数学建模 美赛 2015 A题

Content

I. INTRODUCTION (1)

1.1 Background (1)

1.2 Previous work (1)

1.3 Our work (2)

II. THE DESCRIPTION OF THE PROBLEM (2)

2.1 How do we simulate future epidemic situation? (2)

2.2 How do we consider the influences of medication distribution? (2)

2.3 How do we analyze our results? (2)

III. MODELS (3)

3.1 Notations (3)

3.2 Assumptions (3)

3.3 Improved SEIR Epidemic Model (4)

3.3.1 Ascertainment of the Parameters (6)

3.3.2 Solution and Result (7)

3.3.3 Analysis of the Result (9)

3.4 Medication Distribution Optimization Model (9)

3.4.1 Ascertainment of terminology (10)

3.4.2 Solution and Result (11)

3.4.3 Analysis of the Result (13)

3.5 Medication Delivery Model (13)

3.5.1 Solution and Result (14)

IV. SENSITIVITY ANALYSIS (15)

4.1Influence of β (15)

4.1.1 Influence of βI (15)

4.1.2 Influence of βQ (16)

4.1.3 Influence of βF (17)

4.1.4 Analysis of results (17)

4.2 Time begin intervention (17)

V. CONCLUSIONS (18)

5.1 Conclusions of the problem (18)

5.2 Strengths and weaknesses (18)

5.2.1 Strengths (18)

5.2.2 Weaknesses (19)

VI. FUTURE WORK (19)

VII. REFERENCES (20)

VIII. MEMO (21)

I. Introduction

1.1 Background

Ebola virus disease (EVD), the disease with most fatality rate, spreads by direct contact with body fluids, such as blood, of an infected human or other animals. The current outbreak in west Africa, (first cases notified in March 2014), is the largest and most complex Ebola outbreak since the Ebola virus was first discovered in 1976. Ebola is not only fatal, but also with high risk of transmission. Even the body of Ebola patients are infectious, so improper burials may also cause infection. Another characteristic of Ebola is the latent period, which possibly varies from 4 to 6 days but can top to 29 days long. During the latent period, the infectious has little chance of transmission and appears no symptom.

Recently, drugs aiming at curing Ebola patients has been successfully developed. As a world-focusing virus, Ebola’s spread situation has been studied for a period of time. The new medicine is a great help to control the epidemic situation. Thus at present, we can assist the Ebola eradication process if optimal plan is proposed, which can be reached by building an mathematical simulation model of the epidemic situation. Furthermore, relevant factors, such as the quantity of the medicine needed, possible feasible delivery systems, locations of delivery, speed of manufacturing of the vaccine or drug and other possible ones, can also be addressed on the basis of the former model.

Among all infected countries, Guinea, Liberia and Sierra Leone are the three most affected ones. And there are still newly occurred cases daily. So every beneficial measurements counts. As long as the method is reasonable and scientific, the adoption of which can be of great importance, even save a large number of lives.

1.2 Previous work

The earliest outbreak of Ebola happened at Yambuku, Zaire in 1976, and happened again at Kiewit, Zaire in 1995. Then some relevant researches has been done in the next year. In 1996, Aimee Astacio and other scholars adopted SIR and SEIR model to simulate the spread of Ebola.[1]They fit the parameters based on data acquired from two former outbreaks. Ebola outbroke in Congo and Uganda 10 years later. J.Legrand and others brought out a new model called SEIHFR model modified from premier one.[2]They added two compartments: hospitalized and funeral, and improved the differential equations for Ebola’s features, such as the infectious patient body. The parameters are obtained by knowledge of statistics.

In 2014, Ebola appeared again in three west African countries and the relevant research became more. Caitlin.M.Rivers and others then added intervention into the SEIHFR model, and fit the parameters by former data in Sierra Leone and Libya. More analysis like the upgrade of quarantine and community isolation are also addressed.[3]Network Dynamics & Science

Laboratory of Virginia Tech University made long period simulation of Ebola epidemic in west Africa. [4] Noticeably, CDC released their simulation model as well.[5] The basic of their model is SEIR model with fit parameters. Moreover, a corrective index is used to rectify the deviation caused by lack of data.

