A stochastic epidemic model of COVID-19 disease
AA stochastic epidemic model of COVID-19 disease
Xavier Bardina ∗ Departament de Matem`atiquesUniversitat Auton`oma de Barcelona08193-BellaterraMarco FerranteDipartimento di Matematica “Tullio Levi-Civita”Universit`a degli Studi di PadovaVia Trieste 63, 35121-Padova, ItalyCarles RoviraDepartament de Matem`atiques i Inform`aticaUniversitat de BarcelonaGran Via 585, 08007-Barcelona
Abstract
To model the evolution of diseases with extended latency periods and the presence ofasymptomatic patients like COVID-19, we define a simple discrete time stochastic SIR-typeepidemic model. We include both latent periods as well as the presence of quarantine areas,to capture the evolutionary dynamics of such diseases.
Keywords:
COVID-19, SIR model
MSC:
There exists a wide class of mathematical models that analyse the spread of epidemic diseases,either deterministic or stochastic, and may involve many factors such as infectious agents, modeof transmission, incubation periods, infectious periods, quarantine periods, etcetera (Allen 2003;Anderson and May 1991; Bailey 1975; Daley and Gani 1999; Diekmann et al. 2013).A basic model of infectious disease population dynamics, consisting of susceptible (S), infective(I) and recovered (R) individuals were first considered in a deterministic model by Kermack andMcKendric (1927). Since then, various epidemic deterministic models have been developed, withor without a time delay (see e.g McCluskey 2009 and Huang et al. 2010). At the same time, manystochastic models have been considered: discrete time models (see e.g Tuckwell and Williams 2007; Oli et al. 2006), continuous time Markov chain models and diffusion models (see e.g. Modeand Sleeman 2000). The models obtained in these three categories are of increasing mathematicalcomplexity and allow to study several aspects of the epidemics.Even if the discrete-time models are the simplest ones, they may help to better define the basicprinciples of the contagion and to avoid the constraints due to the own definitions of the moresophisticated models.In this paper we will adapt a simple SIR-type model proposed by Ferrante et al. (2016) andwe will divide the population into several classes to better describe the evolution of the COVIDepidemic. Here we will need to include latency periods, the presence of asymptomatic patients anddifferent level of isolation. To model the evolution of the epidemic, we will describe the evolutionof every single individual in the population, modelling the probability on every day to be infected ∗ Xavier Bardina and Carles Rovira are supported by the grant PGC2018-097848-B-I00. a r X i v : . [ q - b i o . P E ] M a y nd, once infected, the exact evolution of its disease until the possible recovery or the death. Theconstruction of the theoretical model is carried out in Section 2, where we are able to compute theprobability of contagion and an estimate of the basic reproduction number R . Then, in Section3 we answer to five research questions by using a simulation of the evolution of the disease. Theuse of the simulation is justified by the complexity of the model, that prevent to carry out anyfurther exact computation. We are able to see that to stop the epidemic is fundamental to startearly with a severe quarantine and that a late starting date or a more soft quarantine makes thisprocedure almost useless. Moreover, to determine the quarantine it is very important to know thelevel of infectivity of the asymptomatic, since the more infectious they are, the more important isthe quarantine. Finally, as expected, the group immunity plays a very important role to preventthe development of the disease. To model the evolution of epidemics, Tuckwell and Williams (2007) proposed a simple stochasticSIR-type model based on a discrete-time Markovian approach, later generalized by Ferrante et al.(2016) with a SEIHR model. These models, despite their simplicity, are very unrealistic to catchthe characteristic of the COVID-19 disease and for this reason in this paper we introduce a morecomplex system, that we call SEIAHCRD, that better describes this new disease.Assume that the population size is fixed and equal to n , and that the time is discrete, with theunit for the duration of an epoch one day. Every individual, marked by an integer between 1 and n , belongs to one of the following 8 classes: • the class S includes the individuals susceptible to the disease and never infected before; • the class E includes the individuals in a latency period, i.e. individuals that have beeninfected but that are still not infectious or sick; • the class I includes the infectious individuals, that are not yet sick, but that will developlater the disease; • the class A includes the infectious individuals, that are not yet sick, but that will NOTdevelop later the disease, usually referred as Asymptomatic; • the class H includes the infectious individuals, that are sick, but with light symptoms andtherefore at home quarantine; • the class C includes the infectious individuals, that are sick and with severe