A pandemic at the Tunisian scale. Mathematical modelling of reported and unreported COVID-19 infected cases

Starting from the city of Wuhan in China in late December 2019, the pandemic quickly spread to the rest of the world along the main intercontinental air routes. At the time of writing this article, there are officially about five million infections and more than 300 000 deaths. Statistics vary widely from country to country, revealing significant differences in anticipation and management of the crisis. We propose to examine the COVID-19 epidemic in Tunisia through mathematical models, which aim to determine the actual number of infected cases and to predict the course of the epidemic. As of May 11, 2020, there are officially 1032 COVID-19 infected cases in Tunisia. 45 people have died. Using a mathematical model based on the number of reported infected cases, the number of deaths, and the effect of the 18-day delay between infection and death, this study estimates the actual number of COVID-19 cases in Tunisia as 2555 cases. This paper analyses the evolution of the epidemic in Tunisia using population dynamics with an SEIR model combining susceptible cases S(t), asymptomatic infected cases A(t), reported infected cases V (t), and unreported infected cases U(t). This work measures the basic reproduction number R0, which is the average number of people infected by a COVID-19 infected person. The model predicts an R0 = 2.73. Strict containment measures have led to a significant reduction in the reproduction rate. Contact tracing and respect for isolation have an impact: at the current time, we compute that Tunisia has an Rt = 0.42 (95% CI 0.14-0.70). These values depend on physical separation and can vary over time depending on the management of suspicious cases. Their objective estimation and the study of their evolution are however necessary to understand the pandemic and to reduce their unintended damage (due to an absence of symp- toms, or the confusion of certain symptoms with less contagious diseases, or unavailable or unreliable tests).

Epidemic models· Fatality rate· Basic reproduction number· Effective reproduction number· Ordinary differential equations· Bayes Theorem 1 Introduction Given the fragile health systems in most countries, we studied the data available in Tunisia as well as their possible evolution. This work presents a methodology for tracking the epidemic’s numbers. It is based on mathematical models, starting from a simple multiplication via the mortality rate [1], to a study of ordinary differential equations [13] and also to a simulation based on the Bayes theorem for the derivation of the daily reproduction rate [4]. We used a deterministic SEIR model. The main objective of this work is the estimation of the average number of infections one case can generate throughout the infectious period. It is the basic and effective reproduction number of an infectious agent. I. Abdeljaoued-Tej, Ph.D. BIMS Laboratory, LR16IPT09, Institut Pasteur de Tunis, University Tunis-El-Manar, Tunisia; High School of Statistics and Information Analysis, University of Carthage, Tunisia. E-mail: [email protected] . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 23, 2020. ; https://doi.org/10.1101/2020.05.21.20108621doi: medRxiv preprint NOTE: This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice. 2 Ines Abdeljaoued-Tej The COVID-19 morbidity rate in Tunisia, that is, the ratio of the number of people infected to the total Tunisian population, is around 8.73 per 100 000 population. It is the number of infected cases or incidence. The death rate representing the number of people who died from COVID-19 in Tunisia is 0.4 per 100 000 people. The real indicator of the danger of the epidemic is the fatality rate, which represents the proportion of deaths compared to the total number of infected cases. The fatality rate for COVID-19 in Tunisia is 0.044 per 100 000 people. These values are temporary and incomplete. They were based on the tests carried out (the polymerase chain reaction, denoted by PCR, has a sensitivity of 70% [7]). The number of deaths is recorded in hospitals and does not include deaths at home (therefore of cases not reported). But these figures remain an indicator to contain the outbreak and prepare for targeted containment or release of confinement. We use a set of reported data to model the epidemic in Tunisia. It is shared by the Tunisian Ministry of Health and it represents the epidemic transmission in this country. The first case was detected on March 5, 2020. Forty-five dead were reported on May 11, 2020, with 