1.3 Our work

Referring to the SEIHFR model, we come up with the SEIQFR model, and Q covers all effective kinds of quarantine. As the solid medicine is announced to be developed, we focus on the control effect of the medication. Based on our assumptions, we reach a distribution plan for Sierra Leone, and validate the positive effect brought by vaccine. Later on, we do the delivery optimization according to the distribution results. In the end, we do some sensitivity analysis by changing parameters, and find out the most impactful parameter. Some suggestions are mentioned in the memo on the basis of our results.

II. The Description of the Problem

2.1 How do we simulate future epidemic situation?

The Ebola cases data is available from the World Heath Organization. Based on the real data, we can depict the corresponding curve of them. As the differential equation of the transition process is established with unknown parameters, the simulated curve can be reached by fitting the original one. Therefore, the parameters are able to be calculated backwards. Then the epidemic model is completed. With the input of initial values, the simulated curve can be generated.

2.2 How do we consider the influences of medication distribution?

The medication involves two kinds. The first kind is the curative medicine for infected patients. When patients are cured, the parameters for contact rate may decrease, so the epidemic situation is eased. Another kind is the precautionary vaccine for susceptibles, after taking sufficient dose, individuals can probably resist the invasion of the Ebola virus, thus the constitution of population is changed.

2.3 How do we analyze our results?

After the problem solving, we evaluate the stability of the epidemic model by the basic reproduction number. At last, detailed analysis of sensibility are made. Concerning other possible interventions, the values of parameter may fluctuate with them thereby bring changes into the model. For instance, the popularize of the solid medicine may result smaller contact

rate between infected individuals and susceptible individuals. So as the more effective quarantine methods and more scientific burial ways, which can also make a difference on corresponding parameters. Comparing the changes before and after the parameter alteration, we can identify the one parameter with most impact on the result.

III. Models

3.1 Notations

Table1: Notation

Symbol Meaning

βcontact rate

1/αlatent period

1/γaverage period of process

δfatality rate

θthe fraction of infected quarantined

R0basic reproduction number

σthe effective probability of the vaccine

N whole population

S susceptible individuals

E exposed individuals

I infectious individuals

R removed individuals

Q quarantined individuals

F bodies infect on funeral

Im immune individuals

D demand for medication

n the total number of districts

V fitness value

p the period of medication delivery

A i the medication quantity delivered in one site

A total quantity of medication made in one period

d distanc

e between airport and district

c quantities of vaccine delivere

d from airport to district

3.2 Assumptions

The information obtained is mainly from the World Heath Organization. So the data we acquire should be valid.

To ensure the work done is significative. Unmentioned factors are considered not to affect the final results.

The models focus on the present strain of Ebola. So Ebola virus does not mutate in the forecast period.

People born and natural deaths are ignored to reduce the complexity of calculation.

There is relatively strict control of population flow in countries with Ebola outbreak.

Therefore import and export of population are ignored.

To simplify the model, parameters will not change with time.

Suggested by the reports, cured patients cannot get infected again.

Based on fact, in latent period, Ebola patients are not infectious.

No wastage of medication during the delivery to create an ideal environment for transportation.

Medication can be given out equally each day during one period of delivery. Thus the delivery system can remain effective.

3.3 Improved SEIR Epidemic Model

In order to estimate later epidemic situation, we need a model to accomplish the purpose. We implemented the SEIR model for disease transmission. According to the original SEIR model, the whole population (N) can be divided into four compartments. Taken into consideration of the features of Ebola virus disease, two extra compartments are added into the model.

As the name SEIR suggests, there exist four main kinds of individuals:

●Susceptible individuals (S), those who are vulnerable to the disease but have not been

infected yet;

●Exposed individuals (E), those who carry the virus but will not infect others and appear no

symptom;

●Infectious individuals (I), those who are infectious and begin showing certain symptoms; ●Removed individuals (R), those who are killed by the disease or completely cured.

In real situation, as long as the patient is diagnosed as Ebola carrier he will be quarantined in hospital or places alike. And for Ebola patients, they remain infectious even after they die, so the individuals in this situation should be addressed as well. Thus we have other two compartments:

●Quarantined individuals (Q), those who are diagnosed as Ebola carriers and quarantined in

hospital, thus are unable to infect other susceptibles;

●Bodies infect on funeral (F), those who are dead but are still infectious, thus can infect

others during the funeral.

Therefore, our model can be described as the SEIQFR epidemic model.