symptoms, andmost of the time are hospitalized; • the class D includes the deceased individuals; • the class R includes the recovered individuals.Assuming that we start at time 0, we will define for any individual i ∈ { , . . . , n } the family ofstochastic processes Y i Ξ = { Y i Ξ ( t ) , t = 0 , , , . . . } , such that Y i Ξ ( t ) = 1 if the individual i at time t belongs to the class Ξ, where Ξ is equal to S, E, I, A, H, C, D or R , and 0 otherwise. In this way,the total number of individuals in the class Ξ at time t ≥ Y Ξ ( t ) and will beequal to (cid:80) ni =1 Y i Ξ ( t ).Let us now fix the main assumption on the evolution of the epidemic. At time 0 all theindividuals are in S , but one in class I or A and that the evolution of the contagion follows theserules:1. Daily encounters : each individual i , over ( t, t +1], will encounter a number of other individualsequal to N i ( t ) which we will assume to be a deterministic value;2. Contagion probability : if an individual who has never been diseased up to and including time t , encounters an individual in ( t, t + 1] who belongs to the class I or A , then, independentlyof the results of other encounters, the encounter results in transmission of the disease withprobability q I and q A , respectively. 2. Permanence in the classes : any individual, but one, starts from class S. Once infected he/shemoves to class E and so on according to the graph below. The time spent in the classes E , I and A are of r E , r I and r A consecutive days, respectively. These values can be considereddeterministic or stochastic. Any individual who enters the class H remains in this class for r HC consecutive days with probability α or for r HR consecutive days with probability 1 − α .Any individual who enters the class C remains in this class for r CD consecutive days withprobability λ or for r CR consecutive days with probability 1 − λ . As before, the values ofthese four numbers of consecutive days can be considered deterministic or stochastic. Toconclude, we assume that the individuals once in class R reamin there forever, the same asfor the class D .4. Transitions between the classes : we assume that any individual can moves between the classesaccording to the following graph
S E IA H CR Dβ µ − µ α − α − λ λ Here β, µ, α and λ denotes the transition probabilities and the transitions occur at the endof the permanence time spent by the individual in the previous class. Note that µ, α, λ areparameters that depends only on the specific nature of the disease, while β depends on thisand the number of individuals in the classes I and A .Any individual, once infected with probability β , follows one of the four paths described here:(a) he/she transits through the states E, I, H, C, D , where he/she remains, respectively, for r E , r I , r HC and r CD days, after which he/she dies and moves to class D .(b) he/she transits through the states E, I, H, C, R , where he/she remains, respectively, for r E , r I , r HC and r CR days, after which he/she becomes immune and moves to class R .(c) he/she transits through the states E, I, H, R , where he/she remains, respectively, for r E , r I and r HR days, after which he/she becomes immune and moves to class R .(d) he/she transits through the states E, A, R , where he/she remains, respectively, for r E and r A days, after which he/she becomes immune and moves to class R .It is immediate to see that the probability to follow any of these four paths is equal to,respectively, µαλ , µα (1 − λ ) , µ (1 − α ) , (1 − µ ) . In order to evaluate the probability β of contagion at time t of an individual in class S , we willstart defining the probabilities of meeting an individual in the classes S, E, I, A and R as equal.Note that the individuals in classes H and C are in total quarantine and that it is not possible tomeet them and that the individuals in class D are removed from the system. So, we will deal withthree possibly different encounter probabilities: • p I the probability of meeting an individual belonging to the class I ; • p A the probability of meeting an individual belonging to class A; • p S the probability of meeting an individual that is not infectious, that is he/she belongs tothe classes S, E or R . 3ssuming that the probability of meeting any individual is uniform and independent from theabove defined classes, we define these probabilities as p I = y I n − y H − y C − y D − , (1) p A = y A n − y H − y C − y D − ,p S = 1 − p I − p A when y H + y C + y D < n −
1, while p I = p A = 0 , p S = 1 when y H + y C + y D = n −
