouhinou, “Epidemiological models with preva- lence dependent endogenous self-protection measure”, Mathe- matical Biosciences, 229(1):41-49, 2011. [12] L. W. Green, A. L. McAlister, “Macro-Intervention to Support Health Behavior: Some Theoretical Perspectives and Practical Reflections”, Health Education Quarterly, 11(3):323-339, 1984. [13] L. W. Green, J. M. Ottoson, C. Garcı́a, R. A. Hiatt, “Diffusion Theory and Knowledge Dissemination, Utilization, and Integra- tion in Public Health”, Annual Review of Public Health, 30:151- 174, 2009. [14] S. M. Kassa, A. Ouhinou, “The impact of self-protective mea- sures in the optimal interventions for controlling infectious dis- eases of human population”, Journal of Mathematical Biology, 70:213-236, 2015. [15] P. van den Driessche, J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission”, Mathematical Biosciences, 180(1-2):29- 48, 2002. [16] J. Arino, F. Brauer, P. van den Driessche, J. Watmough, J. Wu, “A final size relation for epidemic models”, Mathematical Biosciences & Engineering, 4(2):159-175, 2007. [17] Worldometer, China Population, https://www.worldometers. info/world-population/china-population/, 2020, [24/01/2023]. [18] National Health Commission of the People’s Republic of China, Interpretation of new diagnostic and treatment program for coronavirus pneumonia, 2020, http://www.nhc.gov.cn/yzygj/ s7652m/202002/54e1ad5c2aac45c19eb541799bf637e9.shtml, [24/01/2023]. [19] National Health Commission of the People’s Republic of China, Update on new coronavirus pneumonia situation as of 24:00 on February 13, 2020, http://www.nhc.gov.cn/yjb/s7860/202002/ 553ff43ca29d4fe88f3837d49d6b6ef1.shtml, [24/01/2023]. [20] World Health Organization, Press conference on February 13, 2020, https://twitter.com/WHO/status/1227980048952463360, [24/01/2023]. [21] World Health Organizatin, Report of the WHO-China Joint Mission on Coronavirus Disease 2019 (COVID-19), 2020, https://www.who.int/docs/default-source/coronaviruse/ who-china-joint-mission-on-covid-19-final-report.pdf, [24/01/2023]. [22] Worldometer, Coronavirus statistics in China, https://www. worldometers.info/coronavirus/country/china, [24/01/2023]. [23] C. GU, W. Jiang, T. Zhao, B. Zheng, “Mathematical Recom- mendations to Fight Against COVID-19”, Available at SSRN 3551006, 2020. Biomath 11 (2022), 2212207, https://doi.org/10.55630/j.biomath.2022.12.207 8/8 https://doi.org/10.1056/NEJMoa2001316 https://doi.org/10.1056/NEJMoa2001316 https://doi.org/10.1056/NEJMoa2001316 https://doi.org/10.1056/NEJMoa2001316 https://doi.org/10.1001/jama.2020.2648 https://doi.org/10.1001/jama.2020.2648 https://doi.org/10.1001/jama.2020.2648 https://doi.org/10.1001/jama.2020.2648 https://doi.org/10.1001/jama.2020.2648 https://doi.org/10.1007/978-3-030-85053-1_2 https://doi.org/10.1007/978-3-030-85053-1_2 https://doi.org/10.1007/978-3-030-85053-1_2 https://doi.org/10.1007/978-3-030-85053-1_2 https://doi.org/10.2807/1560-7917.ES.2020.25.3.2000044 https://doi.org/10.2807/1560-7917.ES.2020.25.3.2000044 https://doi.org/10.2807/1560-7917.ES.2020.25.3.2000044 https://doi.org/10.2807/1560-7917.ES.2020.25.3.2000044 https://doi.org/10.1016/j.ijid.2020.01.050 https://doi.org/10.1016/j.ijid.2020.01.050 https://doi.org/10.1016/j.ijid.2020.01.050 https://doi.org/10.1016/j.ijid.2020.01.050 https://doi.org/10.1016/j.ijid.2020.01.050 https://doi.org/10.1371/journal.pone.0230405 https://doi.org/10.1371/journal.pone.0230405 https://doi.org/10.1371/journal.pone.0230405 https://doi.org/10.1001/jama.2020.2565 https://doi.org/10.1001/jama.2020.2565 https://doi.org/10.1001/jama.2020.2565 https://doi.org/10.1016/j.ijid.2020.03.027 https://doi.org/10.1016/j.ijid.2020.03.027 https://doi.org/10.1016/j.ijid.2020.03.027 https://doi.org/10.1016/j.ijid.2020.03.027 https://doi.org/10.1051/ro/2022188 https://doi.org/10.1051/ro/2022188 https://doi.org/10.1051/ro/2022188 https://doi.org/10.1051/ro/2022188 https://doi.org/10.1016/S1473-3099(20)30243-7 