sidarthe: Euler scheme for the SIDARTHE model

In shill1729/odeSolveR: Implementation of simple Euler schemes and fourth-order Runge-Kutta methods for arbitrary first-order ODE systems

Description

A basic Euler scheme is used to numerically solve the ODE system of the novel SIDARTHE model.

Usage

sidarthe(
  parameters,
  initial_conditions,
  t0 = 0,
  tn = 1,
  n = 1000,
  verbose = TRUE
)

Arguments

parameters	named list of parameters see details
initial_conditions	named list of initial conditions, see details
t0	initial time point
tn	the terminal time point
n	number of sub-intervals in time-grid
verbose	whether to plot the populations over time and print parameter+reproductive ratio

Details

The parameters must be a named list containing values

• α, β, γ, δ "represent the transimission rate (i.e. the probability of disease transmission in a singlecontact times the average number of contacts per person) due to contacts between a Susceptible subject and an Infected, a Diagnosed, an Ailing, a Recognised subject. Typically, α is larger than γ (assuming that people tend to avoid contacts with subjects showing symptoms, even though diagnosis has not been made yet), which in turn is probably larger than β and δ (assuming that subjects who have been diagnosed are properly isolated). These parameters can be modified by social distancing policies (e.g., closing schools, remote working, etc.). The risk of contagion due to Threatened subjects, treated in proper ICUs, is assumed negligible."

• ε and θ "capture capture the probability rate of detection, relative to asymptomatic and mildly symptomatic casesrespectively. These parameters, also modifiable, reflect the level of attention on the disease and the numberof tests performed over the population. Note that θ is typically larger than ε, since symptomatic individuals are more likely to get tested."

• ζ and η "denote the probability rate at which an infected subject, respectively not aware and aware of being infected, develops clinically relevant symptoms, and are probably comparable. These parameters are disease-dependent and hardly modifiable."

• μ and ν "respectively denote the rate at which undetected and detected infected subjects develop life-threatening symptoms, and are likely to be comparable if there is no known specific treatment that is effective against the disease, otherwise μ is likely to be larger. These parameters can be reduced by means of improved therapies and acquisition of immunity against the virus."

• τ "denotes the mortality rate (for infected subjects with life-threatening symptoms) and can be reduced bymeans of improved therapies and treatments."

• λ, κ, ξ, ρ and σ "denote the rate of recovery for the five classes of infected subjects, and may differ significantly if an appropriate treatment for the disease is known and adopted to diagnosed patients, while are probably comparable otherwise. These parameters can be increased thanks to improved treatments and acquisition of immunity against the virus."

These descriptions of the model parameters are directly taken from the 2020-03-21 paper "A SIDARTHE model of COVID-19 Epidemic in Italy" by the COVID19 IRCCS San Matteo Pavia Task Force et al. to mitigate any misinterpretation of the input into the model.

(Note the named list must actually have the greek letter names written out, i.e. alpha, beta,...)

Finally the list initial_conditions must contain the following, all as fractions of the total population

• s0 the initial level of susceptible individuals

• i0 the initial level of infected individuals

• d0 the initial level of diagnosed individuals

• a0 the initial level of ailing individuals

• r0 the initial level of recognized individuals

• thr0 the initial level of threatened individuals

• h0 the initial level of heal individuals

• e0 the initial level of extinct individuals

It is best to specify all but s0 and then set the initial susceptible fraction to 1 less the sum of the rest.

Value

list consisting of (1) parameters, a data.frame containing the contact rates, etc and (2) sidarthe, a matrix whose columns consist of the eight (fractions of) populations over time:

• suspected

• infected

• diagnosed

• ailing

• recognized

• threatened

• healed

• extinct

and finally (3) the basic reproductive ratio as defined for this extended SIR model as a consequence of Proposition 2 in the original paper.

References

The Euler scheme implemented by this function is for the model developed in the following paper. It is 8 coupled non-linear ODEs describing dynamics of sub-populations under a pandemic. The descriptions within the novel paper were used to document the initial conditions and parameters of this model in this package.

• "A SIDARTHE Model of COVID-19 Epidemic in Italy" by the COVID19 IRCCS San Matteo Pavia Task Force et al. https://arxiv.org/abs/2003.09861

shill1729/odeSolveR documentation built on March 31, 2021, 10:52 a.m.