1. In theabove formulas, y I , y A , y H , y C and y D denote the number of individuals in classes I, A, H, C and D , respectively.Denoting by j iI , j iA the number of meetings of the i -th individual at time t with individuals inthe classes I and A , respectively, the probability to meet this proportion of individuals is N i ( t ) (cid:88) j iI ,j iA =1 N i ( t )! j iI ! j iA !( N i ( t ) − j iI − j iA )! p j iI I p j iA A p N i ( t ) − j iI − j iA S , where N i ( t ) denotes the daily encounters of the individual i . We can easily derive the probabilityof contagion p j iI + j iA = 1 − ((1 − q I ) j iI (1 − q A ) j iA )where q I and q A denote the probability of transmission of the specific disease for individuals inclasses I and A , respectively, which are usually different. Then the probability of contagion attime t + 1 of a single individual is equal to β i = N i ( t ) (cid:88) j iI ,j iA =1 p j iI + j iA N i ( t )! j iI ! j iA !( N i ( t ) − j iI − j iA )! p j iI I p j iA A p N i ( t ) − j iI − j iA S = 1 − N i ( t ) (cid:88) j iI ,j iA =1 N i ( t )! j iI ! j iA !( N i ( t ) − j iI − j iA )! ((1 − q I ) p I ) j iI ((1 − q A ) p A ) j iA × p N i ( t ) − j iI − j iA S . Substituting (1), we then get β i = 1 − (cid:16) − q I p I − q A p A (cid:17) N i ( t ) = 1 − (cid:16) − q I y I − q A y A n − y H − y C − y D − (cid:17) N i ( t ) when y H + y C + y D < n −
1, while β i = 0 when y H + y C + y D = n − . As done by Tuckwelland Williams (2007), we can use these formulas to simulate the spread of an epidemic under thesegeneral assumptions. Some results, similar to those presented in Tuckwell and Williams (2007),can be found in Ferrante et al. (2016).To conclude, let us now consider the basic reproduction number R , i.e. the expected numberof secondary cases produced by an infectious individual during its period of infectiousness (seeDiekmann et al. 1990). In the present model, this refers to individuals that transits through theclasses I or A . Let us recall the threshold value of R , which establishes that an infection persistsonly if R >
1. As for the SIR-model proposed by Tuckwell and Williams we are not able to derivethe exact explicit value of R , but it is possible to extend their results, when N i ( t ) ≡ N , for all i and t , obtaining that R = µr I q I N + (1 − µ ) r A q A N + O (cid:18) n − (cid:19) . Note that the value of R computed above is the basic reproduction number at the beginning ofthe disease, when there is one infectious individual that has N contacts in a population with n − N some of them won’t be with susceptible individuals and the number of cases producedby an infectious individual will be smaller. At time t , let us call S ( t ) the number of susceptible4ndividuals (class S ) and X ( t ) the number of individuals removed from the population (members ofthe classes H , C and D ). Then, given an individual in class I , set Z i ( t ) his number of contacts withsusceptible individuals during the day t and Q i ( t ) the number of individuals infected by i at theend of day t . Clearly Z i ( t ) follows an hypergeomtric distribution with parameters n − X ( t ) − , S ( t )and N and, given Z i ( t ) = z, z ∈ { , . . . , N } , it can be seen that Q i ( t ) ∼ Bin( z, q I ). ThusE( Q i ( t ) | X ( t ) , S ( t )) = E (cid:16) E( Q i ( t ) | Z i ( t ) , X ( t ) , S ( t )) | X ( t ) , S ( t ) (cid:17) = E( q I Z i ( t ) | X ( t ) , S ( t )) = q I N S ( t ) n − − X ( t ) . Furthermore, using the same ideas in Ferrante et al. (2016), the number of secondary casescorresponding to this individual will be r I q I N S ( t ) n − − X ( t ) + O (cid:18) n − − X ( t ) (cid:19) . Finally, the number of secondary cases produced by one arbitrary infectious individual (that canbe in class I or A ) at time t given S ( t ) and X ( t ) and that we will call R ( t ), will be R ( t ) = ( µr I q I + (1 − µ ) r A q A ) N S ( t ) n − − X ( t ) + O (cid:18) n − − X ( t ) (cid:19) . The COVID19 is a highly contagious disease that has appeared at the end of 2019. From all theinformation that is published every day, often contradictory, in the media, we can extract someproperties of the disease. After the contagion, the virus remains in a latent state for 5-7 days beforethe individual became infectious. Then the individual can begin with symptoms in 3 days or hecan continue asymptomatic, but probably infectious during two weeks. It is not known actuallythe number of asymptomatic people, but it will be probably bigger that the number of people withsymptoms. About 75% of persons with symptoms have just light symptoms that last for two weeksand after that they became recovered. The other 25% get sever symptoms after a first period of7-9 days with light symptoms. A 15% of these patients with severe symptoms die in 4-6 days,while the other 85% will recover after a period of 18-24 days.This disease can be described by the SEIAHRCD model defined before. According to the dataabove described, we can choose the deterministic values for the permanence in the classes r E = 5 , r I = 3 , r A = 14 , r HR = 14 , r HC = 9 , r CR = 20 , r CD = 4and for the probabilities µ = 0 . , α = 0 . , and λ = 0 . . We also have to make an assumption on the ineffectiveness of the individuals when in the classes I and A . We assume that q I = 0 . , and q A = 0 . . That is, we are assuming that asymptomatic individuals are less infectious than individuals whohave symptoms. R , the expected number of secondary cases produced by an infectious individual, is equal withthese parameters to R = N (cid:0) . . (cid:1) + O (cid:18) n − (cid:19) . We see that the threshold value R = 1 is obtained for N = 1 .
