Pandæsim: An Epidemic Spreading Stochastic Simulator
Abstract
:Simple Summary
Abstract
1. Introduction
2. Materials and Methods
2.1. Overview
- Based on the quantities , the rules and their kinetics, compute stochastically at what time each reaction is triggered .
- Let being the next reaction: .
- Apply ; i.e., update the vector by decreasing the quantities of the substrates of and increasing the quantities of its products.
- Update the time: .
2.2. Pandæsim Model
2.3. Simulation Data and Parameters
2.4. Evolution Algorithm
- Soft quarantine: People do not use public transportation at all and do not go to restaurants during the midday break.
- Full quarantine: This corresponds to what actually happened in France; people were confined at home except for a one hour stroll per day in low populated areas (public parks, forests, etc., were forbidden). Again, to reduce the number of parameters, we assumed that the probability of contagion during the stroll was the same as at work. This also allowed us to take into account errands made to get food in more populated places such as groceries or supermarkets.
- First, the infection rate at time t, , is computed as the product of the global daily rate of infection, , by the infection factor of the current location (home, workplace, public transportation) . This infection rate is used the same way the propensity is in the standard SSA.
- Then, for each of the four age slices the deterministic continuous solver computes the average number of individuals of that age that will go from susceptible to asymptomatic state, , as the product of the population in that state and the infection rate at time t:The stochastic discrete solver (SDS) computes stochastic integer numbers such that, on the long run, they will average to the same values as the continuous solver. Even when the population is an integer number of individuals, this product, , is generally a floating point number because the infection rate is itself a floating point number. This number has an integral part (≥0) and a fractional part (between 0 and 1). The (discrete) number of new asymptomatic hosts is then computed as the integer part of the average number, plus 1 if a uniform random number taken into the interval … is below the fractional part:As the difference is 0.5 on the average, the higher the value is, the lower the relative impact of this stochastic discretisation becomes and the result is equivalent to a discrete averaged approach. Conversely, the lower the value is, the more important the stochastic discretisation becomes. This mechanism allows the simulator to automatically choose the best strategy to adapt to the value range of the population [26].
- Finally, when the current time indicates the beginning of a new day, (mod 24), individuals in each state either remain in the same state but shifted by one day, or change to another state. All the states transitions are computed stochastically by the SDS (or deterministically by the DCS) using the method described earlier.
- The population in the asymptomatic state that has on average reached the 5/6 day limit is moved to the first day of the ill state.
- According to the illness duration by age slice parameter, a proportion of the population in the ill state is moved to the hospitalised or to the recovered state. The others remaining in the ill state one more day.
- According to the disease severity by age slice parameter, a proportion of the population in the hospitalised state is moved to the deceased or recovered state. The others remain in the hospitalised state one more day.
The global daily rate of infection is then simply computed by multiplying the constant of propagation of the virus, , by the proportion of the total contagious population:By fitting the simulation results after the beginning of the lockdown to the data gathered from hospital statistics, we empirically found a good estimation of for the SARS-CoV-2 to 0.75. We think that using Pandæsim to model another type of epidemic, only this constant, along with the severity parameters, needs to be changed.
3. Results
4. Discussion
5. Conclusions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
ICU | Intensive Care Unit |
SSA | Stochastic Simulation Algorithm |
ODE | Ordinary Differential Equations |
SDE | Stochastic Differential Equations |
WHO | World Health Organization |
DCS | Deterministic Continuous Solver |
SDS | Stochastic Discrete Solver |
Appendix A
Appendix A.1.
Région | Départements | |||||||
---|---|---|---|---|---|---|---|---|
Ile de France | Paris | 65 | Val-de-Marne | 180 | Val-d’Oise | 70 | Yvelines | 20 |
Seine-Saint-Denis | 140 | |||||||
Hauts de France | Somme | 10 | Nord | 40 | Oise | 10 | Aisne | 10 |
Normandie | Seine-Maritime | 10 | ||||||
Bretagne | Morbihan | 2 | Ille-et-Vilaine | 5 | ||||
Pays de Loire | Loire-Atlantique | 8 | Maine-et-Loire | 8 | ||||
Centre | Eure-et-Loir | 6 | Loiret | 2 | ||||
Aquitaine | Charente-Maritime | 2 | Gironde | 6 | ||||
Occitanie | Haute-Garonne | 2 | Hérault | 10 | Gard | 1 | ||
Corse | Corse-du-Sud | 2 | ||||||
Provence Alpes Cote d’Azur | Bouches-du-Rhône | 40 | ||||||
Auvergne Rhone Alpes | Rhône | 60 | Ardèche | 3 | Drôme | 3 | Loire | 4 |
Bourgogne Franche-Comté | Territoire de Belfort | 3 | Côte-d’Or | 30 | Doubs | 2 | ||
Grand Est | Meuse | 1 | Moselle | 70 | Bas-Rhin | 20 | Haut-Rhin | 60 |
Age | Lethality % | Illness Duration (days) | Local Travel % | Remote Travel % |
---|---|---|---|---|
0–25 | 10 | 7 | 5 | 1 |
26–50 | 15 | 8 | 6 | 1 |
51–75 | 20 | 10 | 6 | 1 |
76+ | 55 | 14 | 0.5 | 0.4 |
Location | Rate % |
---|---|
Home | 0.02 |
Workplace | 2 |
Public transportation | 4 |
Contagious patients % | 1 |
Severe form % | 20 |
Propagation constant | 0.75 |
Region | 0–25 | 26–50 | 51–75 | 76+ |
---|---|---|---|---|
Ile de France | 3,164,218 | 4,177,466 | 2,982,661 | 683,650 |
Hauts de France | 1,597,206 | 1,846,011 | 1,477,121 | 418,273 |
Normandie | 917,615 | 808,834 | 926,890 | 318,070 |
Bretagne | 733,777 | 868,726 | 1026,010 | 300,503 |
Pays de Loire | 1,052,858 | 958,431 | 1,150,747 | 268,259 |
Centre | 652,748 | 617,614 | 717,309 | 256,969 |
Aquitaine | 1,483,728 | 1,435,736 | 1,869,051 | 661,315 |
Occitanie | 1,471,676 | 1,527,461 | 1,873,453 | 594,186 |
Corse | 69,362 | 96,471 | 91,685 | 40,289 |
Provence Alpes Cote d’Azur | 1,270,520 | 1,185,877 | 1,562,820 | 473,619 |
Auvergne Rhone Alpes | 2,152,246 | 2,272,047 | 2,202,878 | 693,612 |
Bourgogne Franche-Comté | 572,106 | 719,857 | 811,137 | 291,031 |
Grand Est | 1,403,834 | 1,556,258 | 1,567,738 | 446,914 |
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Sample Availability:Pandæsim can be downloaded on its dedicated website: https://pandaesim.lri.fr. |
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Amar, P. Pandæsim: An Epidemic Spreading Stochastic Simulator. Biology 2020, 9, 299. https://doi.org/10.3390/biology9090299
Amar P. Pandæsim: An Epidemic Spreading Stochastic Simulator. Biology. 2020; 9(9):299. https://doi.org/10.3390/biology9090299
Chicago/Turabian StyleAmar, Patrick. 2020. "Pandæsim: An Epidemic Spreading Stochastic Simulator" Biology 9, no. 9: 299. https://doi.org/10.3390/biology9090299