Obviously, when time moves on, the value of each compartments will change, therefore another set of variables are needed to indicate the affect of time. Likewise, we define S(t), E(t), I(t), R(t), Q(t) and F(t), which are frictions connected to the corresponding population. For instance, S(t) = S/N.

Fig 1: Compartmental flow

The figure above illustrates how compartments transit into each others. Solid lines are transition traces; dashed lines are infection traces. And sequence numbers are used to identify every process and parameters relevant.

Undoubtedly, there are possibilities that some compartments of individuals may transit into others. So between some of the compartments, there are transition chances described by parameters, which can be obtained by data fitting method. Ebola cases around the world have been collected and released by the World Health Organization, which can be a truthful source of information to set the initial value of the model, and to extract certain parameters.

11221212---[(1-)(1-)(1-)]-[(1-)](1-)-(1-)(1-)(1-)I Q F I Q F Q I D Q DQ IQ D DQ F I IQ F SI SQ SF dS dt

N SI SQ SF dE E dt N dI E I dt dQ I Q dt dF I Q F dt dR I Q F dt ββββββααγθγθδγθδγθγδγδγθδλδγγθδγδγ++?=??++?=???=++???=+???=+???=++?

(1) Above is the differential equation of the transition process. β is contact rate, the ratio of Ebola contact of a compartment (βI refers to the possibilities of susceptibles getting Ebola from infectious individuals); 1a is average latent period; 1is average period of certain process (1Q γrefers to average time before infectious get quarantined); δ is fatality rate (δ1 refers to

fatality rate for infected individuals while δ2 is for quarantined individuals); is the fraction of infected quarantined. Note that λ is a composite of all β transmission terms.

Table 2: Explanation for Parameter in Process

Process Number Parameter Explanation

1 λ

infection rate of S 2 1a

average latent period 3 θ fraction of infected quarantined

4 2DQ δγ?

funeral infection rate of Q 5

1Q γ

duration of traditional funeral 6 δ1 fatality rate of I

7 δ1(1-θ)γD funeral infection rate of I

8 δ2 fatality rate of Q

9 βQ contact rate between S and Q

10 βF contact rate between S and F

11 βI contact rate between S and I

Along with the function (1), table 2 gives the explanations of parameters or combinations of them governing each process.

3.3.1 Ascertainment of the Parameters

One of the essential parts of the model is confirming parameters, which will affect the type and the trend of the epidemic process curve considerably. Theoretically speaking, using the provided data from the World Health Organization, parameters mentioned in formula (1) can be obtained by data fitting. Implementing the morbidity data of Ebola cases into the least square method, we can reach to the estimated values of contact rates mentioned above. The differences between each real value and the value from the differential equation at the same time is noted as f t , according to the least square method, our object function is: 2min ()t F t f =∑ (2)

In F (t ), actually, there are several unknown numbers involving all contact rates. The solution is a set of parameters, which can produce the best fit curve of real data. We call the solution as estimated parameters.

We now get the best fit for real data, but the problem is that any combos of parameters may fit the same curve. Generally, the epidemic model yields a curve of infected individuals at the time that remains low and steady at the prophase, then soars to the summit in short period, and decreases continuously approaching to a small value. Based on the curve we get at present, we can adjust our estimated parameters by comparing with the general regulations of the epidemic curves.

If the later trend of the curve deviates from normal epidemic curves, then another fit is needed.

21min ()()t G t f F t =-∑

(3)

F (t 1) is the best fit we got before.

Therefore we acquire the second best fit parameters. In this way, the suitable set will be eventually reached after rounds of search.

3.3.2 Solution and Result

Taking Sierra Leone for study case, we use MATLAB to solve the problem. When deal with the calculation of parameters, the results are hard to reach perfection. Therefore, the assistance of relevant researches is generously adopted.

Referring to the research of Virginia Tech university [3], we ascertain the proper set as follows: Table 3: Parameter Set

Parameter Value α 0.1

βF 0.107743

βQ 0.076871

βI 0.121417

θ 0.265867

γI 0.05

γD 0.077336

γF 0.233802

γQ 0.242025

γDQ 0.113652

γIQ 0.063019

δ1 0.55

δ2 0.55

The values of parameters are mainly unchanged, while the values of γDQ and γDQ are calculated by

111DQ D Q γγγ=- and 111IQ I Q

γγγ=- respectively.

Then we generate the epidemic simulation curve, which is purely based on natural situation without any intervention from human.