54 and that R = 3 . N = 5 . N i ( t ) ≡ N for any i and5hat therefore β i is constant for any individual. This assumption is strong in the case of a possiblequarantine, but it is still reasonable.In this paper, since the analytical approach to this model is really complicated, we focus on theevolution of the disease by implementing a simulation using environment Maple. All the valuespresented here are the mean computed over 30 repetitions of the simulation and we report also theconfidence intervals.We know very well that our model cannot explain exactly the COVID19 epidemic, since thereare too many unknown aspects about this new disease, but we believe that the study of thebehaviour of our model can help to understand the COVID19 epidemic. More precisely, we willanswer five Research Questions regarding the dynamics of the disease depending on some of theparameters involved. Particularly, we deal with (1) the importance of the number of contacts,(2)-(3) the effectiveness of a quarantine depending on the moment it begins and on its duration,(4) the role of the asymptomatic depending of their level of infectiveness and (5) what happenswith different levels of group immunization.For each of these situations we study six quantities that we consider of major interest:1. Class D: The number of deceased, therefore individuals in class D, after 180 days.2. Max class C: The maximum number of individuals in class C, i.e. with severe symptoms, inone day.3. Total class C: The sum of all the days spent by all the individuals in the class C.4. Max new H: The maximum number of individuals that enter in class H (people with symp-toms) in one day.5. Day max new H: The day when it is reached the maximum number of individuals that enterin class H in one day.6. Prop. infected: The proportion of the population that has been infected after 180 days.For all these quantities we give a table with the mean and the 95% confidence interval for themean in several situations. We also present a plot of one simulation of the number of deaths, thenumber of individuals in class C and the number of individuals that enter in class H each dayfor any of these situations along the 180 days. Note that the number of deceased is consideredassuming that the health service is able to give the same level of assistance whatever the number ofpatients is, but probably in the situations where the health service is more stressed the assistancewill be worse and the number of deceased may increase. We consider the dependence on the number of contacts on the evolution of the disease when N = 10 , , Influence of number of contacts N N
10 5 4 3Class D 189.16 176.03 168.56 159(184.15,194.16) (162.97,189.08) (151.69,185.45)Max class C 564,23 420,03 330,40 225(557.04,571.49) (390.83,449.22) (297.70,363.09)Total class C 12007,33 11524,13 10597,60 9571(11847.58,12167.07) (11432.97,11595.28) (9553.21,11641.98)Max new H 441.16 236.13 172,10 100(432.16,449.38) (219.84,252.41) (155.18,189.01)Day max new H 42.90 64.70 79.53 98(41.77,44.02) (63.11,66.28) (73.87,85.18)Prop. infected 0.9999 0.9489 0.8852 0.8274(0.9998,0.9999) (0.8856,1) (0.7991,0.9712) N = 5 in one of the 30 simulations the epidemic did not go forwardwhen N = 4 it happened in two cases. On the other hand, when N = 3 and with our initialconditions, only in 1 of the 30 simulations the disease went on. For this reason, in the table weonly give the values of such case.Figure 1: Evolution of the number of deceased, new individuals in H and the size of the class C when N = 10 , , The number of deceased and the proportion of infected population are not very different between N = 4 and N = 10. The main difference consists in the velocity in the dissemination of the illnessand so, the maximum of individuals in class C that can require hospitalization. Let us check the effects of a quarantine on the evolution of the disease. We assume that N = 10at the beginning of the spread of the disease and we consider three levels of quarantine, definedby the number of contacts N = 3, N = 2 or N = 0 (the total quarantine). We can also considerwhat happens depending on the moment that the quarantine starts: • when there are 10 deceased (first 10 individuals in class D), • when there is the first deceased (first individual in class D), • when there is the first individual with severe symptoms (first time the class C in not empty). Let us recall that we assume that before the quarantine N = 10. We getTable 2: Influence of a quarantine of N at 10 deaths N
