The initial code for model SIR was based on the work published at gitHub https://github.com/Lewuathe/COVID19-SIR. The author has a webpage of the project https://www.lewuathe.com/covid-19-dynamics-with-sir-model.html All the copyrights for the parts of the code are due to Kai Sassaki https://github.com/Lewuathe. As needed, we are using Apache 2.0 license also.
The code was modified to include data analysis a new models SEAIR-D was developed, which is very new and original with time delays, deaths and other constants. A conversion to a Jupyter Notebook was made and three other codes were included: one for initial conditions optimization by evolutionary algorithm, other for ploting results in a map and last one to see the evolution of evolutionary calculation.
The project has three codes for Covid-19 infection in:
- countries at
./countries - Brazilian states at
./statesBrazil - district regions of Sao Paulo State, Brazil at
./regionsSP
After some time, and sucess, the Institute of Technologies of Sao Paulo State (http://www.ipt.br) modified the code and used it to predict the demand for respirators and hospitals for Sao Paulo State, Brazil. The modified code, a Jupyter Notebook, can be accessed in subfolder ./regionsSP. It calculate infected, recovered and deaths for all administrative regions of Sao Paulo State. The final result can be seen at http://covid19.ats4i.com
It was also developed to predict Covid-19 infection for Brazilian States. You can see that at Jupyter Notebook at ./statesBrazil
If you have interest, ATS (www.ats4i.com) has some posts about Covid-19 pandemics modeling.
This is a open source contribution for the comunity. Please use and contribute!
Clone this repository
git clone https://github.com/gasilva/dataAndModelsCovid19.git
Or use GitHub Desktop https://desktop.github.com/ and File, Clone repository
The project has some codes for Covid-19 infection in countries:
countries_Covid19_v7.ipynb: main codeinitialConditions_YaboxNew_v6.ipynb: code to find initial conditions for the main codehistoryIC.ipynb: plot optimization history while code of initial conditions is runplotMaps.ipynb: plot results in a map (under development)
The Brazlian States and Sao Paulo region follows the same logic.
The countries and initial parameters file is data/param.csv
| country | start-date | range | s0 | e0 | a0 | i0 | r0 | d0 | START | WCASES | WREC | WDTH |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Brazil | 3/2/20 | 200 | 10e6 | 1e-4 | 1e-4 | 200 | 100 | 50 | 50 | .15 | 0.05 | 0.8 |
| China | 1/28/20 | 200 | 50e3 | 1e-4 | 1e-4 | 200 | 100 | 50 | 50 | .15 | 0.05 | 0.8 |
| Italy | 2/28/20 | 200 | 2e6 | 1e-4 | 1e-4 | 200 | 100 | 50 | 50 | .15 | 0.05 | 0.8 |
| US | 2/20/20 | 200 | 20e6 | 1e-4 | 1e-4 | 200 | 100 | 50 | 50 | .15 | 0.05 | 0.8 |
| India | 3/10/20 | 200 | 10e6 | 1e-4 | 1e-4 | 200 | 100 | 50 | 50 | .15 | 0.05 | 0.8 |
The optimized initial conditions are taken from the file data/param_optimized_Yabox_HistMin.csv
| country | start-date | range | s0 | e0 | a0 | i0 | r0 | d0 | START | WCASES | WREC | WDTH |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Brazil | 03/03/20 | 200 | 32509903 | 0 | 0 | 71 | 477 | 329 | 286 | 0.4003 | 0.3898 | 0.1397 |
| China | 01/29/20 | 200 | 674008260 | 0 | 0 | 200 | 100 | 50 | 50 | 0.3926 | 0.4784 | 0.4224 |
| Italy | 02/29/20 | 200 | 525450 | 0 | 0 | 129 | 71 | 157 | 302 | 0.2731 | 0.2884 | 0.2461 |
| US | 02/19/20 | 200 | 23452805 | 0 | 0 | 135 | 167 | 472 | 194 | 0.7096 | 0.1996 | 0.3107 |
| India | 03/10/20 | 200 | 37519697 | 0 | 0 | 265 | 362 | 81 | 128 | 0.9216 | 0.3379 | 0.4319 |
The Brazlian States and Sao Paulo region follows the same logic.
