HIV modeling
Building a new stochastic SEIR model to deal with probability distributions
This model is going to focus on individual level disease dynamics to model the disease propagation.
It models DHS dataset which contains a homogeneous population between 15 and 49 years old.
The basic idea of this model is to have a dataframe with the number of rows equal to the population size (each individual is a row) and two columns:
- State column to describe the state of each individual (S, I, or D)
- Year column to save the day of transition of the individual into that state
- Age column to know the age of the individuals
However, the population-level rates of transmission still apply here i.e. a person goes from S → I following two points:
- the effective contact rate $\beta$, which is itself given by:
- the number of contacts the person has per unit time (given by $r$)
- the chance of an S contracting the disease from a contact with an I (given by $\rho$)
- the chance a given contact is with an I - infectious individual (the higher the number of I, the higher the chance)
This is done stochastically.
Once a person becomes I, their progression is unique to them. This progression is calculated in advance for computational reason, but it allows to use the time ditributions we want.
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!pip install plotly==4.14.3
import pandas as pd
import numpy as np
import math
import plotly.graph_objects as go
import plotly.express as px
from scipy.stats import expon
from scipy.stats import gamma
from scipy.stats import weibull_min
from numpy.random import default_rng
rng = default_rng()
import tqdm
import time
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# Need this new function for model below:
def make_df(p, num_I, num_R):
df = pd.DataFrame(np.full((p,1), 'S').T[0], columns=['State'])
df['Year'] = 0
df['Age'] = (np.random.random(p)*35+15).astype(int)
tochange=df.loc[rng.choice(p, size=num_I+num_R, replace=False),'State'].index
df.loc[tochange[0:num_I],'State'] = 'I'
df.loc[tochange[num_I:num_I+num_R],'State'] = 'R'
return df
# Modelling the decrease of beta over time
#np.random.random(size=(p,days))
#np.log(4)
j=12
over = 10
#10/np.cumsum(np.ones(100))
b1 = 0.25 # original beta = beta value before epidemic
b2 = 0.05 # end beta = beta at the end of epidemic
b2 + (b1/np.exp((j+(b1*2.9)-over+1)/(b1*27)))
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def seir_model_stoch(beta, beta2, p, num_I, num_R, years, T_Infectious, ART, control):
################################
#### Explanation of inputs ####
################################
#### As seen in SSA, beta has a starting value, but after a certain threshold (as soon as incidence or prevalence reaches a certain threshold) behaviours change and beta decreases
# beta is initial value of beta at start of epidemic (usually 0.3, but can range from 0.2 to 0.5 as seen in SSA)
# beta2 is final value (usually around 0.05)
# p is total number of individuals in population
# num_I is initial number of PLWHA in population (for simulations start with something between 1 and 10 depending on size of p)
# num_R is initial number of people deceased from HIV/AIDS
# years is number of years you want to run simulation for
# T_infectious is distribution of progression of HIV in an individual (use 'gamma' for HIV)
# ART is to emulate ART usage:
# ART == 0 means no ART
# ART == 1 means ART stops evolution of I to R but does not stop spread from I to S
# ART == 2 means ART stops both I to R, and S to I
# control sets the threshold at which beta above will decrease
# control == 0 means no control i.e. beta never decreases
# control == 1 means beta decreases once incidence is 15 per 1 thousand population
# control == 2 means beta decreases once incidence is 30 per 1 thousand population
################################