1032 total number of infected cases: the dataset is available in Table 2, and Fig. 8 gives the geographical distribution by Tunisia’s regions. The three phases of COVID-19 epidemics as defined in [11] can be decomposed as a linear phase, exponential growth, and decreasing stage. The linear growth in the number of reported cases (from March 5 to March 24) is where the number of daily reported cases is almost constant day after day. The second phase of the epidemic cor- responds to an exponential increasing phase, it starts on March 25, 2020. The third phase of the epidemic starts on April 17, 2020. It corresponds to a time-dependent exponentially decreasing transmission rate, due to major public interventions and social distancing measures. In practice, epidemiological data typically permit the estimation of theeffective reproduction numberRt. If the basic reproduction numberR0 is greater than 1, countries need to take action because the progression of the virus could lead to an exponential phase. If it is less than 1 then new cases per day go to zero. Changes in transmission speed over time are recommended to bring it down the critical threshold of 1. We address the following fundamental issues concerning this epidemic in Tunisia: How will the epidemic evolve concerning the number of reported cases and unreported cases? How will the number of unreported cases influence the severity of the epidemic? What is the effect of the massive containment policies (decided on March 25, 2020, in Tunisia) on the epidemic? To answer these questions, we developed mathematical models that recover from data of reported cases and the number of unreported cases for the COVID-19 epidemic. 2 Methods The average number of people contaminated by a COVID-19 infected person is central in studying epidemics. This number is denoted byR0: if it is less than 1, a patient infects on average less than one individual, and the disease disappears from the population over time. Conversely, if R0 > 1, the disease can spread in the population and become epidemic. This number, therefore, makes it possible to determine if the epidemic can spread in the population, at what speed (doubling time), and with what magnitude. Its calculation requires the use of more or fewer complex models. Compartmental models were created in the 1930s by Kermack and McKendrick [5]. The principle is to divide the population into epidemiological classes such as those susceptible to infection, those who are infectious, and those who have recovered and acquired immunity or are deceased. The state of the epidemic is determined by knowing the size of these three classes of the population. At each point in time t, we can consider the number S(t) of susceptible people, the number I(t) of infected people, and the number R(t) of recovered (and therefore immune) or deceased people. Each of these quantities is a function of time t, which we cannot measure at all times, but it is measured regularly at certain times (for example daily or hourly), and these measures provide both an abstract model of this function and a daily assessment of the epidemic. The functions S(t), I(t) and R(t) are linked to each other bylaws that describe how they influence each other. For example, their sum S(t) +I(t) +R(t) represents the total population which therefore remains constant over time. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 23, 2020. ; https://doi.org/10.1101/2020.05.21.20108621doi: medRxiv preprint A pandemic at the Tunisian scale 3 The SEIR model is a little more elaborate. It is obtained from the SIR model by adding a new epidemio- logical class to take into account the duration of incubation, namely exposed people (infected non-infectious) who are therefore not contagious, represented by the function E(t). The purpose of the model is to predict forward in time the future number of cases in a time-line of the epidemic from early reported case data. A typical SEIR (susceptible, exposed, infectious, removed) model is given in [10]. Assume that infected individuals were not infectious during the incubation period [25]. Assume population growth rate and death rate are zero. Assume people exhibit consistent behaviours before and during epidemic phase [26]. Assume no quarantine or other mitigation intervention is implemented. All these assumptions directly influence the model, which is dependent on several parameters. See [20] for a more precise idea of the evolution of these assumptions. Sophisticated SEIR/SIRU models are developed in [21], but we choose to follow the approach of [11] to have a starting point of view