SEAIR-D - Original variation, proposed by this author, from SEIR model for susceptible, exposed, asymptomatic, infected and deaths
- S(t) are those susceptible but not yet infected with the disease
- E(t) are those exposed to the virus
- A(t) is the number of cases asymptomatic
- I(t) is the number of infectious individuals
- R(t) are those individuals who have healed
- K(t) are those individuals who have died
: Effective contact rate [1/min]
: Fraction of effective contact rate [1/day]
: Recovery(+Mortality) rate γ=(a+b) [1/min]
: recovery of healed [1/min]
: mortality rate [1/min]
: is the rate at which individuals move from the exposed to the infectious classes. Three parameters
,
and
are used along the days. The transition from
to
and from
to
are made by a sigmoid function. Its reciprocal
is the average latent (exposed) period. [1/day]
p: is the fraction of the exposed which become symptomatic infectious sub-population.
(1-p): is the fraction of the exposed which becomes asymptomatic infectious sub-population.
: medicine and hospital improvement rate [1/day]
: direct mortality rate [1/day]
The inclusion of asymptomatic cases in SEIRD model generates SEAIRD equation system:
The last equation does not need to solved, because
The original σ is divided in three factors σ, σ0 and σ01 along time. They are linked by a sigmoid function to smooth the non-linearity.
How the final σ is calculated. It is a linear combination of two σ values. There is a development of 4 days to change from one value to another. The optimization also finds the startTdate to make the jump in addtion to the two values of σ.
sigma=sg.sigmoid2(t-startT,t-startT2,sigma0,sigma01,sigma02,t-int(size*3/4+0.5))
How the sigmoid function is calculated. Numba package compiles the function and make it faster. The parallelization does not work but a LRU Cache is used to save memory and used it to evaluate repeated calculations.
from functools import lru_cache
from numba import njit
import math
import numpy as np
@njit #(parallel=True)
def sigmoid(x,vari,varf):
rx=1 / (1 + math.exp(-x))
return vari*(1-rx)+varf*rx
@lru_cache(maxsize=None)
@njit
def sigmoid2(x,xff,vari,varf,varff,half):
if half<=0:
return sigmoid(x,vari,varf)
else:
return sigmoid(xff,varf,varff)
The function format has this shape by considering σ0=0 and σ01=1 or σ0=1 and σ01=0 with startT=5.
It is a new completely development model but inspired on the paper below. However, it does not have same equations and parameters:
https://www.hindawi.com/journals/ddns/2017/4232971/#references
and
The mathematical Models are based in Lotka-Volterra equations, it is like a predator-prey type of model.
Source: https://triplebyte.com/blog/modeling-infectious-diseases
A simple mathematical description of the spread of a disease in a population is the so-called SIR model, which divides the (fixed) population of N individuals into three "compartments" which may vary as a function of time, t:
- S(t) are those susceptible but not yet infected with the disease
- I(t) is the number of infectious individuals
- R(t) are those individuals who have recovered (dead+healed)
β : Effective contact rate [1/min]
γ: Recovery(+Mortality) rate [1/min]
The SIR model describes the change in the population of each of these compartments in terms of two parameters, β and γ. β describes the effective contact rate of the disease: an infected individual comes into contact with βN other individuals per unit time (of which the fraction that are susceptible to contracting the disease is S/N). γ is the mean recovery rate: that is, 1/γ is the mean period of time during which an infected individual can pass it on.
The differential equations describing this model were first derived by Kermack and McKendrick [Proc. R. Soc. A, 115, 772 (1927)]:
Here, the number of 'recovery' englobes both recovered and deaths. This parameter is represented by γ.
The SIR model code is based on
https://www.lewuathe.com/covid-19-dynamics-with-sir-model.html
https://github.com/Lewuathe/COVID19-SIR
https://www.kaggle.com/lisphilar/covid-19-data-with-sir-model
https://triplebyte.com/blog/modeling-infectious-diseases
The γ is split in two by γ = a + b, where a is the rate of recoveries, and b is the rate of death. Since the death rate seems to be linear (1.5% in China, for example), this linear decomposition of γ is precise enough.
Some facts about SIR model:
- The number of Susceptible individuals can only decrease
- The number of Recovered can only increase
- The number of Infectious individuals grows up to a certain point before reaching a peak and starting to decline.
- The majority of the population becomes infected and eventually recovers.