10 3 2 0Class D 189.16 186.96 183.80 184.30(184.15,194.16) (181.17,192.74) (178.04,189.55) (179.55,189.04)Max class C 564.23 564.60 557.46 560.66(557.04,571.49) (557.04,571.41) (551.60,563.31) (552.22,569.09)Total class C 12007.33 12013.86 11887.86 11960.53(11847.58,12167.07) (11828.88,12198.83) (11775.82,11999.89) (11823.23,12097.82)Max new H 441.16 439.73 440.66 441.90(432.16,449.38) (431.53,447.92) (434.07,447.24) (435.30,448.49)Day max new H 42.90 42.56 43.93 43(41.77,44.02) (41.70,43.41) (42.90,44.95) (42.08,43.91)Prop. infected 0.9999 0.9984 0.9967 0.9934(0.9998,0.9999) (0.9983,9984) (0.9965,0.9968) (0.9933,0.9934)
It is clear that the efficacy of these quarantine, including the total quarantine, is almost null,since at this level of the disease the 99% of the population has been infected (case N = 0).7igure 2: Evolution of the number of deceased, new individuals in H and the size of the class C when thequarantine with N = 3 , Table 3:
Influence of a quarantine of N after the first death N
10 3 2 0Class D 189.16 174.50 157.60 105.83(184.15,194.16) (168.55,180.44) (149.26,165.93) (89.12,122,53)Max class C 564.23 433.23 386.80 365.40(557.04,571.49) (401.39,465.06) (339.66,433.93) (311.49,419.37)Total class C 12007.33 11176.66 10028.40 6860.66(11847.58,12167.07) (10947.57,11405.74) (9453.42,10603.37) (5791.85,7929.46)Max new H 441.16 347,50 343,06 367,73(432.16,449.38) (312.17,382.82) (299.08,387.03) (324.30,407,15)Day max new H 42.90 42.70 40.10 41.03(41.77,44.02) (40.67,44.72) (38.41,41.78) (39.50,42.55)Prop. infected 0.9999 0.9401 0.8412 0.5660(0.9998,0.9999) (0.9268,0.9533) (0.7975,0.8848) (0.4769,0.6550)
Figure 3:
Evolution of the number of deceased, new individuals in H and the size of the class C when thequarantine with N = 3 , In this case we can note the efficacy of a quarantine if it is rigorous, with a significant differencebetween N = 3 and N = 2. Note that in the total quarantine, there will be more that 100 deceasedsince at this level of the disease more of the 50% of the population has been infected (case N = 0). In this case the quarantine is effective. For N = 2 we reduce the mortality of about 50% and for N = 0 the number of deceased is just 20. Note that when the first individual enters in C there areabout 1000 individuals infected. 8able 4: Influence of a quarantine of N after the first enter in C N
10 3 2 0Class D 189.16 162.80 106.70 20.96(184.15,194.16) (157.16,168.43) (98.49,114,98) (15.51,26.40)Max class C 564.23 272.93 130 75.50(557.04,571.49) (259.34,286.51) (104.41,155.58) (56.90,94,09)Total class C 12007.33 10382,53 6828,43 1337,87(11847.58,12167.07) (10247.95,10517.10) (6430.56,7226.29) (1006.86,1668.87)Max new H 441.16 149.36 96.96 117.13(432.16,449.38) (134.02,164.69) (68.38,125.53) (92.21,143,04)Day max new H 42.90 50.33 45.60 32.90(41.77,44.02) (45.74,54.91) (35.97,55.22) (31.36,34.43)Prop. infected 0.9999 0.8662 0.5709 0.1105(0.9998,0.9999) (0.8597,0.8726) (0.5388,0.6029) (0.0839,0.1370)
Figure 4:
Evolution of the number of deceased, new individuals in H and the size of the class C when thequarantine with N = 3 , Let us consider now the duration of the quarantine. We consider quarantines, beginning at firstindividual in C, of duration 20, 60 and 120 days. We also deal with two levels of quarantine,beginning from a number of contacts of N = 10 we pass to a quarantine with N = 3 or N = 1. N = 3Table 5: Influence of a quarantine (from N = 10 to N = 3) of duration k days after the first enter in C. k Evolution of the number of deaths, new individuals in H and the size of the class C when thereis a quarantine of k days. When the quarantine is done with N = 3, its effects in the number of deceased are not veryimportant although we do a long quarantine of 120 days. Nevertheless, we are able to reducethe maximum in class C. To value the importance of the quarantine we need a more restrictedquarantine. N = 1Table 6: Influence of a quarantine (from N = 10 to N = 1) of duration k days after the first enter in C. k Figure 6:
Evolution of the number of deaths, new individuals in H and the size of the class C when thereis a quarantine of k days. We can see here that with a strong quarantine ( N = 1) and with a long quarantine ( k = 120),the disease can be stopped. In the two other cases, k = 20 or k = 60 the final numbers of deceasedare very similar, since after a short stop the disease returns to growth up arriving at the samelevels of the initial outbreak and having two clear peaks.10 .4 RQ4: How does the infectivity of asymptomatic individuals influ-ence the spread of the disease? One of the main problems in the study of the COVID19 disease is the role of the asymptomatic.In our general framework we have supposed that the infectivity of an asymptomatic individual(class A) is one fourth of the infectivity of an infective individual that will have symptoms (classI), that is, q A = 0 .
