The Finial Task-veronica

The Finial Task: SIR ModelVeronica1、 Effectiveness of immunizationInfectivity iIllness dPopulation NInitial Infectious Population(1)Assuming the vaccine is 100% effective①The contact number for polio is 7. In order to make sure that the epidemic will not occur, the infection rate should be less than therecovery rate, that is, cid(1-ve)<1, 7· (1-v)<1, v>. So the fraction of the population that should be vaccinated would be more than 6/7.②The contact number for measles and pertussis is 18. As a result, the fraction of the population that must be vaccinated would be more than 17/18.(2)Assuming the vaccine is 90% effective①The contact number for polio is 7. If we do not want the epidemic to occur, the infection rate should be less than the recovery rate, which means that cid(1-ve)<1, 7· (1-0.9v)<1, v>60/63. So the fraction of the population that must be vaccinated should be more than 60/63.②The contact number for measles and pertussis is 18. So the fraction of the population that must be vaccinated will be more than 85/81, that is to say, although we can have all the people vaccinated, the epidemic of these diseases cannot stop spreading. T hat’s why measles and pertussis persist while polio has been effectively eliminated.(3)Assuming that half of the population has been immunized, the initial recovered population will change from zero to 5000, and the curve will be much flatter, which means it will take more time to occur, and will last for a long time.According to the formula: cid (1-ve) <1,<1, v>2/3, so 2/3 ofthe population should be immunized to prevent an epidemic.Susceptible Population S10,0007,5005,0002,5000024681012141618202224262830Time (Day)P e o p l eSusceptible Population S : current1Susceptible Population S : current2、 Effectiveness of quarantineInfectivity iIllness dPopulation N<Average Durationof Illness d>(1) Assuming the quarantine time is half of a day, and the quarantine contact rate is zero. Quarantine fraction for current1, 2, 3, and 4 are 0.2, 0.4, 0.5, and 0.6 respectively. From the following graph, we can see that when the quarantine fraction is less than 0.5, there will be no epidemic.Susceptible Population S10,0007,5005,0002,5000Ti me (Day)P e o p l eSuscepti bl e Popul ati on S : current4Suscepti bl e Popul ati on S : current3Suscepti bl e Popul ati on S : current2Suscepti bl e Popul ati on S : current1(2)Assuming that the contact rate of quarantined individuals is 3①By testing various quarantine fractions like 0.2, 0.4, 0.6, 0.8, 1(they are the numbers of the fraction0,1,2,3,4), we get that the contact rate changes from 0 to 3, andeven perfect quarantine can not prevent an epidemic from happening.Susceptible Population S10,0007,5005,0002,500006121824303642485460Time (Day)P e o p l eSusceptible Population S : fraction0Susceptible Population S : fraction4Susceptible Population S : fraction3Susceptible Population S : fraction2Susceptible Population S : fraction1②The quarantine fraction is 1. By testing various quarantine time like 0.5, 0.3, 0.1, 0.07 (they are the numbers of the time1,2,3,4 in the graph), we get that the faster the infectious people are moving to quarantine, the higher the susceptible population will be. When the time is less than 0.07, there is something wrong with the model. But it still can not prevent the epidemic.So we get to know that if the contact rate of the quarantined individuals cannot reduce to a certain number, an epidemic would still happen no matter how fast the infectious people are moved to quarantine and no matter how high the quarantine fraction is.Susceptible Population S10,0007,5005,0002,500006121824303642485460Time (Day)P e o p l eSusceptible Population S : time4Susceptible Population S : time3Susceptible Population S : time2Susceptible Population S : time1Finally, quarantine and vaccine are both effective, but it is convenient to implement quarantine. Therefore, in an emergency, just-in-time quarantine is possible, and vaccine would be impossible to prevent the disease, because the latter needs a lot of time and money to produce.3、Loss of immunityInfectivity iIllness dPopulation NLoss of immunity makes recovered people become the susceptible population after a period of time. By changing the average duration of immunity, we have the following graph. It can be concluded that when the average duration of immunity becomes shorter, the continuous occurrence of the epidemic would be more likely to happen. So when the immunity is not permanent, it is difficult to eradicate a disease.Susceptible Population S10,0007,5005,0002,5000024681012141618202224262830Ti me (Day)P e o p l eSuscepti bl e Popul ati on S : current3Suscepti bl e Popul ati on S : current2Suscepti bl e Popul ati on S : current1Suscepti bl e Popul ati on S : current04、Extending the SIR modelInfectious PopulationInfectives iaPopulation I>Assuming the vaccinated fraction is zero, the average duration of immunity is 10,000. Then run the model, we have the following graghs.Susceptible Population S10,0007,5005,0002,500006121824303642485460Time (Day)P e o p l eSusceptible Population S : outputAsymptomatic Infectives A8,0006,0004,0002,000006121824303642485460Time (Day)Asymptomatic Infectives A : outputSymptomatic Population I2,0001,5001,000500006121824303642485460Time (Day)P e o p l eSymptomatic Population I : outputRecovered Population R10,0007,5005,0002,500006121824303642485460Time (Day)P e o p l eRecovered Population R : outputObviously an incubation period makes the epidemic occur more easily. From the charts, there is nearly 79% of the susceptible population being uninfected while 1% of the population was symptomatic in about 8 days; and about 5% of the susceptible population is uninfected while 10% of the population becomes sick in 13 days.