The the Python notebook of
https://www.kaggle.com/lisphilar/covid-19-data-with-sir-model#Scenario-in-Italy
A well posed explanation about SIR Model is given by:
So we can add a new variable k, (Kill rate), and add to the system of equations. Therefore:
- S(t) are those susceptible but not yet infected with the disease
- I(t) is the number of infectious individuals
- R(t) are those individuals who have healed
- K(t) are those individuals who have died
β : Effective contact rate [1/min]
γ: Recovery(+Mortality) rate γ=(a+b) [1/min]
a: recovery of healed [1/min]
b: mortality rate [1/min]
The last equation does not need to solved, because
The SIR-D model code is based on the contribution of Giuliano Belinassi, from IME-USP, Brazil
Lewuathe/COVID19-SIR#13 (comment)
Source: https://triplebyte.com/blog/modeling-infectious-diseases
- S(t) are those susceptible but not yet infected with the disease
- E(t) are those exposed to the virus
- I(t) is the number of infectious individuals
- R(t) are those individuals who have recovered (deads+healed)
β : Effective contact rate [1/min]
γ: Recovery(+Mortality) rate γ=(a+b) [1/min]
σ: is the rate at which individuals move from the exposed to the infectious classes. Its reciprocal (1/σ) is the average latent (exposed) period.
The last equation does not need to solved, because
About SEIR models:
https://idmod.org/docs/malaria/model-seir.html
Codes from the book of Modeling Infectious Diseases in Humans and Animals Matt J. Keeling & Pejman Rohani, Chaper 2.6, SEIR model
http://homepages.warwick.ac.uk/~masfz/ModelingInfectiousDiseases/Chapter2/Program_2.6/index.html
This code for countries has data from Repository by Johns Hopkins CSSE
https://github.com/CSSEGISandData/COVID-19
For Brazilian States and Sao Paulo State Regions the data is from Brazil.io
https://data.brasil.io/dataset/covid19/_meta/list.html
Anderson, R. M., May, R. M. , Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1991
Cotta R.M., Naveira-Cotta, C. P., Magal, P., Modelling the COVID-19 epidemics in Brasil: Parametric identification and public health measures influence medRxiv 2020.03.31.20049130; doi: https://doi.org/10.1101/2020.03.31.20049130
De la Sen, M., Ibeas, A., Alonso-Quesada, S.,Nistal, R., On a New Epidemic Model with Asymptomatic and Dead-Infective Subpopulations with Feedback Controls Useful for Ebola Disease, Discrete Dynamics in Nature and Society, Volume 2017, Article ID 4232971, 22 pages, Special Issue, Epidemic Processes on Complex Networks, https://doi.org/10.1155/2017/4232971
Glaser, A., Dynamics and Control of Infectious Diseases, Lecture, WWS556d, Princeton University, April 9, 2007. http://www.princeton.edu/~aglaser/lecture2007_diseases.pdf
Hamzaha, F.A.B.,Laub, C.H., Nazric, H., et al. CoronaTracker: World-wide COVID-19 Outbreak Data Analysis and Prediction CoronaTracker Community Research Group, [Submitted]. Bull World Health Organ. E-pub: 19 March 2020. doi: http://dx.doi.org/10.2471/BLT.20.255695
Keeling, M.J., Rohani, P., Modeling Infectious Diseases in Humans and Animals, Princeton University Press, Release Date: September 19, 2011, Princeton University Press, ISBN: 9781400841035 https://www.kobo.com/us/en/ebook/modeling-infectious-diseases-in-humans-and-animals Python, C++, Fortran, Matlab codes availables at: http://homepages.warwick.ac.uk/~masfz/ModelingInfectiousDiseases/index.html
Kermack William Ogilvy and McKendrick A. G. 1927, A contribution to the mathematical theory of epidemicsProc. R. Soc. Lond. A115700–721 http://doi.org/10.1098/rspa.1927.0118
Prem, K., Liu, Y, Russell, T.W. et al, The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China: a modelling study,The Lancet Public Health,2020,ISSN 2468-2667, https://doi.org/10.1016/S2468-2667(20)30073-6 and http://www.sciencedirect.com/science/article/pii/S2468266720300736
Trawicki, M. B., Deterministic Seirs Epidemic Model for Modeling Vital Dynamics, Vaccinations, and Temporary Immunity, Mathematics 2017, 5, 7; https://doi.org/10.3390/math5010007
William Ogilvy Kermack, A. G. McKendrick and Gilbert Thomas Walker 1997A contribution to the mathematical theory of epidemicsProc. R. Soc. Lond. A115700–721 https://doi.org/10.1098/rspa.1927.0118