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165 | -- "SEIR" model for simulating the outbreak of the Coronavirus
-- disease (COVID-19).
-- This program solves a vectorial ordinary differential equation
-- (or a system of ordinary differential equations).
--
-- * The unknown is a vector containing the values S, E, I, R.
-- The letters stands for:
-- S: Susceptible
-- E: Exposed
-- I: Infectious
-- R: Recovered
-- * There is a propagation of population in the
-- direction S ---> E ---> I ---> R, plus new
-- infections: ^------<---
--
-- Related publication:
-- Nowcasting and forecasting the potential domestic and
-- international spread of the 2019-nCoV outbreak originating
-- in Wuhan, China: a modelling study
--
-- https://www.thelancet.com/journals/lancet/article/PIIS0140-6736(20)30260-9/fulltext
--
-- Simplification here:
-- - no flights: L_{W,I}, L_{W,C}, ... = 0
-- - zoonotic force = 0.
with Ada.Characters.Handling, Ada.Text_IO, Ada.Integer_Text_IO;
procedure COVID_19 is
type Real is digits 15;
package PFIO is new Ada.Text_IO.Float_IO (Real);
type Status is (Susceptible, Exposed, Infectious, Recovered);
type Status_Vector is array (Status) of Real;
function "*" (l : Real; v : Status_Vector) return Status_Vector is
r : Status_Vector;
begin
for i in v'Range loop r (i) := v (i) * l; end loop;
return r;
end "*";
function "+" (a, b : Status_Vector) return Status_Vector is
r : Status_Vector;
begin
for i in a'Range loop r (i) := a (i) + b (i); end loop;
return r;
end "+";
inv_incubation_period : constant := 1.0 / 5.2;
inv_infective_period : constant := 1.0 / 2.9;
-- We solve numerically x' (t) = f (x (t), t) over the time step h.
--
procedure Evolution (xt : in out Status_Vector; reproductive_number : Real; h : Real) is
--
function f (x : Status_Vector) return Status_Vector is
n, inv_n, s_to_e, e_to_i, nb_infected_over_period, susc_rate : Real;
--
begin
-- Count the population at time t.
n := 0.0;
for s in Status loop
n := n + x (s);
end loop;
inv_n := 1.0 / n;
nb_infected_over_period := x (Infectious) * inv_infective_period;
-- Some Susceptible persons get the virus
-- from Infectious people and become Exposed.
susc_rate := x (Susceptible) * inv_n; -- As proportion of the population.
s_to_e := susc_rate * reproductive_number * nb_infected_over_period;
-- Exposed persons become Infectious after incubation.
e_to_i := x (Exposed) * inv_incubation_period;
-- Infectious people recover after infective period. -> Recovered.
-- This rate is already computed: nb_infected_over_period;
return
(Susceptible => -s_to_e,
Exposed => s_to_e - e_to_i,
Infectious => e_to_i - nb_infected_over_period,
Recovered => nb_infected_over_period
);
end f;
k1, k2, k3, k4 : Status_Vector;
begin
-- Runge-Kutta, Order 4
k1 := f (xt);
k2 := f (xt + h * 0.5 * k1);
k3 := f (xt + h * 0.5 * k2);
k4 := f (xt + h * k3);
xt := xt + h * (1.0 / 6.0) * (k1 + 2.0 * k2 + 2.0 * k3 + k4);
end Evolution;
type Scenario is (No_Lockdown, Lockdown, Lockdown_in_two_Steps);
procedure Simulation (s : Scenario) is
use Ada.Text_IO, Ada.Integer_Text_IO, PFIO;
x : Status_Vector;
dt : Real;
reproductive_number : Real;
n_iter : Integer;
out_step : Integer;
rf : File_Type;
sep : constant Character := ';';
use Ada.Characters.Handling;
basic_reproductive_number : constant := 3.5;
begin
dt := 1.0;
n_iter := 365;
out_step := 1;
x :=
(Susceptible => 1_000_000.0,
Exposed => 0.0,
Infectious => 1.0, -- Patient 1.
Recovered => 0.0
);
-- Status numbers at time t = 0.
Create (rf, Out_File, "covid_19_" & To_Lower (Scenario'Image (s)) & ".csv");
Put (rf, "t");
for l in Status loop
Put (rf, sep);
Put (rf, Status'Image (l));
end loop;
New_Line (rf);
for i in 0 .. n_iter loop
if i mod out_step = 0 then
Put (rf, i);
for l in Status loop
Put (rf, sep);
Put (rf, x (l), 4, 5, 0);
end loop;
New_Line (rf);
end if;
case s is
when No_Lockdown =>
reproductive_number := basic_reproductive_number;
when Lockdown =>
if i < 40 then
reproductive_number := basic_reproductive_number;
else
reproductive_number := 1.0;
end if;
when Lockdown_in_two_Steps =>
if i < 40 then
reproductive_number := basic_reproductive_number;
elsif i < 60 then
reproductive_number := 2.0;
else
reproductive_number := 1.0;
end if;
end case;
Evolution (x, reproductive_number, dt);
end loop;
Close (rf);
end Simulation;
begin
for s in Scenario loop
Simulation (s);
end loop;
end COVID_19;
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