Simulations of the spread of COVID-19 and control policies in Tunisia
SSimulations of the spread of COVID-19 andcontrol policies in Tunisia
Slimane BenMiled and Amira Kebir , BIMS Lab, Institut Pasteur de Tunis, Universit´e de Tunis elManar; [email protected] IPEIT, Universit´e de Tunis; [email protected]
Abstract
We develop and analyze in this work an epidemiological model forCOVID-19 using Tunisian data. Our aims are first to evaluate Tunisiancontrol policies for COVID-19 and secondly to understand the effect ofdifferent screening, quarantine and containment strategies and the ruleof the asymptomatic patients on the spread of the virus in the Tunisianpopulation. With this work, we show that Tunisian control policies areefficient in screening infected and asymptomatic individuals and that ifcontainment and curfew are maintained the epidemic will be quickly con-tained.
On March 11, 2020, WHO announced that the COVID-19 epidemic hadpassed the pandemic stage, indicating its autonomous spread over severalcontinents. Since March 22, Tunisia has experienced a turning point andgeneral health containment has begun. Tunisia’s strategy of containmentand targeted screening corresponds to the first WHO guidelines, the aimbeing to detect clusters by diagnosing only suspicious persons and thento trace the people who came into contact with the positive cases. Thismethod is now showing its limitations. The mass screening carried out insome countries shows that asymptomatic patients or those who developonly a mild form of the disease may exist in significant numbers. So whatis the rule of the asymptomatic patients on the spread of the virus in theTunisian population and does containment and mass screening strategiesare sufficient to control the spread of the virus in the Tunisian population?In this work, a mathematical epidemiological model for COVID-19 isdeveloped to study and predict the effect of different screening, quarantine,and containment strategies on the spread of the virus in the Tunisianpopulation. This model is more detailed than the classical model (SIR)but it remains very simple in its structure. Indeed, all individuals areassumed to react on average in the same way to the infection, there are a r X i v : . [ q - b i o . P E ] M a y QAs I DR α As + α I τ β γ µτ γµ Figure 1: The model flow chart no differences in age, sex, contacts. The model is calibrated and fitted toTunisian data.In what follows, we present the model and its assumptions. Then wecalibrate different parameters of the model based on the Tunisian data andcalculate the expression of the basic reproduction number R as a functionof the model parameters. Finally, we carry out simulations of interven-tions and compare different strategies for suppressing and controlling theepidemic. COVID-19 is a respiratory disease that spreads mainly through the res-piratory droplets expelled by people who cough. So the transmission isusually direct from person to person. Infection is considered possible evenwhen in contact with a person with mild symptoms. In fact, in the earlystages of the disease, many people with the disease have only mild symp-toms. It is, therefore, possible to contract COVID-19 through contactwith a person who does not feel sick. Subsequently, in this work, we con-sider susceptible individuals, noted S , who are infected first go through astage where they are infected but asymptomatic, noted As for unreportedasymptomatic infectious. This stage appears to be particularly importantin the spread of COVID-19. The individuals then develop symptoms andbecome symptomatic infectious, so either enter directly into a quarantinestage, noted Q , corresponds to reported symptomatic infectious individ-uals, or go through a moderate or severe infectious stage, noted I forunreported symptomatic infectious and then can go through the quaran-tine stage or not. Finally, the infection ends and the individuals are thenimmunized, denoted R or dead, denoted D . This life cycle can be repre-sented using the following flow chart (1) followed by table 1 that lists themodel parameters.The quarantine is assumed to act on the As and I stages. Indeed, we Notation Description Value Reference t Initial epidemic time − .
172 fitted S Number of susceptible at time t As Number of asymptomatic at time t .
317 fitted I Number of infected at time t .
050 fitted α Transmission rate by asymptomatic 4 . e −
08 fitted α = fα Transmission rate by infected f β Rate at which asymptomatic infectious 1 / τ Rate at which asymptomatic enter in quarantine 0 .
400 fitted τ Rate at which infected enter in quarantine 0 .
778 fitted γ Rate of recovery 0 .
045 estimated from data µ Rate of mortality 0 .