of the epidemic. It produces a result closer to the realistic basic reproductive numberR0. The study we are presenting is based on a refined SEIR model, tested at the beginning of the COVID-19 pandemic in China or France [13], and in Palestine [2], which takes into account three classes of infected populations represented by functions A(t), U(t) and V (t). Any of these infected populations can transmit the virus to susceptible people S(t). The first class, represented by the function A(t), is made up of people infected but who do not know it, called asymptomatic. The other infected people, called symptomatic, are divided into two classes represented by the functions U(t) and V (t). The function U(t) represents symptomatic people not listed by the public authorities. They are infected people with symptoms but who do not know they are carrying the virus. They are therefore not officially recognised as infected (either because the COVID-19 test turned out to be negative, or because they simply escaped the various checks put in place). The population of those who are symptomatic, infected, and who tested positive for the virus are represented by the function V (t). The other laws which describe the evolution of the epidemic can be represented by the following diagram in Fig. 1: Fig. 1 Diagram flux: refined SEIR model . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 23, 2020. ; https://doi.org/10.1101/2020.05.21.20108621doi: medRxiv preprint 4 Ines Abdeljaoued-Tej The infected people are thus divided into asymptomatic A(t) and symptomatic U(t) and V (t). Only those infected who have tested positive for the virus, represented by the function V (t), are known to the public authorities. These are the ones that appear in the daily reports of the epidemic. However, the three population classes A(t), U(t) and V (t) are carriers of a highly contagious infection. Draconian control is applied to classV (t) to avoid contact with classS(t) of healthy people who are likely to be infected. However, the latter can be infected by both classes A(t) andU(t), with a transmission rate β: it represents the rate of susceptible people who become infected when in contact with contaminated persons. In this refined model, there are thereforeβS(t)(A(t) +U(t)) newly infected people. The model also takes into account the number of days n from which symptoms appear on carriers of the virus (estimated at 7 days for COVID-19). After this period of n days, only the proportionf of asymptomatic people become symptomatic and test positive. They are added to the class V (t). The rest of the asymptomatic population, namely (1 −f)A(t), escapes the mesh of control and is added to the class U(t). Every day, classA(t) therefore contributes to class V (t) by fA(t)/n people and to class U(t) by (1−f)A(t)/n people. Finally, part of the symptomatic U(t) and V (t) is removed after a certain number of days with a rate η. We obtain, by noting the inverse of n byν: dS(t) dt =−βS (t) [A(t) +U(t)], dA(t) dt =βS (t) [A(t) +U(t)]−νA (t), dU(t) dt = (1−f)νA (t)−ηU (t), dV (t) dt =fνA (t)−ηV (t). This system is supplemented by initial data S(t0) = S0> 0, A(t0) = A0 > 0, V(t0) = 0 and U(t0) = U0≥ 0, where,t0 is the beginning date of the epidemic, t≥t0 is time in days. The parameters are listed in Table 1. Symbol Interpretation Method t0 Time at which the epidemic started fitted S0 Number of susceptible at time t0 (Tunisian population) fixed A0 Number of asymptomatic infectious at time t0 fitted U0 Number of unreported symptomatic infectious at time t0 fitted β Transmission rate fitted 1/ν Average time during which asymptomatic infectious are asymptomatic fixed f Fraction of asymptomatic infectious that become reported symptomatic infectious fixed fν Rate at which asymptomatic infectious become reported symptomatic fitted (1−f)ν Rate at which asymptomatic infectious become unreported symptomatic fitted 1/η Average time symptomatic infectious have symptoms fixed Table 1 Parameters and initial conditions of the model We assume that the removal rate ν is the sum of the removal rate of reported symptomatic infectious individuals, and of the removal rate of unreported symptomatic infectious individuals due to all other causes, such as mild symptoms, or other reasons. The cumulative number of reported symptomatic infectious cases at time t is denoted by CR(t). We assume that CR(t) has the following special form: CR(t) =χ1 exp(χ2t)−χ3. We obtain the model starting time of the epidemic t0: CR(t0) = 0 ⇔ χ1 exp(χ2t0)−χ3 = 0 ⇒ t0 = ln(χ3)− ln(χ1) χ2 . We fixS0 to 11 818 619, which corresponds to the total population of Tunisia. We assume that the variation in S(t) is small during the period considered, and we fix ν,η,f . We can estimate the parameter β and . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 23, 2020. ; https://doi.org/10.1101/2020.05.21.20108621doi: medRxiv preprint A pandemic at the Tunisian scale 5 the initial conditions U0 and A0 from the cumulative reported cases CR(t). We then construct numerical simulations and compare them with data. We obtain A(t) = A0 exp(χ2(t−t0)) and A0 = χ2/(fν ). We must have U(t) = U0 exp(χ2(t−t0)). So, by substituting these expressions into previous identities, we obtain: χ2 A0 =βS0 (A0 +U0)−νA0, χ2 U0 = (1−f)ν I0−ηU , β = χ2 +ν S0 ∗ η +χ2 (1−f)∗ν +η +χ2 , and U0 =A0((1−f)ν)/(η +χ2). We fix β such that the value χ2 becomes the dominant eigenvalue of (βS0−ν βS 0 ν(1−f) −η ) , and (A 0,U 0) is the positive eigenvector associated with this dominant eigenvalue χ2. Thus, we apply im- plicitly the Perron–Frobenius theorem. Moreover, the exponentially growing solution ( A(t),U (t)) that we consider (which is starting very close to (0 , 0)) follows the direction of the positive eigenvector associated with the dominant eigenvalue χ2. The need to impose lock-down comes from the classes of infected but unreported, that is, A(t) andU(t): A(t) =A0 exp(χ2(t−t0)) and U(t) =U0 exp(χ2(t−t0)) These two functions are not known at any given time t. Their objective estimation and the study of their evolution are however necessary to understand the pandemic and to reduce their unintended damage (due to an absence of symptoms, or the confusion of certain symptoms with less contagious diseases, or unavailable or unreliable tests). The basic reproductive number becomes: R0 = βS0 ν ∗ (1 + (1−f)ν η ). It is a single measure that does not adapt to changes in behaviour and restrictions. By definition, R0 is an intrinsic property of an outbreak and does not change once calculated, as it is calculated assuming a fully susceptible population. On the other hand, the effective reproduction rateRt is a relevant indicator to qualify contagiousness. Every day t, a new case count of infected people gives us a clue about the current value ofRt. It figures that the value of Rt a day t is related to the value of Rt−1 and every previous value ofRt−j (from day t−j to t− 1). As a pandemic evolves, increasing restrictions change Rt. If the effective reproduction numberRt is smaller thanR0, then it reduces the number of infected people, and it reduces the number of future infected people. Stopping containment, resuming international flights without any physical distancing measure, will affect Rt (which is likely to rise above 1). In order to compute Rt we apply a modified version of a solution used in [4] and trained by Kevin Systrom, Adam Lerer and Frank Dellaert 1. It introduces a process model with Gaussian noise to estimate a time-varyingRt. It uses Bayes’ rule to updateRt based on how many new cases have been reported each day: P (Rt|k) = P (k|Rt)∗P (Rt) P (k) , whereP (k|Rt) is the likelihood of seeing k new cases givenRt times,P (Rt) is the prior beliefs of the value ofRt without the data, and P (k) the probability of seeing this many cases in general. To make it iterative: every day that passes, we use yesterday’s prior P (Rt−1) to estimate today’s prior P (Rt). We assume the distribution of Rt to be a Gaussian centred around Rt−1, so P (Rt|Rt−1) = N (Rt−1,σ ), where σ is a hyper-parameter. So on day one: P (R1|k1)∝P (R1)∗L(R 1|k1). 1 https://github.com/k-sys/covid-19/blob/master/Realtime%20R0.ipynb . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 23, 2020. ; https://doi.org/10.1101/2020.05.21.20108621doi: medRxiv preprint 6 Ines Abdeljaoued-Tej By definition, the likelihood functionL(R1|k1) is conditioned on the observed k1 and it is a function of the unknown parameterR1. On day two: P (R2|k1,k 2)∝P (R2)∗L(R 2|k2) = ∑ R1 P (R1|k1)∗P (R2|R1)∗L(R 2|k2). On day t: P (Rt|k1,k 2,...,k t−1,k t)∝P (Rt)∗L(R t|kt) = ∑ Rt−1 P (R1|k1)∗P (R2|k2)∗···∗ P (Rt−1|kt−1)∗P (Rt|Rt−1)∗L(R t|kt) A likelihood function says how likely we are to see k new cases, given a value of Rt: Given an average arrival rate of λ new cases per day, the probability of seeing k new cases is distributed according to the Poisson distribution: P (k|λ) =λk exp (−λ)/k!. There’s a connection between Rt and λ: λ =kt−1∗ exp (η(Rt− 1)) wherekt is the number of infected cases at dayt, and 1/η is the serial interval defined as an average residence time when infectious individuals recover or die [22]. 