An Agent-Based Modelfor optimizing strategies against

contagious disease spread

Daniele Carli

June 28, 2016

1

Contents

1 Introduction 31.1 The Epidemic Models for Spread of Diseases . . . . . . . . . . 31.2 ABM variations on the deterministic models . . . . . . . . . . 31.3 Different Strategies . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.1 First the infected: priority to cure . . . . . . . . . . . 41.3.2 Prevention is better than cure: priority to immunization 41.3.3 Follow my orders: enable percentage . . . . . . . . . . . 5

1.4 For a fistful of Dollars: Budget and total costs . . . . . . . . . 51.5 A social problem: the Anti-Vaccinists . . . . . . . . . . . . . . 5

2 The Code 62.1 The setup button . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 The go button . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 The civilian’s duties . . . . . . . . . . . . . . . . . . . 72.2.2 The behaviours of the doctors . . . . . . . . . . . . . . 92.2.3 The model’s update . . . . . . . . . . . . . . . . . . . . 10

3 The Results 113.1 The Economic Impact . . . . . . . . . . . . . . . . . . . . . . 113.2 The social impact . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 The anti-vaccinists effect . . . . . . . . . . . . . . . . . . . . . 163.4 Segregation? The get-away effects . . . . . . . . . . . . . . . 193.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Further Applications 224.1 Improve the model . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Analyze real Data . . . . . . . . . . . . . . . . . . . . . . . . . 22

2

1 Introduction

1.1 The Epidemic Models for Spread of Diseases

By modeling the infectious diseases, it’s possible to predict how they willspread and their future course: this allows to evaluate ad hoc strategies andcontrol the epidemic spreading. A predominant method to model them is toregroup population into compartments (which represents their health status)by labeling them with a letter. The simplest case is the SI model, in whichthere are only two possible states: Susceptible to infection (S) and Infected(I).The number of elements in every class may vary during time, because theEpidemic Models are dynamic and they evolve during time according todifferential equations which determine the number of transitions from onecompartment to another.For the correct specification of the model, the equations must contain thetransition rates between compartments. For example: between S and I, thetransition rate is a parameter which takes into account the probability of be-come infected by contact between a Susceptible and an Infected. There aremany other variants, produced by adding other conditions, each with differ-ent features and different related compartments as Recovered (R), Exposed(E), etc...In this article we will develop and analyze a stochastic variant of the SEIRSdeterministic model with an Agent-Based Model, to evaluate the best strate-gies to contain costs and control the disease spread.The code will be described in the next section.

1.2 ABM variations on the deterministic models

The principal difference between a deterministic model and an Agent-BasedModel is that deterministic ones are valid only in case of sufficiently largepopulations, whereas for small populations the stochastic fluctuations can berelevant and can determine the oubreak or the lapse of the disease.To create this model we used NetLogo, a free, open-source multi-agent pro-grammable modeling environment, so as to introduce the stochastic be-haviour. This model therefore doesn’t evolve according to differental equa-tions, but according to the actions specified for the agents.In this model there are only two agents: civilians (who can be sick and by

3

which the disease spreads) and doctors (who cannot be infected, but theycan cure or immunize civilians). Both of them moves randomly around theworld: civilians spreading the infection, doctors damming it.Every Susceptible civil who gets infected, becomes Exposed: he can transmitthe infection, but his infectiousness is at a low rate because he is not symp-tomatic, so doctors cannot realize he is sick. After a short period (10 days),he became Infected (symptomatic): his infectiousness is higher than beforeand doctors will be able to locate and recover him. If the civil is Recovered,his infectivity will drop drastically and he cannot be able to move aroundthe world. After the recover period (set to 10 days), he will turn Susceptible.If a civil will be not recovered, exceeded the illness duration there will bea chance for him to heal spontaneously, or he will die. Instead, if a civil isVaccinated, he cannot turn sick during the entire duration of vaccination.

1.3 Different Strategies

The main objective of this work is to find the best strategy to vanquish thedisease, both in Economic and Social terms.To do this, the Model provide three different doctors strategies and the usercan choose which one will be used, selecting them by the behaviour chooser.Doctors will follow the instructions.

