Viraly - Virality concepts demo

Introduction

This is a simple command line program that simulates an epidemic in 4 different models:

1. permanent infection, infinite population: exponential growth
2. permanent infection, finite population: logistic growth
3. temporary infection with duration T, finite population: epidemic curve
4. temporary infection with guassian duration of avg T and stdev L, finite population: epidemic curve

An epidemic is the propagation of something (a disease, a phrase, a brand, an idea, ...) over a population by means of interactions between elements. A technical summary document can be found here. There is a bokeh based web frontend on the /web folder an instance of which is available with examples for:

• a free epidemic
• a free epidemic with seasonal effects
• a managed 3 stage epidemic
• a 3 stage simulation of a sudden contact shock
• a simulation of R0 uncertainty on a highly immunized population exposed to a very contagious virus

Requirements

Scipy, Numpy and Matplotlib

Usage

python3 viraly.py "h,p,T,L,I,h2,p2,tint,tmax,M,N0,DR"
python3 viraly.py "h,p,T,L,I,h2,p2,tint,tmax,M,N0,DR,progressive,ttime"

Parameters and their meaning

h           # average number of contacts per unit of time
p           # probability of transmission during a contact
T           # average duration of infections
L           # standard deviation of the normal distribution of the infection duration
I           # incubation time
h2          # average number of contacts per unit of time under contention
p2          # probability of transmission during a contact under contention
tint        # simulation time with initial parameters (i.e., before contention)
tmax        # total simulation time
M           # population size
N0          # initial number of infections
DR          # death rate
progressive # [optional] whether or not the change of parameters at time tint should be progressive
ttime       # [optional] the parameters transition time (if progressive == True)

The parameters must be given via command line in the order listed above as quoted comma-separated list.

Notes:

• h and p are the parameters that drive propagation
• they are presented as independent for physical intuition purposes but only the product hp matters in practice
• the simulation includes two phases: [ 0, tint [ with parameters (h,p) and [ tint, tmax [ with parameters (h2,p2)
• if tint == tmax the simulation reduces to a single phase with parameters (h,p)
• if progressive == True the transition between phases is done using a linear variation (h,p) -> (h2,p2)
• the Basic Reproduction Number (R0) is given by hpT
• for model 3 the infections remain constant if hp = 1/T and decrease to zero if hp < 1/T
• for model 4 hp needs to be slightly lower for the situations above to occur

Examples

python3 viraly.py "4,0.1145,15,3,1,2,0.02,120,120,10276617,4,0.03"
python3 viraly.py "4,0.1145,15,3,1,2,0.02,24 ,120,10276617,4,0.03"
python3 viraly.py "4,0.1145,15,3,1,2,0.02,24 ,120,10276617,4,0.03,True,7"

The first example simulates a free epidemic for 120 days, whereas the second example simulates an epidemic for 120 days with sudden change of h and p (contention) at time t=24. The third example is equal to the second but with a linear change of parameters over the course of 7 days.

Outputs

• plot with active cases, new cases, recoveries an deaths
• plot with acumulated cases and acumulated deaths
• plot with the usual SIR variables: Susceptible, Infected and Removed (Recovered or Dead)
• plot with comparison of models: exponential, logistic and epidemic (with fixed recovery time) and epidemic2 (with gaussian recovery time)
• plot with evolution of the effective reproduction number R(t) over time
• console output for the preferred model with Active Cases, New Cases, Removals, Susceptibles and R(t), plus misc stats (peak, totals, ...)

Configuration

The boolean global variable PREFER_MOD4 controls whether or not model4 is the preferred model for console output and plots. If PREFER_MOD4 is False the preferred model is model3.

Example outputs

Example 1: output for model 4 with a sudden parameter change (contention) at t=24 such that h₂p₂T < 1:

Example 2: output for model 4 with a sudden parameter change (contention) at t=24 such that h₂p₂T > 1:

Example 3: same as above but as a free epidemic (no parameter change):

Example 4: SIR plot for the case above:

Example 5: comparison of models for the case above:

Example 6: output for model 3 with critical choice of parameters hp = 1/T so that the epidemic is in the limit of propagation:

Example 7: same as above for model 4 where due to the gaussian recovery hp needs to be slightly lower:

Example 8: output of early erratication case due to sub critical hp for model 4:

Bonus section

There are extra parameters not documented here which can be checked on the source code. With those parameters a 3 stage simulation can be performed and this command line program can be integrated with an external application as demonstrated on the web frontends mentioned in the introduction.

Disclaimer

This is an experiment related to the math of virality and should not be used for decisions related to real world public health situations.