3 Data description The data available in [6] gives the number of COVID-19 positive cases per day, the number of deaths due to COVID-19, and the number of reported recoveries (see Table 2). The daily growth rate of the infected cases in Fig. 2 is given by: ( Cj−Cj−1)/Cj−1 in day j, where Cj is the number of reported COVID-19 cases at timej. The median of the daily growth rate of reported cases in Tunisia from March 5 to 11 th May 2020 is equal to 0.24%. The total number of deaths is 45. It represents a fatality rate of 4.36% of total reported cases. Fig. 2 Data on reported infected cases and death cases in Tunisia were collected from January 23, 2020. The first infected case was reported on March 5. On May 11 the daily number of reported cases was zero. Based on the Ministry of Health data, we can say that the first wave of infection has passed. Patients who die on a given day j were infected much earlier, so the mortality rate denominator should be the total number of patients infected at the same time as those who died [24]. To measure a death ratio, a delay of 18 days is used between the mortality number and the infected number. This delay is justified by the days between infection and death, which is about 17 to 19 days [8]. The relationships be- tween deaths a given day j and the number of cases 18 days before (j− 18) are linear for Tunisia, as shown in Fig. 3. The mortality rate is defined byT18 =Mj/Cj−18 whereMj is the total number of deaths at timej. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 23, 2020. ; https://doi.org/10.1101/2020.05.21.20108621doi: medRxiv preprint A pandemic at the Tunisian scale 7 The median mortality rate during this period is equal to 0.12. The progression of the number of cases reported over the last eighteen days P18 is known in each country. It depends on the rate of containment and its effectiveness: P18 = Cj/Cj−18. Fig. 3 shows the progression rate of COVID-19 in Tunisia. The median of P18 is equal to 1.86 and it is used to compute the estimated number of reported and unreported cases [1]. We estimate the number of cases in Tunisia as of May 11 th, 2020 by modulating the estimated mortality rate: For example, if the mortality rate is 2%, then the actual cases on May 11 is estimated at 2 555. Fig. 3 Relationships between deaths a given day j and the number of cases 18 days before j− 18 is linear for Tunisia. The progression rate P18 is equal to 25.28 on April 2 and decreases to 1.14 on May 11, 2020. On the other hand, we scale the number of deaths per 100 000 population due to COVID-19 related to the number of registered infected people per 100 000 population. Tunisia appears to be doing rather well, compared to other countries [2], with a ratio of deaths per 100 000 population to infected cases per 100 000 population equal to 0.05. The incubation period for COVID-19 is very important: it is a time during which individuals have been infected but are not yet infectious themselves. 4 Results The first infected case in Tunisia was documented on March 5, 2020. According to data, the time t = 0 will correspond to 5 March. It is used as the first day of the forecast. Fitting Tunisian data from March 25 to April 1st enables to compute the cumulative number CR(t) of reported symptomatic infectious cases at timet: CR(t) =χ1 exp(χ2 t)−χ3. We findχ1 = 1.885,χ2 = 0.209, and χ3 = 2.054. The straight line in the right side of Fig. 4 corresponds to t→ ln(χ1) +χ2t. We first estimate the value of χ3 and then use a least square method (MSE) to evaluate χ1 and χ2. We observe that the data for Tunisia provides a good fit for CR(t) (explained variance score equal to 0.99). From now on, we fix the fractionf = 0.6 of asymptomatic that become reported symptomatic infectious. The average time during which asymptomatic infectious cases are symptomatic is equal to 1 /ν = 1/7. The average time symptomatic infectious have symptoms is equal to η = 1/7. The values 1/η = 7 days and 1/ν = 7 days are taken from information concerning earlier coronaviruses, and are used now by medical authorities [18]. Thus, the value of t0 = 0.409 means that the starting time of the epidemic is 5 March. We obtainR0 = 2.73, β = 2.76 10−8,I0 = 3.8, and U0 = 0.3. We ran numerical simulations and compare them with data. In the following, let take into account the fact that very strong isolation measures have been imposed for all Tunisia since 25 March. Specifically, since March 12, families in Tunisia were required to stay at home. To take into account such a public intervention, we assume that the transmission of COVID- 19 from infectious to susceptible individuals stopped after 25 March. Therefore, we consider the following model: for t≥t0, β(t) =β if t∈ [t0, 25] and β(t) = 0 for t> 25. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 23, 2020. ; https://doi.org/10.1101/2020.05.21.20108621doi: medRxiv preprint 8 Ines Abdeljaoued-Tej Fig. 4 In the left side figures, the dots correspond to t→ CR(t), and in the right side figures, the dots correspond to t→ ln(CR(t))−χ3, where CR(t) is taken from the cumulated confirmed cases. For a total population N equal to 11 818 619, we suppose that S0 = 11.81 106 is the susceptible population to be infected by COVID-19. Without these strict control policies, the number of infected cases certainly would have been different. Fig. 5 shows the number of t→ CR(t), the unreported cases t→U(t), and the data corresponding to the confirmed cumulated cases for Tunisia, without containment measures. Fig. 5 We useχ1 = 1.885,χ2 = 0.209,χ3 = 2.054,t0 = 0.409 and S0 = 11.81 106. Which givesR0 = 2.73,β = 2.76 10−8, I0 = 3.8, and U0 = 0.3. In the right figure, the number of symptomatic reported cases if containment has not occurred can be estimated to 472 457 i.e. 4% of the total population of Tunisisa. They would have been infected with more or less severe symptoms. The reproduction numberR0 = 2.73 is quite high and the damage of COVID-19 came from the virulence of the symptoms. In the literature, the value ofR0 varies from 1.4 to 3.9 [17]. A more sophisticated model is developed in [3] to examine the effects of an outbreak with control policies, it gives anR0 in the same range. The measure of the effective reproduction number Rt under the current severe containment conditions in Tunisia, using the Bayes’ theorem, gives Rt = 0.42 with CI 0.14 - 0.70 at May 11, 2020 (for an optimal σ = 0.2 and η = 1/7). Fig. 6 shows that it leads to small outbreaks that eventually become extinguished. To avoid overfitting on any one state, we choose the σ that maximizes P (k) over every state. To do this, we add up all the log-likelihoods per state for each value of σ then choose the maximum: Fig. 7 indicates the optimal result. 5 Discussion The current emerging trend in mathematical epidemiology towards real-time predictive methods will allow for a shift to more quantitative surveillance and prevention policy at the earliest stages of the epidemic. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 23, 2020. ; https://doi.org/10.1101/2020.05.21.20108621doi: medRxiv preprint A pandemic at the Tunisian scale 9 Fig. 6 The variance σ is optimised on smoothed daily data. When σ = 0.20 and η = 1/7,Rt = 0.42 (mean CI 0.14-0.70). These results are quite unstable and are highly dependent on the number of cases detected. The downward trend could change at any time, especially as populations change their behaviour. This will result in more consistent and expanded surveillance of emerging infectious diseases and improved design of health interventions and logistical allocations as epidemics develop. An epidemic outbreak of a new human coronavirus COVID-19 occurred in Tunisia. The unreported cases and the disease transmis- sion rate are useful information. We estimate an actual number of infected cases in Tunisia based on the 18-day effect from infection to death and a mortality rate equal to 2%. We find that the number of cases on May 11 is at least equal to 2 555 infected cases, the worst case is about 4% of the population. In fact, according to the SEIR model with asymptomatic and symptomatic compartments, where the transmission rate β = 2.76 10−8, the total number of reported cases is 472 457 population near the turning point May 15. On May 11, the number of cases was 1032 infected, 45 dead, and a total population of 11 818 619. The effective production rate puts Tunisia in the leading of its neighbours. It shows very good management of the epidemic by Tunisia, especially on the number of deaths, Recovery, and Infected cases. The estimation ofR0 is achieved by fitting to the data (during the exponential period). This value may differ due to acquired immunity and other factors. The reproductive number R0 is equal to 2.73 for a fixed fractionf = 0.8 of symptomatic cases that are reported and for a removal rateν = 7. Effective reproduction numberRt is given by matching the data, day by day, using Bayes’ theorem. On May 17, Rt is equal to 0.42 (95% CI 0.14-0.70) where η = 1/7 and σ = 0.2. The effective reproduction number Rt has reduced to less than 1: public measures were taken, such as isolation, quarantine, and public closings, strict travel restrictions, face mask for everyone, and reliable targeted testing. These measures exacerbate the spread of this disease and mitigate