1.3.1 First the infected: priority to cure

With this choice, doctors will first cure the symptomatic civilians they en-counter, instead of immunizing. Specifically, they will recover one civil at atime, and if there are more than one symptomatic civil around them, theywill recover the civil who is sick since more time.If there are no symptomatic civilians around them, doctors will immunizeone random civil around them.

1.3.2 Prevention is better than cure: priority to immunization

The second strategy is cruel: doctors will first immunize, recovering only incase there are no healthy civilians around them. Symptomatic people will benot taken into account until there will be only symptomatic civilians aroundthe doctor, so he will recover the one who is sick since more time.

4

1.3.3 Follow my orders: enable percentage

This strategy lets the user to choose his favorite behaviour: enables thepercentage-of-cure slider, allowing the user to select with which probabilitydoctors will cure or immunize. Doctors will cure (if possible) according tothe probability specified by the slider and they will immunize in the othercases.

1.4 For a fistful of Dollars: Budget and total costs

Since the model’s target is to find the best strategy also in economical terms,all the actions are submitted to the condition of a daily budget (septable bythe user with the daily-budget slider): if a doctor has no sufficient remainingbudget to implement an action, he will do nothing.Every action has a different cost: immunization has a uniquely cost, whereasthe cure has a cost for every day in which the civil is recovered (that includesmedicines and hospitalization costs). The individual cost for the immuniza-tion is septable by the user with the immunization cost slider, whereas thedaily cost through the cure cost slider.

1.5 A social problem: the Anti-Vaccinists

Last purpose of this work is to understand if (and how) the Anti-vaccinistscan affect an epidemic spreading. To do so, we implemented for every civila random evenly distributed opinion about the vaccine. Civilians can prop-agate their opinion and persuade other people: anyone who has a certainopinion (over a fixed thereshold) about the vaccine will turn into an anti-vaccinist and he will refuse the vaccine.The anti-vaccinists scenario can be choose or not by the Anti-vaccinist switcher.

5

2 The Code

2.1 The setup button

Now, we have to populate the world: clicking on the setup button, all theprevious simulations will be blanked and a new simulation will be setted.First of all the model will set up the constants, then the agents will becreated: as many civilians as specified in the number-people slider, everyonein a random position.The user can choose how many of them will be Exposed with the initial-sick slider, the others will be healthy and Susceptible; then, the model willcreate a number of doctors equal to as set by the number-doctors slider,also in a random position. Then the global variables and the display will beupdated: civilians and doctors (and anti-vaccinists, in the case) will be easilydistinguishable by their shape; the civilian health will be distinguishable bythe different color.

(a) (b) (c) (d) (e) (f) (g)

Figure 1: different shapes and colors that represents the species and thecivilian’s status of health. (a) Susceptible (b) Exposed (c) Infected (d) Re-covered (e) Vaccinated (f) Doctor (g) Anti-vaccinist.

2.2 The go button

When the environment is set, the simulation can start. The go button liststhe actions actuable by civilians and doctors at every tick, that represents aday.The entire code of the go button is long, so it is divided according to themain sections.

6

2.2.1 The civilian’s duties

The go button starts with the list of the actions of the civilians:

to go

ask civilians [

get-older

if sick? [infect]

if not recovered?

[ifelse Get-away

[ ifelse not symptomatic? [get-away!][move]]

[ move] ]

if Against-vaccine [change-opinion] ]

Civilians will first perform the get-older action: if a civil is vaccinated (Im-mune), his remaining time of immunity will decrease one per tick, whereas ifhe’s sick (Exposed or Infected) he will try to infect some other civil and thetime since he become sick will be updated. When his sick-time will reach 10days he will turn Infected (and symptomatic, so his infectiousness will reach80%) and then also his symptomatic-time will be updated at every tick, untilit reaches the maximum duration of the illness: at this time, the civil willheal spontaneously or he will die, acordingly to the probability set in thechance-heal slider.If a civil is recovered, than his recover-time will decrease one per tick and,when reached zero, he will get healthy and susceptible. Endlessly, if the car-rying capacity of the model is not reached, every non-recovered civil will tryto reproduce with a probability set to 0.1%.

to get-older

if immune? [ set remaining-immunity remaining-immunity - 1 ]

if sick? [ set sick-time sick-time + 1

if sick-time > 10 [get-symptomatic] ]

if recovered?