##### Set up the dataframe #####
################################
# Initialize population dataframe with data given by user
df = make_df(p, num_I, num_R)
# This variable is used to track daily value of beta if it varies over time
xxbeta=np.array([],dtype=float)
# Initialize the arrays to return
# Below are numbers of S, I, R total
S=np.array([],dtype=int)
I=np.array([],dtype=int)
R=np.array([],dtype=int)
# Below are the daily additions in S, I, R
Spd=np.array([],dtype=int)
Ipd=np.array([],dtype=int)
Rpd=np.array([],dtype=int)
# Beta values to track spread
b=beta
b2=np.array([],dtype=float)
b1=b
# Signal to initiate decrease of beta
over = 0
# signal to end transmission and deaths due to ART
art1 = 0
art2 = 0
# Stochastic model so use random values to decide on progression
rand = np.random.random(size=(p,years))
# Depending if you want exponential, gamma, or Weibull distribution for T_Infectious
# Uses distributions found on blog part 3
if T_Infectious == 'expon':
ItoR = expon.rvs(loc=0,scale=10,size=p)
elif T_Infectious == 'gamma':
ItoR = gamma.rvs(4,loc=3,scale=2,size=p)
else:
ItoR = weibull_min.rvs(2.3, loc=2, scale=20.11, size=p)
################################
####### Simulation code ########
################################
# Iterate over every day the simulation is run
for j in range(0,years-1):
# Record daily beta values
xxbeta=np.append(xxbeta, b[j])
# First we get the index of the individuals that will change state today:
# Random number tells you which 'S' have been exposed on this day
if ART < 2:
StoI_index = df.loc[(df.State == 'S') & (df.Age < 49) & (rand[:,j] < b[j]*len(np.where(df.State=='I')[0])/(len(np.where(df.State=='I')[0])+len(np.where(df.State=='S')[0])))].index
StoS_index = df.loc[(df.State == 'S') & (df.Age < 49) & (rand[:,j] >= b[j]*len(np.where(df.State=='I')[0])/(len(np.where(df.State=='I')[0])+len(np.where(df.State=='S')[0])))].index
elif ART == 2:
if art2 == 0:
StoI_index = df.loc[(df.State == 'S') & (df.Age < 49) & (rand[:,j] < b[j]*len(np.where(df.State=='I')[0])/(len(np.where(df.State=='I')[0])+len(np.where(df.State=='S')[0])))].index
StoS_index = df.loc[(df.State == 'S') & (df.Age < 49) & (rand[:,j] >= b[j]*len(np.where(df.State=='I')[0])/(len(np.where(df.State=='I')[0])+len(np.where(df.State=='S')[0])))].index
elif art2 == 1:
StoI_index = df.loc[(df.State == 'S') & (df.Age > 55)].index # cannot happen so put an impossible condition like df.Age > 55 to emulate
StoS_index = df.loc[(df.State == 'S') & (df.Age < 49)].index # anyone S under 49 will stay S
StoRem_index = df.loc[(df.State == 'S') & (df.Age >= 49)].index
# For each row, if a person has been a certain number of years in I, they will go to R (progression to AIDS and death)
# This follows ItoR variable which is either exponential or gamma distributed according to above
ItoRem_index = df.loc[(df.State == 'I') & (df.Age >= 49)].index
if ART == 0: #don't use ART
ItoR_index = df.loc[(df.State == 'I') & (j-df.Year >= ItoR) & (df.Age < 49)].index
ItoI_index = df.loc[(df.State == 'I') & (j-df.Year < ItoR) & (df.Age < 49)].index
elif ART > 0:
if art2 == 0:
ItoR_index = df.loc[(df.State == 'I') & (j-df.Year >= ItoR) & (df.Age < 49)].index