Fig 2: Real case & Simulated case

The real data we acquire is until 01/02/2015, which should be fit by our model with proper parameters. In our model, I + Q should fits the real infected cases, and we can see an accordance between the real total cases and simulated I + Q cases in early time.

Later on, the simulated curve appears to climb up faster than the real one, which is probably because that intervention methods are taken afterwards so as to ease the epidemic aggravation to some extent.

Fig 3: Future epidemic simulation

The figure above is the simulation of the epidemic situation from 03/2014 to 08/2015, which period lasts long enough to allow us to observe the whole trend of the curve. There is an absence of the compartment S. It is because that the quantity of S is much larger than the other’s, the presence of S curve conceals the main features of other curves. And satisfactorily, the

simulated situations of all other compartments fit the general regulations well. Thus the model is valid enough to yield reasonable results.

3.3.3 Analysis of the Result

Inspecting the results yields from the model, we find them matching our expectation. At the prophase, the curve simulated can fit the real one, and the later trend of it is in accordance of general epidemic curves. To be specific:

The curve for E : the peak value appears first, for it is the source of the transition to other compartments. After the relevant period, people exposed will gradually become infected.

The curve for I : following the transition sequence, I peaks right after E . There is a dip of cases from E to I , which is due to the other transition traces of I .

The curve for Q : as the next stage of transition, the peak value of Q approximately halves of that of I . The other half died for the fatality of quarantined is set as 0.55.

The curve for F : the value of F fluctuates with its source, so it has a similar trend and a lag from other curves.

Stability analysis

A key parameters describing the natural spread of an infection is the basic reproduction number R 0, which are defined as the number of secondary infections generated by an infected index case in the absence of control interventions.[6][7]

If R 0 drops below unity, the epidemic eventually stops, otherwise the epidemic will maintain at certain district (developing as an endemia), which we do not wish.

As the parameters are determined, R 0 can also be calculated by:

2210000(1)=

Q

Q DQ IQ I F I Q F F R R R R θγβγδγδβδβγ??+-=++++?? (4)

where Δ is confirmed as: 11=+(1)(1)(1)Q D I θγγθδγθδ??-+--

So we get the basic reproduction number of Sierra Leone is R 0 = 1.89, which exceeds unity. The result infers that without any intervention, the spread of Ebola epidemic will not stop by itself.

3.4 Medication Distribution Optimization Model

In the epidemic model, as long as the parameters are determined, then the later spread situation of Ebola virus can be estimated. To control the aggravation of the epidemic, medication should be provided properly. This model solves how the materials should be distributed before delivery.

First, the immune population is calculated by the vaccinated population and the successful vaccination rate. Then we define the demand of medication, which should be offered through

fixed sites. To meet the optimization objective (minimizing the sum of I in each district after a short period of days), a corresponding vaccine offering plan can be reached by intelligent algorithm. The solution is a plan describing ideal quantities of vaccine gave out in each site per day (vaccinating speed).

3.4.1 Ascertainment of terminology

● Immune Population

To address the problem, we introduced 1/γv as the vaccine period, like 3 days of vaccine and once per day. σ is the effective probability of the vaccine, for we consider the probability of people vaccinated acquiring the resistant ability may not be 100％ but somewhat near it. Considering the people taking vaccine are either susceptibles or exposed, the population range of vaccine is S + E , while the vaccine has no effect on exposed individuals, so the effective rate is S /(S + E ). When offering vaccine, there is a limit on the capability of one site per day, which is indicated as A i /p , so the number of new immune people per day is

=i v A S Im S E p

γσ?+ (5) where A i is the medication quantity delivered in one site; p is the period of medication delivery. The properties of Im group are the same as the recovered group, thus R becomes bigger, which is one impact brought by vaccine. Simultaneously, S becomes smaller.

● Medication Demand

The medication needed in each district is depended on the epidemic stage. To be specific, a district with bigger E(t) and I(t) is on a more serious stage than those with smaller. Therefore, to achieve our objective, more medication should be distributed to districts on more serious stage. The needs for one person is set as one unit. The whole demand for medication is named as D . It is assumed that the site can equally give out all vaccine during one delivery period, while the constraint is manufacture speed, that is A i

● Vaccine Manufacture Speed

Another feature in this model is the medication supplement. Only with steady and sufficient manufacture process of vaccine, can the timely and sufficient delivery be assured.