[set recover-time recover-time - 1

if recover-time = 0

[get-healthy

set recovered? false] ]

if symptomatic?

[set symptomatic-time symptomatic-time + 1

if symptomatic-time > duration [heal-or-die] ]

reproduce

end

After that, sick civilians will move around the world, trying to infect someoneelse: if they are Exposed (then infected but not symptomatic) they can

7

transmit the infection with a probability as specified in the infectiousnessslider, but only to healthy and not immune civilians who are in contact withthem (on the same patch). Instead, if they are symptomatic (Infected), theycan infect other civilians in a radius of 1 patches around them, this time witha probability of 80% (set as constant in the code).Obviously, if a civil is already sick or recovered or immune, he can’t getinfected.

to infect

ifelse symptomatic? [if random-float 100 < infectivity

[ask other civilians in-radius 1 with

[not sick? and not immune? and not recovered?]

[get-sick]]]

[if random-float 100 < infectiousness

[ask other civilians-here with

[not sick? and not immune? and not recovered?]

[get-sick]]]

end

Furthermore, if the Get-away switcher is turned on, non-symptomatic peoplewill perform the get-away! function and they will flee from the symptomaticpeople they encounter. This behaviour is impemented as a choice and notdirectly in the code because it can produce a strong impact on the diseasespread, as we shall see later.

to get-away!

ifelse count civilians in-radius 2 with [symptomatic?] != 0

[face one-of civilians in-radius 2 with [symptomatic?] right 180 fd 1]

[move]

end

Instead, if the against-vaccine switch is turned on, the model contemplatesthe anti-vaccinists and an opinion is randomly evenly distributed among thepopulation as a number between 0 and 100. If the opinion number is over 80,the civil will turn into an anti-vaccinist and he will refuse the immunization;on the contrary, if the opinion number is under 20, the civil will be a vaccin-ist supporter and he will try to convince other civilians that immunizationis good.Civilians will influence each other, modifying their opinion about the vac-cine, depending on how many against-vaccine or vaccine supporters thereare around them (in a radius specified by the radius-of-opinion slider).

8

to change-opinion

if opinion < 80 and opinion > 20

[if count civilians in-radius radius-of-opinion with [opinion >= 80] + 1

>= count civilians in-radius radius-of-opinion with [opinion <= 20]

[set opinion opinion + 1] if count civilians in-radius radius-of-opinion with [opinion >= 80]

<= count civilians in-radius radius-of-opinion with [opinion <= 20] + 1

[set opinion opinion - 1]]

end

2.2.2 The behaviours of the doctors

The go button also comprehend the list of actions of the doctors. The numberof doctors in the model cannot change: they are created with the setup buttonin a number equal to that specified with the number-doctors slider and cannotget sick or die. They move randomly in the world and can cure or immunizethe civilians around them.For every tick, doctors can choose one civil around them to cure or immunizehim, depending on the behaviour selected in the behaviour chooser.If the choice is priority to cure, doctors will cure a symptomatic civil in aradius of three patches around them. If there are not symptomatic civiliansaround them, they will immunize a random civil around them.If the choice is priority to immunization, doctors will first immunize one not-symptomatic civil around them (in a radius of three patches) for every tick,and if there are only symptomatic people, they ll cure one of them.Otherwise, user can select the enable percentage option, that allows the userto set with which likelihood a doctor will cure people around him basing onthe percentage of cure slider.In any case, doctors will not act if they have not enough remaining budgetto immunize or cure (if the daily remaining budget is lower than the cure orimmunization cost).

ask doctors [

move

if remaining-budget > 0 [

if behaviour = "priority_to_cure"

[ifelse count civilians in-radius 3 with

[symptomatic? and not recovered?] > 0

[if remaining-budget >= cure-cost [cure]]

[if remaining-budget >= immunization-cost

[immunize]]]

if behaviour = "priority_to_immunization"

9

[ifelse count civilians in-radius 3 with

[not symptomatic? and not recovered?] > 0

[if remaining-budget >= immunization-cost [immunize]]

[if remaining-budget >= cure-cost [cure]]]

if behaviour = "enable_percentage"

[ifelse random-float 100 <= percentage-of-cure

[if count civilians in-radius 3 with

[symptomatic? and not recovered?] != 0

[if remaining-budget >= cure-cost [cure]]]

[if count civilians in-radius 3 with

[not symptomatic? and not recovered?] != 0

[if remaining-budget >= immunization-cost

[immunize]]]]]]

When a doctor cure some civil, the civil will stay steady (recovered) for10 days, his infectivity will drop from 80% to 10% and he will turn blue.Obviously, a doctor will cure only symptomatic civilians and immunize theothers, unless they are anti-vaccinists.