the final size of the epidemic [21]. It is contained in Tunisia, provided that quality of care and social distancing measures are in place. This protection may involve several factors, such as genetic predispositions, the immune status of populations, the history of vaccinations, and in particular what is known as training immunity, which is currently being discussed when BCG vaccination is involved [15]. Compared to other countries, there is a certain level of protection that must be understood (in the biological and immunological sense) to take advantage of it for better control of this disease or others that will arise. Finding a vaccine, testing the miracle cure would reduce evolution. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 23, 2020. ; https://doi.org/10.1101/2020.05.21.20108621doi: medRxiv preprint 10 Ines Abdeljaoued-Tej In Tunisia, there is strict containment, but also contact tracing and respect for isolation: 24% of the cases are foreign and the rest are domestic 2. The reopening of the borders without severe distancing measures could have an impact and consequences for the epidemic outcome: Tunisia is heavily exposed from different Chinese airports (Guangdong province and weakly to Zhejiang province) as the main source of entry risk [9]. Several infection waves are expected, and they could be spread over several years [16]. The mathematical models outlined in this research could help anticipate the damage to the health care system that they could cause promptly. Using models applied to hospital and death data, further work will be carried out to estimate the impact of containment, as well as the current immunity of the population [19]. In terms of public health implications, to stop the outbreak, at least 58-76% of transmissions should be blocked. And we will no longer talk about this pandemic when 2/3 of the population will have been in contact with the virus. In the meantime, confinement could become a toxic situation for our democracies: For societies like Tunisia the ease with which entire sections of the population living in precarious conditions are endangered [23]. The extraordinary real-time media coverage of the progression of this pandemic has weighed heavily on the behaviour of all concerned, not only ordinary people but also politicians and even scientists [14]. The mathematics of emergency medicine can yield very relevant results.

1. Abdeljaoued-Tej, Ines, and Marc Dhenain. ”Estimation of Tunisia COVID-19 infected cases based on mortality rate.” medRxiv (2020). 2. Abdeljaoued-Tej, Ines. ”COVID-19 data analysis and modeling in Palestine.” medRxiv (2020). 3. Benmiled, Slimane, and Amira Kebir. ”Simulations of the spread of COVID-19 and control policies in Tunisia.” arXiv preprint arXiv:2005.00750 (2020). 4. Bettencourt, Luis MA, and Ruy M. Ribeiro. ”Real time bayesian estimation of the epidemic potential of emerging infectious diseases.” PLoS One 3.5 (2008). 5. Diekmann, Odo, Johan Andre Peter Heesterbeek, and Johan AJ Metz. ”The legacy of Kermack and McKendrick.” Publications of the Newton Institute 5 (1995): 95-115. 6. Dong, Ensheng, Hongru Du, and Lauren Gardner. ”An interactive web-based dashboard to track COVID-19 in real time.” The Lancet infectious diseases (2020). 7. Fang, Yicheng, et al. ”Sensitivity of chest CT for COVID-19: comparison to RT-PCR.” Radiology (2020): 200432. 8. Flaxman, Seth, et al. ”Report 13: Estimating the number of infections and the impact of non-pharmaceutical interventions on COVID-19 in 11 European countries.” (2020). 9. Gilbert, Marius, et al. ”Preparedness and vulnerability of African countries against importations of COVID-19: a mod- elling study.” The Lancet 395.10227 (2020): 871-877. 10. Keeling, Matt J., and Pejman Rohani. Modeling infectious diseases in humans and animals. Princeton University Press, 2011. 11. Liu, Zhihua, et al. ”Understanding unreported cases in the COVID-19 epidemic outbreak in Wuhan, China, and the importance of major public health interventions.” Biology 9.3 (2020): 50. 12. Magal, Pierre, and Glenn Webb. ”The parameter identification problem for SIR epidemic models: identifying unreported cases.” Journal of mathematical biology 77.6-7 (2018): 1629-1648. 13. Magal, Pierre, and Glenn Webb. ”Predicting the number of reported and unreported cases for the COVID-19 epidemic in South Korea, Italy, France and Germany.” Italy, France and Germany (March 19, 2020) (2020). 14. McKibbin, Warwick J., and Roshen Fernando. ”The global macroeconomic impacts of COVID-19: Seven scenarios.” (2020). 