ItoI_index = df.loc[(df.State == 'I') & (j-df.Year < ItoR) & (df.Age < 49)].index
elif art2 ==1:
ItoR_index = df.loc[(df.State == 'I') & (df.Age > 55)].index # cannot happen so impossible condition
ItoI_index = df.loc[(df.State == 'I') & (df.Age < 49)].index
RtoRem_index = df.loc[(df.State == 'R') & (df.Age >= 49)].index
RtoR_index = df.loc[(df.State == 'R') & (df.Age < 49)].index
# Use indexes collected above to populate per day values
Ipd = np.append(Ipd,len(StoI_index))
Rpd = np.append(Rpd,len(ItoR_index))
# Now we use the indexes collected above randomly to change the actual population dataframe to the new states
df.loc[ItoR_index, ['State','Year']] = ['S', j]
df.loc[ItoR_index, 'Age'] = df.loc[ItoR_index, 'Age'] + 1
df.loc[ItoI_index, 'Age'] = df.loc[ItoI_index, 'Age'] + 1
df.loc[StoI_index, ['State','Year']] = ['I', j]
df.loc[StoI_index, 'Age'] = df.loc[StoI_index, 'Age'] + 1
df.loc[StoS_index, 'Age'] = df.loc[StoS_index, 'Age'] + 1
df.loc[RtoR_index, 'Age'] = df.loc[RtoR_index, 'Age'] + 1
df.iloc[ItoRem_index] = ['S', j, 15]
df.iloc[StoRem_index] = ['S', j, 15]
df.iloc[RtoRem_index] = ['S', j, 15]
# Append the S, I, and R arrays
S=np.append(S,len(np.where(df.State=='S')[0]))
I=np.append(I,len(np.where(df.State=='I')[0]))
R=np.append(R,len(np.where(df.State=='R')[0]))
# Code below for control measures to reduce beta values
if control == 1:
if (I[-1]/p > 0.006):
art1 = 1
if over == 0:
over = j
if art1 == 1:
if j > over + 15:
art2 = 1
if over != 0:
b = beta2 + (b1/np.exp((j+(b1*2.9)-over+1)/(b1*27)))
if control == 2:
if (I[-1]/p > 0.01):
art1 = 1
if over == 0:
over = j
if art1 == 1:
if j > over + 15:
art2 = 1
if over != 0:
b = beta2 + (b1/np.exp((j+(b1*2.9)-over+1)/(b1*27)))
xxbeta2 = ((S[j-1]+I[j-1])/I[j-1])*Ipd[j]/S[j-1]
b2 = np.append(b2, xxbeta2)
Ipd[0]+=num_I
Rpd[0]+=num_R
return S, I, R, Spd, Ipd, Rpd, xxbeta, b2, over
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# Define parameters for stochastic model
years = 50
p = 100000
num_E = 0
num_I = 50
num_R = 0
beta_stoch = [0.17,0.17,0.26,0.26,0.36,0.36]
#beta_stoch = np.linspace(0.2,0.5,num=10)
#beta_stoch = [0.1,0.1,0.1,0.1,0.1,0.1]
beta_stoch2 = [0.05,0.05,0.05,0.05,0.05,0.05]
#beta_stoch = beta_stoch2
control= [1,1,1,1,2,2]
n = len(beta_stoch)
#results_stoch = []
# Run n stochastic simulations
for i in tqdm.tqdm(range(n)):
res = seir_model_stoch(beta_stoch[i]*np.ones(years),beta_stoch2[i], p, num_I, num_R, years, 'gamma', 0, control[i])
results_stoch.append(res)
#collapse_hide
fig = go.Figure()
for i in range(len(results_stoch)):
#fig.add_trace(go.Scatter(name='Beta_stoch1', x=np.arange(len(results_stoch1[0])), y=results_stoch1[6], line={'dash':'dot','color':'yellow'}, legendgroup="Sim_"+str(i)))
fig.add_trace(go.Scatter(name='Beta_meas'+str(i), x=np.arange(len(results_stoch[i][0])), y=results_stoch[i][7], line={'dash':'dot','color':'yellow'}, legendgroup="Sim_"+str(i)))
fig.add_trace(go.Scatter(name='I_stoch'+str(i), x=np.arange(len(results_stoch[i][0])), y=results_stoch[i][1]/p, line={'dash':'dot', 'color':'red'}, legendgroup="Sim_"+str(i)))
fig.add_trace(go.Bar(name='Ip_stoch'+str(i), x=np.arange(len(results_stoch[i][0])), y=results_stoch[i][4]*10/p, legendgroup="Sim_"+str(i)))
fig.add_trace(go.Scatter(name='R_stoch'+str(i), x=np.arange(len(results_stoch[i][0])), y=results_stoch[i][2]/p, line={'dash':'dot', 'color':'green'}, legendgroup="Sim_"+str(i)))
fig.update_layout(
xaxis_title = 'Years',
yaxis_title = 'Proportion of population',
title={
'text':r'$\text{Stochastic HIV SIR model}$',
'x':0.5,
'xanchor':'center'
}
)
fig.show()