2.2.3 The model’s update

Endlessly, the go button contains commands to update display, global vari-ables and to set the daily budget available for the doctors, subtracting thedaily hospitalization costs from how much is set in the daily-budget slider.At last, there are two stopping criteria: if the disease is eradicated or themodel reach the 5th year.

update-global-variables

update-display

set remaining-budget

daily-budget - cure-cost * count civilians with [recovered?]

if not any? civilians with [sick?] [ stop ]

if (ticks > 1825) [ stop ]

tick

end

Moreover, other important parametrs that could change the dynamics of thespread can be modified by the sliders that are present in the interface. Forexample, the initial infectiousness of the Exposed civilians can be modifiedby the corrispondent slider, or the maximum duration of the Infected (symp-tomatic) state, or the duration of immunity (the vaccine duration).Furthermore, we can set the daily costs for the hospitalization an the costsfor the immunization, or also the daily budget available.

10

3 The Results

What will be the best strategy against a disease? We can highlight twoimportant target: to fight the infection optimizing the economic impact or thesocial impact. All the results can be viewed in the interface, where there aremany monitors and plots: we can view the time (in years), which percentageof civilians is infected and which is vaccinated (or which percentage refusethe vaccine, if the model admits anti-vaccinists), and how many civilians diedduring the simulation. Also, the user can see the total cost and the remainingbudget and view the trends of the differents health status and the costs forimmunization and cure, or have a look on the simulated world. This is howthe user sees the interface.

Figure 2: the model’s interface

3.1 The Economic Impact

What could be the best strategy to save money and at the same time to tryto eradicate the disease? First of all, we want to know what could be thebest trategy to save money. To do this analysis, we set the same illness andcost parameters, varying only the daily budget and the behaviour of doc-tors. We will analyze, for every new set of parameters, the results obtained

11

through five different behaviours of doctors: priority to immunization, prior-ity to cure and enable percentage with 25, 50 and 75% of percentage-of-cure.Basing on the seasonal flu parameters, we set an high infectivity (up to80%), with a probability of die for the illness set to 10% (90% of probabilityto heal spontaneously). With these parameters, we expect an endemic situ-ation where the infection never disappear and may kill some people, but ina restricted percentage. Moreover, we expect that the best economic choicewould be the immunization: is cheaper than the cure, and may preserve thepopulation to be infected also with a restricted budget, that is primarly setto 1000$ per day.

Figure 3: the priority to immunization scenario, with a daily budget of1000$

Figure 4: the priority to cure scenario, with a daily budget of 1000$

As we expected, the best scenario occurs with the priority to immunizationbehaviour: doctors have not enough budget to cure many people, so thedisease will spread and many people will die whole with high costs, whereas

12

immunization is cheaper and doctors can immunize all the civilians they can,preventing the spread and also ensuring the demographic growth. Unfortu-nately, if we analyze the data, we can see that the social impact is very high,with more than 300 deaths, also acting priorizing the cure. How can bepossible a so much high dead rate? It’s a consequence of the low budget:if we see the trend of the costs, we can notice that in the priority to curescenario the costs reach the daily threshold and here they stop despite thetrend was increasing, sign that the doctors would try to cure someone elsebut they have not enough budget. If we rise the daily budget, than we expecta decrease of the deaths and possibly an eradication of the illness. We setthe budget to 2000$, and this is what happened.

Figure 5: the priority to immunization scenario, with a daily budget of2000$

Figure 6: the priority to cure scenario, with a daily budget of 2000$

The enable percentage scenarios were not reported, because their trends andcosts are very similar to the other two scenarios: in any case we had a

13

growth of population, sign that the method is good and preserves a hugepart of health population. The illness remains endemic and we still havea huge number of dead and a high total cost: since the number of dead issimilar in both cure and immunization behaviour, the best one that mini-mize the economic impact is again the priority to immunization, that admita cheaper strategy with similar results to the priority to cure. In the curescenario, however, we can see an alternance between cure and immunization:after the first period where the illness is not yet symptomatic and the doc-tors immunize (since there are not symptomatc people to cure), than thereis an outbreak of the infection, and the correspective outbreak of the curecosts. Also in this case, despite the budget is high, doctors can’t cure all thecivilians who need it and in this period the daily cost reaches the maximum.