05 and q I = 0 .
2. Here we consider two other cases: in the first both probabilitiesare equal ( q A = q I = 0 .
2) and in the second case the asymptomatic individuals are not infectious( q A = 0). We also consider the case without quarantine and the case with quarantine (from N = 10to N = 3) after the first individual in class C. Table 7:
Influence of q A without quarantine q A Figure 7:
Evolution of the number of deaths, new individuals in H and the size of the class C with different q A without quarantine. We can see that the behaviour of the disease is very similar in the three cases and the maindifference just consists in the velocity of its spread.
We can see the importance of knowing the role of asymptomatics in implementing quarantine. Themore infectious the asymptomatic are, the more effective the quarantine is.
We consider finally the group immunity. We consider three cases, where the immunity group is of50%, 70% and 90%, respectively. We can see that an immunity of the 50% is not enough to controlthe disease since almost all the individuals without immunity become infected after the 180 days.11able 8:
Influence of q A with quarantine q A Figure 8:
Evolution of the number of deaths, new individuals in H and the size of the class C with different q A with quarantine. On the other hand, with an immunity of the 70% the disease is well controlled. Moreover ifthe immunity regards the 70% of the population, in 11 over 30 trials there aren’t any deaths andin one of the trials the disease does not infect any person.Table 9:
Influence of group immunity
Level 0% 50% 70%Class D 189.16 90.53 18.00(184.15,194.16) (87.07,93.98) (12.78,23.21)Max class C 564,23 207.63 20.96(557.04,571.49) (202.28,212) (14.95,26.96)Total class C 12007.33 5804.80 1124.06(11847.58,12167.07) (5680.10,5929.49) (797.15,1450.96)Max new H 441.16 120.76 12.40(432.16,449.38) (118.41,123.10) (9.06,15.73)Day max new H 42.90 61.30 94.26(41.77,44.02) (58.94,63.65) (77.36,111.19)Prop. infected 0.9999 0.4848 0.0990(0.9998,0.9999) (0.4842,0.4853) (0.0713,0.1266)
If we consider an immunity of the 90% only in 8 of the 30 trials there have been infected peoplewith a maximum of 12 individuals infected and no one deceased.12igure 9:
Evolution of the number of deaths, new individuals in H and the size of the class C when thereis group immunity.
Like any mathematical model, the model presented in this paper does not exactly describe COVID19disease, even if it shares with this most of its characteristic. Furthermore, for this new disease manyaspects are still unknown, such as the level of infectivity of asymptomatic patients.Our goal is to understand how a disease similar to the COVID19 spreads over in a closedpopulation and to answer to some specific research questions. What we have obtained can besummarized as follows:1. Reducing the number of contacts of each individual the spread of the disease slows down, thepressure on the health system reduces, but we end up with a similar number of deceased.2. It is basic to start the quarantine at least when the first severe patient is detected. Waitingfor the first deceased leads to many more additional deceased.3. To raise the quarantine it is very important to know the level of infectivity of the asymp-tomatic. The more infectious they are, the more important is the quarantine.4. To stop the disease you must perform a strict and long quarantine and you must start thisas soon as possible.5. Group immunity is very important to prevent the development of the disease.Even if most of the answers are the expected ones, we have obtained these through a soundstochastic epidemic model, that despite of its simplicity probably is able to catch most of thepeculiarity of this disease.We believe that a more sophisticated version of this model and more elaborated simulationscan allow us to answer to more complex questions, but probably it will be better to wait when adeeper knowledge on this disease will be available.