(1)The contact rate of the symptomatic population is 0. The infectivity is 0.25.Setting different incubation times, we get that the longer the incubation time, the quicker and bigger of the disease to spread. The asymptomatic infective curve will reach a higher top, and take a longer time to decline. The symptomatic population is small and constant. The recovered population increases in a steady way. So the incubation period results in inefficiency of the quarantine. Because when the incubation period is long, there will be many people in the asymptomatic stage without being quarantined, but they are as active as healthy people, a lot of other people will be infected by them gradually. So with less symptomatic people being quarantined and a lot of asymptomatic people, the quarantine will become ineffective and the ability of the population to enjoy herd immunity will be worse.Susceptible Population S10,0007,5005,0002,500006121824303642485460Time (Day)P e o p l eSusceptible Population S : 2 days Susceptible Population S : 10 days Susceptible Population S : 18 days Susceptible Population S : 100 daysAsymptomatic Infectives A10,0007,5005,0002,500006121824303642485460Time (Day)Asymptomatic Infectives A : 2 days Asymptomatic Infectives A : 10 days Asymptomatic Infectives A : 18 days Asymptomatic Infectives A : 100 daysSymptomatic Population I4,0003,0002,0001,0000Time (Day)P e o p l eSymptomatic Population I : 2 days Symptomatic Population I : 10 days Symptomatic Population I : 18 days Symptomatic Population I : 100 daysRecovered Population R10,0007,5005,0002,500006121824303642485460Time (Day)P e o p l eRecovered Population R : 2 days Recovered Population R : 10 days Recovered Population R : 18 days Recovered Population R : 100 days(2)The contact rate of the symptomatic population is 0. Incubation time is 18 days The following graphs show that the higher the infectivity, the quicker the epidemic to spread, and the faster the decline trend will be.Susceptible Population S10,0007,5005,0002,5000Time (Day)P e o p l eSusceptible Population S : infectivity3Susceptible Population S : infectivity2Susceptible Population S : infectivity1Asymptomatic Infectives A10,0007,5005,0002,500006121824303642485460Time (Day)Asymptomatic Infectives A : infectivity3Asymptomatic Infectives A : infectivity2Asymptomatic Infectives A : infectivity1Symptomatic Population I2,0001,5001,000500006121824303642485460Time (Day)P e o p l eSymptomatic Population I : infectivity3Symptomatic Population I : infectivity2Symptomatic Population I : infectivity1Recovered Population R10,0007,5005,0002,500006121824303642485460Time (Day)P e o p l eRecovered Population R : infectivity3Recovered Population R : infectivity2Recovered Population R : infectivity15、Just-in-Time Immunization(1)The vaccine effectiveness is 100%, the fraction vaccinated is 100%, the time to deploy vaccine is 2 days, and the average duration of the disease is 4 days. The threshold for immunization is 5%. According to the graph below, about 77% of the population eventually gets the disease. Compared with the case without JIT vaccination, it can reduce the fraction of the infected people.Asymptomatic Infectives A8,0006,0004,0002,000006121824303642485460Time (Day)Asymptomatic Infectives A : 5%Asymptomatic Infectives A8,0006,0004,0002,000006121824303642485460Time (Day)Asymptomatic Infectives A : no vaccination Asymptomatic Infectives A : 5%When the threshold is 1%, the asymptomatic infective will be half less than the previous situations, which means it is more effective to stop the epidemic from the beginning.Asymptomatic Infectives A8,0006,0004,0002,00006121824303642485460Time (Day)Asymptomatic Infectives A : 1%Asymptomatic Infectives A : no vaccinationAsymptomatic Infectives A : 5%(2)The vaccine effectiveness is 95%, the fraction vaccinated is 80%, the time to deploy vaccine is 7 days, and the average duration of the disease is 4 days. Comparing the previous graph and the following one, we can know that when the effectiveness is low, the fraction vaccinated is small, the time to deploy vaccine is long. As a consequence, the JIT immunization program will be less effective no matter how early you implement it.Asymptomatic Infectives A8,0006,0004,0002,00006121824303642485460Time (Day)Asymptomatic Infectives A : no vaccination2Asymptomatic Infectives A : thre1%Asymptomatic Infectives A : thre5%In conclusion, it is probably more effective to use JIT vaccination in the condition that when diseases mostly occur in urban areas where the population is dense and the contact rate is high, and when diseases are spreading through daily contact or inhalation like influenza. The ineffective situation may happen when diseases occur in rural areas or diseases can not spread via people’s daily contact, which means the infectivity is low, like HIV.。