15. Miller, Aaron, et al. ”Correlation between universal BCG vaccination policy and reduced morbidity and mortality for COVID-19: an epidemiological study.” MedRxiv (2020). 16. Neher, Richard A., et al. ”Potential impact of seasonal forcing on a SARS-CoV-2 pandemic.” Swiss medical weekly 150.1112 (2020). 17. Read, Jonathan M., et al. ”Novel coronavirus 2019-nCoV: early estimation of epidemiological parameters and epidemic predictions.” MedRxiv (2020). 18. Rothe, Camilla, et al. ”Transmission of 2019-nCoV infection from an asymptomatic contact in Germany.” New England Journal of Medicine 382.10 (2020): 970-971. 19. Salje, Henrik, et al. ”Estimating the burden of SARS-CoV-2 in France.” Science (2020). 20. Tang, Biao, et al. ”An updated estimation of the risk of transmission of the novel coronavirus (2019-nCov).” Infectious disease modelling 5 (2020): 248-255. 21. Tang, Biao, et al. ”Estimation of the transmission risk of the 2019-nCoV and its implication for public health interven- tions.” Journal of Clinical Medicine 9.2 (2020): 462. 22. Thompson, R. N., et al. ”Improved inference of time-varying reproduction numbers during infectious disease outbreaks.” Epidemics 29 (2019): 100356. 23. Alon, Titan M., et al. The impact of COVID-19 on gender equality. No. w26947. National Bureau of Economic Research, 2020. 2 https://covid-19.tn/ . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 23, 2020. ; https://doi.org/10.1101/2020.05.21.20108621doi: medRxiv preprint A pandemic at the Tunisian scale 11 24. Verity, Robert, et al. ”Estimates of the severity of coronavirus disease 2019: a model-based analysis.” The Lancet Infectious Diseases (2020). 25. Wu, Joseph T., Kathy Leung, and Gabriel M. Leung. ”Nowcasting and forecasting the potential domestic and interna- tional spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study.” The Lancet 395.10225 (2020): 689-697. 26. Zhao, Dahai, et al. ”A comparative study on the clinical features of COVID-19 pneumonia to other pneumonias.” Clinical Infectious Diseases (2020). Ines Abdeljaoued-Tej, mathematician, Higher School of Statistics and Information Analysis, University of Carthage, and BIMS Laboratory, LR16IPT09, Institut Pasteur de Tunis, University of Tunis El Manar, Tunisia. Appendix Fig. 7 The varianceσ is optimised on smoothed daily data from March 18 to May 17, 2020. When η = 1/7, we obtain the optimal value σ = 0.2. This result is highly dependent on the input smoothed data. Table 2: The total number of infected, deaths, and recovery cases from March 3 to May 11, 2020, in Tunisia. Begin of Table 2 Date Infected Deaths Recovered 3/4/20 1 0 0 3/5/20 1 0 0 3/6/20 1 0 0 3/7/20 1 0 0 3/8/20 2 0 0 3/9/20 2 0 0 3/10/20 5 0 0 3/11/20 7 0 0 3/12/20 7 0 0 3/13/20 16 0 0 3/14/20 18 0 0 3/15/20 18 0 0 3/16/20 20 0 0 3/17/20 24 0 0 . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 23, 2020. ; https://doi.org/10.1101/2020.05.21.20108621doi: medRxiv preprint 12 Ines Abdeljaoued-Tej Continuation of Table 2 Date Infected Deaths Recovered 3/18/20 29 0 0 3/19/20 39 1 0 3/20/20 54 1 0 3/21/20 60 1 0 3/22/20 75 3 1 3/23/20 89 3 1 3/24/20 114 4 1 3/25/20 173 5 2 3/26/20 197 6 2 3/27/20 227 6 2 3/28/20 278 8 2 3/29/20 312 8 2 3/30/20 312 8 3 3/31/20 394 10 3 4/1/20 423 12 5 4/2/20 455 14 5 4/3/20 495 18 5 4/4/20 553 18 5 4/5/20 574 22 5 4/6/20 596 22 5 4/7/20 623 23 25 4/8/20 628 24 25 4/9/20 643 25 25 4/10/20 671 25 25 4/11/20 685 28 43 4/12/20 707 31 43 4/13/20 726 34 43 4/14/20 747 34 43 4/15/20 780 35 43 4/16/20 822 37 43 4/17/20 864 37 43 4/18/20 864 37 43 4/19/20 879 38 43 4/20/20 884 38 148 4/21/20 884 38 148 4/22/20 909 38 190 4/23/20 918 38 190 4/24/20 922 38 194 4/25/20 939 38 207 4/26/20 949 38 216 4/27/20 967 39 279 4/28/20 975 40 279 4/29/20 980 40 294 4/30/20 994 41 305 5/1/20 998 41 316 5/2/20 1009 42 323 5/3/20 1013 42 328 5/4/20 1018 43 406 5/5/20 1022 43 482 5/6/20 1025 43 591 . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 23, 2020. ; https://doi.org/10.1101/2020.05.21.20108621doi: medRxiv preprint A pandemic at the Tunisian scale 13 Continuation of Table 2 Date Infected Deaths Recovered 5/7/20 1026 44 600 5/8/20 1030 45 638 5/9/20 1032 45 660 5/10/20 1032 45 700 5/11/20 1032 45 727 End of Table 2 Fig. 8 Distribution of infected cases in the 24 Tunisian regions, on May 11, 2020. The most affected regions are Kebili (south-west: 68.17 per 100 000 population), Tataouine in the south (24.09 per 100 000 population), and Tunis (22.06 per 100 000 population). Jendouba (north-west) is the least affected with 0.25 per 100 000 population. . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted May 23, 2020. ; https://doi.org/10.1101/2020.05.21.20108621doi: medRxiv preprint