https://doi.org/10.1016/S1473-3099(20)30243-7 https://doi.org/10.1016/S1473-3099(20)30243-7 https://doi.org/10.1016/S1473-3099(20)30243-7 https://doi.org/10.1016/j.mbs.2010.10.007 https://doi.org/10.1016/j.mbs.2010.10.007 https://doi.org/10.1016/j.mbs.2010.10.007 https://doi.org/10.1177/109019818401100308 https://doi.org/10.1177/109019818401100308 https://doi.org/10.1177/109019818401100308 https://doi.org/10.1146/annurev.publhealth.031308.100049 https://doi.org/10.1146/annurev.publhealth.031308.100049 https://doi.org/10.1146/annurev.publhealth.031308.100049 https://doi.org/10.1146/annurev.publhealth.031308.100049 https://doi.org/10.1007/s00285-014-0761-3 https://doi.org/10.1007/s00285-014-0761-3 https://doi.org/10.1007/s00285-014-0761-3 https://doi.org/10.1007/s00285-014-0761-3 https://doi.org/10.1016/S0025-5564(02)00108-6 https://doi.org/10.1016/S0025-5564(02)00108-6 https://doi.org/10.1016/S0025-5564(02)00108-6 https://doi.org/10.1016/S0025-5564(02)00108-6 https://pdfs.semanticscholar.org/5f90/b2811b1f5fbc2881f3be1852c2eebb135cfb.pdf https://pdfs.semanticscholar.org/5f90/b2811b1f5fbc2881f3be1852c2eebb135cfb.pdf https://pdfs.semanticscholar.org/5f90/b2811b1f5fbc2881f3be1852c2eebb135cfb.pdf https://www.worldometers.info/world-population/china-population/ https://www.worldometers.info/world-population/china-population/ https://www.worldometers.info/world-population/china-population/ https://www.worldometers.info/world-population/china-population/ http://www.nhc.gov.cn/yzygj/s7652m/202002/54e1ad5c2aac45c19eb541799bf637e9.shtml http://www.nhc.gov.cn/yzygj/s7652m/202002/54e1ad5c2aac45c19eb541799bf637e9.shtml http://www.nhc.gov.cn/yzygj/s7652m/202002/54e1ad5c2aac45c19eb541799bf637e9.shtml http://www.nhc.gov.cn/yzygj/s7652m/202002/54e1ad5c2aac45c19eb541799bf637e9.shtml http://www.nhc.gov.cn/yzygj/s7652m/202002/54e1ad5c2aac45c19eb541799bf637e9.shtml http://www.nhc.gov.cn/yzygj/s7652m/202002/54e1ad5c2aac45c19eb541799bf637e9.shtml http://www.nhc.gov.cn/yzygj/s7652m/202002/54e1ad5c2aac45c19eb541799bf637e9.shtml http://www.nhc.gov.cn/yjb/s7860/202002/553ff43ca29d4fe88f3837d49d6b6ef1.shtml http://www.nhc.gov.cn/yjb/s7860/202002/553ff43ca29d4fe88f3837d49d6b6ef1.shtml http://www.nhc.gov.cn/yjb/s7860/202002/553ff43ca29d4fe88f3837d49d6b6ef1.shtml http://www.nhc.gov.cn/yjb/s7860/202002/553ff43ca29d4fe88f3837d49d6b6ef1.shtml http://www.nhc.gov.cn/yjb/s7860/202002/553ff43ca29d4fe88f3837d49d6b6ef1.shtml http://www.nhc.gov.cn/yjb/s7860/202002/553ff43ca29d4fe88f3837d49d6b6ef1.shtml https://twitter.com/WHO/status/1227980048952463360 https://twitter.com/WHO/status/1227980048952463360 https://twitter.com/WHO/status/1227980048952463360 https://twitter.com/WHO/status/1227980048952463360 https://www.who.int/docs/default-source/coronaviruse/who-china-joint-mission-on-covid-19-final-report.pdf https://www.who.int/docs/default-source/coronaviruse/who-china-joint-mission-on-covid-19-final-report.pdf https://www.who.int/docs/default-source/coronaviruse/who-china-joint-mission-on-covid-19-final-report.pdf https://www.who.int/docs/default-source/coronaviruse/who-china-joint-mission-on-covid-19-final-report.pdf https://www.who.int/docs/default-source/coronaviruse/who-china-joint-mission-on-covid-19-final-report.pdf https://www.who.int/docs/default-source/coronaviruse/who-china-joint-mission-on-covid-19-final-report.pdf https://www.who.int/docs/default-source/coronaviruse/who-china-joint-mission-on-covid-19-final-report.pdf https://www.worldometers.info/coronavirus/country/china https://www.worldometers.info/coronavirus/country/china https://www.worldometers.info/coronavirus/country/china https://www.worldometers.info/coronavirus/country/china https://doi.org/10.2139/ssrn.3551006 https://doi.org/10.2139/ssrn.3551006 https://doi.org/10.2139/ssrn.3551006 https://doi.org/10.55630/j.biomath.2022.12.207 Introduction Mathematical modelling Parameter estimation and numerical simulations Parameter estimation Numerical simulations Discussion References