003 estimated from data assumed that the state can detect asymptomatic individuals by for exam-ple random screening, and then positive testing ones go into quarantine.Let τ , respectively τ , be the quarantine rate for As class, respectively I .We assumed that the asymptomatic individual, As , turn out to be I at a rate β . We further assumed that quarantined individuals, Q andinfected individuals, I either die at a rate of µ per unit of time or becomerecovered/immune, R , at a rate of γ per unit of time.Finally, we assume that each healthy individual is infected proportion-ally by As asymptomatic individuals, with a rate of α and by I infectedindividuals, with a rate of α . As I state is constituted with the moder-ate or severe state, they are more contagious than the As state, therefore, α < α .Therefore, our model consists of the following system of 6 ordinarydifferential equations: dSdt = − α SAs − α SIdAsdt = α SI + ( α S − β − τ ) AsdIdt = βAs − ( τ + γ + µ ) IdQdt = τ As + τ I − ( µ + γ ) QdRdt = γ ( I + Q ) dDdt = µ ( I + Q ) (1)With an initial condition at time t = t defined as following: S ( t ) = S > , As ( t ) = As > , I ( t ) = I > , Q ( t ) = 0 , R ( t ) = 0 , D ( t ) = 0One of the advantages of the basic reproduction number R conceptis that it can be calculated from the moment the life cycle of the infec-tious agent is known. We calculate the R for our model using the NextGeneration Theorem [3] (see section A.1), R = α S β + τ + α βS ( β + τ )( τ + γ + µ ) (2) he first term of the expression of (2) corresponds to infections gen-erated by asymptomatic types (healthy carrier to mild symptoms). Thesecond term corresponds to secondary infections caused by moderate orsevere symptomatic infection. With this expression, we can see that thereare several ways to lower the R and thus control the epidemic. For exam-ple, we can reduce the number of susceptible people (decreasing α and α )by confining the population, reducing contacts, and wearing masks. Wecan also reduce the rate of contact with an infected person by increasingquarantine rates ( τ and τ ) by isolating asymptomatic or symptomaticinfected persons through mass screening. The estimation of the different parameters of the model is done in threesteps (see section A.2). In the first step, we will estimate the start date ofthe epidemic, t , the initial states As ( t ) and I ( t ) as well as the infectionrates α and α . In the second step, we estimate the mortality rate, µ , andthe recovery rate, γ . In the third step, we evaluate the parameters τ and τ by an optimization method. The program is available for download .We used the Tunisian Health Commission data-set of reported datato model the epidemic in Tunisia. It represents the daily new-cases, death,and recoveries in Tunisia. The first case was detected on March 2, 2020.To estimate the initial conditions As ( t ) and I ( t ) and parameters α and α , we fix S = 11694720, which corresponds to the total popula-tion of Tunisia and assume that the variation in S ( t ) is small during theperiod consider. We also fix the parameters β, γ, τ and τ . For this esti-mation, we adapt the method developed by [4] in our case. Let CRt ) thecumulative number of reported infectious cases at time t , defined by, CR ( t ) = (cid:90) tt τ As ( t ) + τ I ( t ) dt (3)Let’s assume that CR ( t ) = χ exp( χ t ) − χ with χ, χ and χ threepositive parameters that we estimate using log-linear regression on casesdata (see figure 2 and table 2). Table 2: Parameters of the estimated cumulative number of reported infectiouscases
CR χ χ χ t R .
208 0 .
210 0 . − .
401 0 . We obtain the model starting time of the epidemic t by assuming that CR ( t ) = 0 and therefore equation (3) implies that: t = 1 χ (ln( χ ) − ln( χ )) . (4) https://github.com/MayaraLatrech/covid19_sasymodel.git https://covid-19.tn/ a) Fitted CR . (b) Fitted log ( CR ). Figure 2: The fitted cumulative number of reported infectious cases CR = χ exp( χ t ) − χ to Tunisian data using Linear regression.Figure 3: In this figure, we plotted the number of individuals in quarantinewithout curfew (in blue), Q ( t ) ( i.e. the number of diagnosed cases per day),and the Tunisians data (in orange). For now, we assume that α = fα with f a fixed parameter big-ger than 1 and let’s τ = τ /τ . Then, by following using the approachdescribed in the step 1 of section A.2, we have: I ( t ) = βχ + τ + βτ + γ + µ χ χ τ (5) As ( t ) = (1 − βτχ + τ + βτ + γ + µ ) χ χ τ (6) α = ( χ + β + τ )( fβχ + τ + γ + µ + 1) S (7) R = ( χ + β + τ )( β + τ ) (1 + fβτ + γ + µ )(1 + fβχ + τ + γ + µ ) (8)In figure 3, we plotted the number of individuals in quarantine withoutcurfew predicted by the ODE model, Q ( t ) i.e. the number of diagnosedcases per day and compared to the Tunisians data. We observe that fromthe 20th day onwards, the simulated curve deviates from the observeddata. This deviation is due to the epidemic control policies put in placebetween 20 and 25 March (closure of cafes shops and the introduction ofa curfew). t can be seen that if the curfew had not been installed, the countrycould have had thousands of additional cases during the month of April. In figure 4 we study the effect of the curfew installed by the Tunisian statesince March 20, 2020, on the number of infected people reported by thestate.Figure 4 shows the effect of two curfew strategies on the dynamicsof the epidemic: a 12-hour curfew (the chosen Tunisian policy) and an18-hour curfew (a more restrictive policy). During the period of curfew,the rate of infection α is divided by 100. In Figure 4(a), it can be seenthat, for the chosen policy to maintains a 12-hour curfew for 100 days,the epidemiological peak in terms of the number of reported infected isreached after about 50 days with a value equal to 953. After the peak, weobserve a slow decrease in the number of reported infected persons. Onthe other hand, in the more restrictive case of18 hours curfew, the peakwould be reached more quickly after 27 days with a more rapid decrease.These values should be compared with the 774 cases given by [1] and thefact that the epidemiological peak was reached around April 29, 2020,after about 58 days with a reported infected number equal to 975.We represent on figure 4(b), the number of deaths by time, it appearedthat the peak of the deaths is shifted to the peak of the infected forabout 50 days, this shift corresponds to the hypothesis that we madeon the duration between the beginning of the symptoms and the deaths( i.e.