3.2 The social impact

When the economic side is not important and the doctors can operate in theway they prefer, we can concentrate on the social impact of the disease: ob-viously, if we let the doctors act as they want with a very huge daily budget(5000$), we can be sure that the best social strategy will be the cure.

Figure 7: the priority to cure scenario, with a daily budget of 5000$

As we can see, doctors were not limited by their budget, and if they have tocure a civil they can do it. In this way, despecting on the total costs, theillness had a lower social impact, with only 96 died instead of the over 300 ofthe immunization strategy. Which can be the best way between a low socialimpact and a low economic impact (limited budget)? The only way to im-prove the strategy is varying the parameters: to keep costs low the only wayis to lower the behaviour costs, in particular lowering the daily hosptalizaion

14

Figure 8: the priority to immunization scenario, with a daily budget of5000$

costs or lowering the duration of the hospitalization. An important resultalso can be reached increasing the vaccine duration: in this way the immu-nization will be more durable and we can rise the percentage of immunizedcivils, avoiding the disease spread. In this scenario, with an immunizationperiod of 60 days, we used a 2000$ per day budget, expecting results similarto which we had for the 40-days immunization, with no great difference ontrends.

Figure 9: the priority to immunization scenario, with a daily budget of2000$ and the vaccine duration of 60 days

We can see an important result: in this case, differently from the 40 daysscenario, results are better for the priority of cure behaviour than the immu-nization: as always immunization is cheaper, but with the priority of curewe can eradicate the illness with few deaths and at a not so high total cost,despite the trends are not so different. The main difference between the two

15

behaviour is that when the percentage of infected is low, in the firts case(immunization) doctors prefer to immunize instead of cure and the illness isnot totally defeated, whereas in the second case (cure), even if the percentageof infected is low, doctors who can’t find a symptomatic civilians, fold on theimmunization. Then, when they encounter a symptomatic, they recover him,and so the infection disappear.

Figure 10: the priority to cure scenario, with a daily budget of 2000$ andthe vaccine duration of 60 days

But why this doesn’t happened also in the ”40 days vaccination” scenario,where the disease was not eradicated? The answer is that with an higherperiod of immunization the infection could not spread so easily, so the per-centage of infected remains at a low rate, permitting to doctors to cure allthe infected a few at a time. In other words, there is not an outbreak ofpeople that needs cure, as we can see in the cost trend, and if the doctorshave to cure someone they have enough budget to do it.

3.3 The anti-vaccinists effect

Now we want to see how the anti-vaccinists can influence the disease spread.In the previous cases, the presence of anti-vaccinists was not very influential:with an high rate of infectiousness and a high probability of dying fromthe disease (10%), the civilians were easily infected and the anti-vaccinistseasily died. Instead, anti-vaccinists can have a great impact in case of anot so contagious and not dangerous disease, because they can significativelyincrease costs and deaths. To prove this we changed the disease parameters,setting the infectiousness to 30%, with a probability of dying from the diseaseof 2% (we set the chance heal slider to 98%). Since the disease is trivial, the

16

budget available to the doctors is very low, set at 200$ per day. The anti-vaccinists affects mostly the behaviours that contemplates the majority ofvaccinated, so now we analyze only the priority to immunization scenario,comparing the results between the regular case and the anti-vaccinists case.

Figure 11: the priority to immunization scenario, with infectiousness set to30 %, a daily budget of 200$, 2% mortality and without anti-vaccinists

Figure 12: the priority to immunization scenario, with infectiousness set to30 %, a daily budget of 200$, 2% mortality and with anti-vaccinists

As we can see, the results are very different: in the regular case the infectionbecome endemic but with the vaccination we prevent an outbreak; with theanti-vaccinists, instead, many civilians refuse the immunization and this leadsto an epidemic outbreak. Despite the probability of dying from the disease isvery low, we have a significative increase of deaths: 167, nearly seven timesmore than the regular scenario, also because with a low budget doctors cannotcure many people. What can happen if the probability of die from the disease

17

was 10%, as the previous simulations? We think it would be an massacre:than, we increased the daily budget to 2000$, so as to permit to the doctorsto cure, if they want.