Then the constraint mainly comes from the total quantity of medication made in one period (p ), which we set as A . As is mentioned before, there is a limited number of delivering vaccine in one site per period, which is indicated as A i and we have A i

● Medication Distribution

Our objective of this model is the medication distribution. That is, the values of A i . Our objective function is

(

)1min ()n p j j I =∑

(6)

where n is the total number of districts concerned; the values of I is used that for one period later. And we have

1 ,=1,2,,n s.t. 0n

i i i A A i D A =?=???>≥?∑

(7)

The connection between I and A i can be referred to formula (1) and (4).

Here we introduce Particle Swarm Optimization (PSO) to solve the problem, which is an intelligent algorithm used to find optimal solution. According to the formula (5), fitness value (V ) used to evaluate possible distributions is

(

)1n

p j j V I ==∑

So the objective is indicated by the fitness value, which is the sum of I for several days (a period) later.

Fig 4: Flow chart of MPSO algorithm

MPSO algorithm is illustrated as flow chart above. The modified algorithm enables faster running speed.[8] We may set proper iteration times to balance between the running time of the program and the quality of the solution. More times it updates, the better is the solution.

3.4.2 Solution and Result

We take Sierra Leone, a country with 14 districts, as a study case, so we have n = 14.

Other settings are: A = 1,000,000; p = 4; 1/γv = 3 and σ = 1. Based on epidemic model, we

calculate I in all 14 districts of Sierra Leone as the initial value:

Table 4: Initial value of I in initial value

District Kailahun Kenema Kono Bombali Kambia Port Loko Koinadugu I for 4th Feb 565 502 246 990 150 846 104

District Tonkolili Bo Bonthe Moyamba Pujehun Freetown

Rural Area I for 4th Feb 448 314 5 205 31 2029 1118 The initial values are extracted from the last day’s real data. So we get results:

Fig 5: Pie chart of vaccine distribution in Sierra Leone

The pie chart illustrates how vaccine should be delivered to each district. Theoretically, according to this plan, four days later, the sum of I should be a relevant small number. The results may not reach the overall best solution, for the iteration times are limited and the new plans generated are basically random. And the unit of quantities of vaccine is one share for one person at a time. For example, the optimal quantities delivered to West Rural Area is 209508 shares, which tops among all other districts of Sierra Leone. It is mainly resulted from the largest density of infected population there, that is the infected stage of the district.

Fig 6: Influence of vaccine on I in Sierra Leone

The bar chart shows there is a clear connection between I and vaccine quantities. And when offered more vaccine, I tends to become smaller, thus the vaccine has certain effect on reducing infected population.

3.4.3 Analysis of the Result

In the part of ‘Medication Demand’, we mentioned that there is a relation between the districts’ epidemic stage and the quantities of vaccine for each dis tricts. Comparing figure 5 and table 4, it is obvious that our results is reasonable. Bombali, Port Loko, Freetown and Western Rural Area, all has the large number of I, and large quantities of vaccine as well. Bontheand Pujehun, which with the lowest number of I, also have the lowest number of vaccine.

The initial value of this model are extracted from the statistical data on WHO website. From the statistical data of Sierra Leone, we could draw a conclusion that the epidemic stage of Sierra Leone is serious now. So that the influence of vaccine maybe more effective on the early epidemic stage.

3.5 Medication Delivery Model

In distribution model we ascertained the medication distribution for every district. If the departure location is confirmed, a medication delivery system is therefore developed. We build a delivery model to get an optimal plan of transporting corresponding amount of medication to each district.

Departure Locations

To address the meditation delivery system, we have to determine the locations for material receiving. A proper pick of locations will definitely benefits the delivery system.

Our model is based on real world situation, so we choose international airports for medication transportation. The airports receive the supply from medication manufacture sites, and send them to the capitals of each district to meet the optimal distribution plan.

Here we ignored the delivery process among manufacture sites and airports. Because the manufacture sites are located in other continents, the delivery system is affected by many unknown factors, which makes the problem rather complicated.

Transportation System

Now the sending sites and the receiving sites are determined, so are the quantities needed in each receiving sites. The optimal delivery system should has the shortest total travelled distance. It is described as our model objective:

11min ()n m ij ij i j c d ==∑∑

(8)

and

1

1

s.t. 0n

ij j

i m

ij i j ij

c y c x c ==?=???=???≥??