30 days). We note that the simulated death curve overestimatesthe observed curve. This may be due to the mortality of unreportedinfected and therefore the surplus corresponds to unreported COVID-19 mortality, which makes the optimization of the model’s variables fordeaths imprecise.Moreover, the model predicts 228 deaths at the end of the epidemicin the current case (12 hours curfew). In the case where the curfew was18 hours, the number of deaths would be 84. This information should betaken with caution, because at the time the simulations were made thenumber of deaths was low.Finally, we notice that the ratio between the undeclared cases (asymp-tomatic and symptomatic) represents between 45% at the beginning of theepidemic for less than 10% at the end of the epidemic (see figure 5).
We study in this section the effect of more intensive mass screening, i.e. by increasing τ and τ , on the basic reproduction number, R , and onthe number of declared infected, Q (see figures 6, 7).In figure 6, we can see that no matter how intensively we screen, wehave R >
1. Moreover, we note that to minimize the basic reproductionnumber, it is necessary to increase the massive screening of the class of a) Effect of curfew on number of re-ported infected. (b) Effect of curfew on number ofdeath. Figure 4: Effect of containment on number of reported infected and deathFigure 5: Ratio of undeclared cases (asymptomatic and symptomatic)Figure 6: Effect of quarantine rates, τ and τ , on the basic reproduction number R . 7 symptomatic infections. Similarly in figure 7, we can see that the increasein the mass screening effort allows a more rapid decrease in the numberof cases (see figure 7(a)). This is probably since cases are detected earlier,they do not contribute to the contamination of the healthy ones.Indeed, mass screening has an indirect effect on recruitment in theinfect compartment. More specifically, we assume that β + τ does notvary when τ changes. Consequently, the effort of mass screening onasymptomatic patients cannot exceed this value, and then any additionaleffort beyond β + τ will be passed on to healthy patients and thereforewill be useless.It is observed that the calibration of the model on the Tunisian datausing Metropolis-Hastings (MH) algorithm, gives a value of τ = 0 . i.e. a 20% decrease in the number ofcases (see figure 7(b)). (a) Effect of mass screening on num-ber of reported infected. (b) Effect of mass screening on num-ber of death. Figure 7: Effect of different scenarios of mass screening policies on the tunisianpopulation
It can be noted that since April, 15, 2020, Tunisia has succeeded in slowingdown the speed of propagation thanks to and containment and a curfew.With this work, we suggest that if containment and curfew are main-tained, short-term projections could be more optimistic. The fact thatthe epidemic is quickly contained tends to show that the number of un-declared infected is low, which may suggest that our model is efficient forthe evaluation of undeclared cases. In fact, we show that at the time ofthe epidemiological peak, the number of unreported infected persons con-stitutes at most 1 / uarantine, τ , at a high value. This expresses the good performance ofthe control policy of the Tunisian government. Indeed, in Tunisia, thecontrol policy consists of an intense isolation campaign targeting sick in-dividuals and their relatives. An effort of testing was carried out in atargeted manner, similar to snowball sampling. This approach enabled tohave a major screening effort on infected and asymptomatic individuals.Finally, the model was successfully able to predict the time of the peakat the end of April. A Materials and Methods