Figure 13: the priority to immunization scenario, with infectiousness set to30 %, a daily budget of 2000$, 10% mortality and without anti-vaccinists

Figure 14: the priority to immunization scenario, with infectiousness set to30 %, a daily budget of 2000$, 10% mortality and with anti-vaccinists

Effectively, the number of dead is not so different from the regular case, onlytwo times greater, but what is really changed is the total cost: the modelwith the anti-vaccinists ended with a total cost nearly seven times greaterthan the regular one.

18

3.4 Segregation? The get-away effects

This particular behaviour was not primarly implemented in the code, be-cause we fear the results it can produce: turning on this switch, all thenon-symptomatic civilians, every time they encounter a symptomatic civil ina radius of two patches around them, will face him and move in the oppositedirection (if they encounter more than one infected they will face a randomone of them).This can likewise simulate the behaviour of the segregation, where the symp-tomatic are avoided.These are in fact the results: also with a very high infectious disease andwithout many budget, the disease disappear soon, in any case and with ev-ery behaviour adopted by the doctors.To prove that we set the daily budget at 250$, a ridiculous budget comparedto the past simulations.Here we report also the results from the simulations with the percentage ofcure set to 25%, because it was one of the scenarios with the longer durationbefore the illness disappearance.

Figure 15: the priority to immunization scenario with a daily budget of250$ and the get-away switch tured on

For the priority to immunization scenario the duration was less than halfyear, and as the percentage of cure scenario there were no spread of the ill-ness, that is suddenly defeated.The trend change only with the priority to cure behaviour: the disease canoutbreak, because the doctors begin to cure and leave out the immunization.In this case the limited daily budget doesn’t allows doctors to cure all thepeople, but anyway after some time the disease can be defeated, due to the

19

Figure 16: the enable percentage scenario with a probability of cure se to25%, a daily budget of 250$ and the get-away switch tured on

death for the infection, the doctor’s cure and due to the get-away behaviour.Because of this, this function will be no further applied.

Figure 17: the priority to cure scenario with a daily budget of 250$ andthe get-away switch tured on

3.5 Conclusions

There are many behaviours that can lead to important results: to save moneythe best one is the immunization, but it depends on the severity of thedisease and the budget available. For example, with a high fatal disease anda limited budget, the priority to immunization is a cruel choice: it means letthe symptomatic die and try to preserve the health of the other civilians.With a greater budget instead the doctors will be able to immunize and curecivilians keeping down the deaths and the costs.

20

Furthermore, is very important to try to extend the duration of vaccines,because a long lasting vaccine would lead to a greater percentage of peopleimmunized, and then all the strategies applicated would lead to good results.Endlessly, the anti-vaccinists can be a real social problem: if they can spreadtheir opinion and convince a lot of people not to vaccinate, the consequencemight be serious, also in the case of not so dangerous diseases: they can leadto a massive refuse of the vaccination, causing an increase of the deaths andthe costs due to the hospitalization.

21

4 Further Applications

4.1 Improve the model

This model is already an extension of the NetLogo Virus Model (WilenskyU., 1998).The previous model was basic, without doctors and with civilians that canonly be Infected, Susceptible or become Immune after healing from the dis-ease. Some interesting improvings of this model can be the addition of othersliders for a better control of the disease parameters, as the radius of infec-tiousness of a symptomatic civil, the recover duration and the infectiousnessof symptomatics.Another interesting variation could be making the immunity imperfect: inthis way, depending uniquely by a probability or also depending on the ageof the civil who gets vaccinated, the immunity could be not effective, or canlast for less time, or other modifications congenial to the programmer.

4.2 Analyze real Data

An important result could be to use this model to analyze real-life illnessesspreading, trying to recreate their parameters and conditions to produce atruthful simulation of them. In this way, it will be possible to adjust also thevaccine and cure costs, the recover duration and the other parameters andhave a real estimate of the best economic or social strategy for every specificdisease.

22