∑∑ (9) where i identifies different airports; j identifies different districts; d ij is the distance between airport i and district j ; x i is the quantities of medication sent from airport i ; y i is the quantities of medication received by district j ; c ij is the quantities of medication delivered form airport i to district j .

The solution should be a matrix of c ij , which describes the whole delivery system.

3.5.1 Solution and Result

There are two international airports in Sierra Leone: SherBro International Airport and Lungi International Airport. We acquire the geographic coordinates of them, alone with the information of capitals of each district. So the distances among them can be calculated. Distance(km) n ma Kono Bombali a Loko gu

SherBro International Airport

229.6 151.2 210.5 158.9 181.6 140.5 245.8 Lungi International Airport

290.8 235.5 244.6 129.9 63.51 47.86 210.5

Distance(km) Tonkoli

li

Bo

Bonth

e

Moyam

ba

Pujehu

n

Freeto

wn

Rural

Area

SherBro International

Airport

146.2 97.89 1.59 70.52 90.51 132 107.6 Lungi International

Airport

138.1 176.2 143.1 97.98 215.1 15.28 34.52 The airports are only for sending out vaccine, and the capitals are only for receiving materials. So the distances needed exclude those among capitals.

We implement Lingo to solve the problem, and we get the delivery system as follows:

Table 6: Delivery system

Vaccine delivery amount Kailahu

n

Kene

ma

Kono Bombali

Kambi

a

Port

Loko

Koinadu

gu

SherBro International

Airport 60240 44925

4971

5

161781 0 0 22859

Lungi International

Airport

0 0 0 0 2842 137868 2815

Vaccine delivery amount Tonkoli

li

Bo

Bonth

e

Moyam

ba

Pujehu

n

Freeto

wn

Western

Rural

Area

SherBro International

Airport 54310 35045

1449

5

50093 1518 0 0

Lungi International

Airport

0 0 0 0 0 141948 209508 The table above indicates our final plan for Sierra Leone. Theoretically, the plan produced by the algorithm is the best solution of the problem.

IV. Sensitivity Analysis

4.1Influence of β

βI refers to the possibilities of susceptibles getting Ebola from infectious individuals, with βQ and βF meaning from quarantined individuals and funeral respectively. Although these three parameters all refer to the effective contact rate, the influence power on epidemic model of each other is different, because of the different infection source and other factors. Now we give a sensitivity analysis by changing the value of these parameters, then we get the evaluation of each contact rate.

Continuing our work on Sierra Leone and comparing with the results in first model, we reduce 0.2 in βI, βQ and βF respectively and observe the change under three circumstances.

4.1.1 Influence of βI

The value of βI in SEIQFR model decreases by 0.2, while other parameters are unchanged. We use MATLAB to draw the prediction curves with changed and unchanged parameters, the figure is as follows:

Fig 7: The spread situation with βI changed

It is obvious that the value of peak decreased by 100 thousands approximately, and the arrival time of peak is delayed about 3 months after changing the value of βI. The decrease of peak means that the epidemic situation is controlled effectively with the decrease of infection rate of I. And the delay of peak time gives us more chance to take other measures to control the spread of Ebola.

4.1.2 Influence of βQ

With only the value of βQ decreasing by 0.2, we compare the result and that in model one.

Fig 8: The spread situation with βQ changed

We find that the curve tendency is similar with former one. The difference between the values of two peaks is about 5 thousands, and the time delayed is 50 days.

4.1.3 Influence of βF

With only the value of βF decreasing by 0.2, we compare the result and that in model one.

Fig 9: The spread situation with βF changed

The difference between curve peak is about 2.5 thousands, and the time delayed is about 20 days.

4.1.4 Analysis of results

The dip of peak value between changed and former ones is due to a change of I. And the difference between peak time of changed and former curves means the deadline for we to take controlling measures. Both the peak value dip and the peak time delay are the bigger the better. So we can draw the conclusion that the influence of βF is the least, while βI is the greatest, which means that the contact rate between S and I is the most impactful parameter for controlling the epidemic.

4.2 Time begin intervention

When the outbreak happens, measurements will be took by the government. We focus on the intervention time of quarantine. To compare the difference caused by various intervention time, we simulate three situations with the same level of quarantine but only at different moment. And one case without intervention for reference.