A.1 Computation of the basic reproduction num-ber R We use the next generation matrix to drive the basic reproduction num-ber R [3]. In the system (1) we have two infected compartments repre-sented by the second and third equations of the system. Therefore, at theinfection-free steady state, i.e. for a small ( As, I ) and S = S , the linearepidemic subsystem is : dAsdt = S ( α As + α I ) − ( β + τ ) AsdIdt = βAs − ( τ + γ + µ ) I (9)If we set X = ( As, I ) T as the vector of infected, F is the matrix thatrepresents the production of new infections and T the matrix of transferinto and out of the compartment by transmission, mortality, quarantine,and recovery, then the matrix form of the linear epidemic subsystem is:˙ X = ( F − T ) X Where : F = (cid:18) α S α S (cid:19) and T = (cid:18) β + τ − β τ + γ + µ (cid:19) .Therefore, the next generation matrix is : F T − = S ( β + τ )( τ + γ + µ ) (cid:18) α ( τ + γ + µ ) + α β α ( β + τ )0 0 (cid:19) Knowing that the basic reproductive number R is the largest eigen-value of the next-generation matrix, then: R = α S β + τ (1 + α βα ( τ + γ + µ ) ) A.2 Parameter estimation
Step 1:
In this part, to estimate the initial conditions As ( t ) and I ( t )and parameters α and α we adapt the method developed by [4]. Let = γ + µ and CR ( t ) the cumulative number of reported infectious casesat time t , defined by, CR ( t ) = (cid:90) tt τ As ( t ) + τ I ( t ) dt Let’s assume that CR ( t ) = χ exp( χ t ) − χ with χ, χ and χ threepositive parameters.By assuming that, CR ( t ) = 0 equation (3) implies that:exp( χ t ) = χ χ and then t = 1 χ (ln( χ ) − ln( χ )) . (10)Using equation (3), we have also: CR (cid:48) ( t ) = τ As ( t ) + τ I ( t ) (11)= χ χ exp( χ t ) (12)Let’s note τ = τ τ and H ( t ) = As ( t ) + τ I ( t ). Then we have, H ( t ) = H ( t ) exp( χ ( t − t )) , with H ( t ) = χ χ τ . (13)In order to simplify the calculus, we will use the normalized functions, AsH and IH . We have: As ( t )) H ( t ) = 1 − τ I ( t ) H ( t ) . (14)Rewriting the third equation (1), with H variable, dIdt = βH − ( τ + βτ + γ ) I (15)By assuming, that I ( t ) = I ( t ) exp( χ ( t − t )) and substituting inequation (15), we obtain: χ I ( t ) = βH ( t ) − ( τ + βτ + γ ) I ( t ) (16)Equation (16), implies I ( t ) H ( t ) = βχ + τ + βτ + γ (17)By using equation (14) and (17), we obtain: As ( t )) H ( t ) = χ + τ + γ χ + τ + βτ + γ (18)Let’s assume that α = fα with f a fixed parameter bigger than 1.The parameter α is evaluated using As ( t ) = As ( t ) exp( χ ( t − t )) andthe second equation of (1) at t , we obtain: χ As ( t ) H ( t ) = α S ( f I ( t ) H ( t ) + As ( t ) H ( t ) ) − ( β + τ ) As ( t ) H ( t ) ⇔ (19) α = χ + β + τ ( f I ( t ) As ( t ) + 1) S (20) nd therefore using equations (17) and (18), α = ( χ + β + τ )( fβχ + τ + γ + 1) S (21) Step 2
We hereby propose to estimate γ and µ . We notice that, R ( t ) = γµ D ( t ), for all t >
0. Let ρ = µγ , ρ is estimate using dead and recoveriesdata.Let p the fraction of infectious (quarantined or not) that become re-ported dead ( i.e. − p become reported recovered). Thus ρ = p ˆ µ (1 − p )ˆ γ ,with 1 / ˆ µ the average time to death and 1 / ˆ γ the average time to recover.Therefore, p = ρ ˆ γ ˆ µ + ρ ˆ γ . (22) Step 3
Parameters τ et τ was estimated using Metropolis-Hastings(MH) algorithm developed in the pymcmcst python package [5] References [1] Abdeljaoued-Tej, I., & Dhenain, M.. Estimation of Tunisia COVID-19 infected cases based on mortality rate. MedRxiv, ,[2] Anderson, R. M.& May, R. M. Infectious Diseases of Humans: Dy-namics and Control.
Oxford Science Publications , 1992.[3] Diekmann, O.; Heesterbeek, J. A. P.; Metz, J. A. J. On the defi-nition and the computation of the basic reproduction ratio R0 inmodels for infectious diseases in heterogeneous populations.
Journalof Mathematical Biology , (4), 365–382.[4] Liu, Z.; Magal, P.; Seydi, O.; Webb, G. Understanding UnreportedCases in the COVID-19 Epidemic Outbreak in Wuhan, China, andthe Importance of Major Public Health Interventions. Biology , , 50.[5] Miles, P. R. pymcmcstat: A Python Package for Bayesian InferenceUsing Delayed Rejection Adaptive Metropolis. em Journal of OpenSource Software, (38) 147,(38) 147,