Version: 1.8.1

# 7. Differential Equations

This step illustrates how to use differential equations.

## Formulation​

We are interested in the spreading of the disease inside the buildings. In order to model it, we will use differential equations. So, we will need to:

• Add two global variables to define the building epidemic properties (`beta`) and numerical integration parameter (`h`).
• Add new variables for the buildings (`I`, `S`, `T`, `t`, `I_to1`) to manage epidemic;
• Define differential equations for disease spreading inside buildings.
• Add one behavior for buildings for the spreading of the disease. ## Model Definition​

### global variables​

We define two new global variables used in the disease spreading dynamic inside the buildings: (i) `beta` is the contamination rate, and `h` is the integration step (used in the `solve` statement).

``global  {    ...    float beta <- 0.01;    float h <- 0.1;    ...}``

### building​

In order to define the disease spread dynamics, we define several variables that will be used by the differential equations:

• `I`: float, number of people infected in the building.
• `S`: float, number of people not infected in the building.
• `T`: float, the total number of people in the building.
• `t`: float, the current time of the equation system integration.
• `I_to1`: float, the remaining number of people infected (float number lower between 0 and 1 according to the differential equations).
``species building {    ...    float I;    float S;    float T;    float t;       float I_to1;     ...}``

Then, we define the differential equations system that will be used for the disease spreading dynamic. Note that to define a differential equation system we use the block `equation` + name. These equations are the classic ones used by SI mathematical models.

``species building {    ....    equation SI{     diff(S,t) = (- beta * S * I / T) ;    diff(I,t) = (  beta * S * I / T) ;    }    ...}``

At last, we define a new reflex for the building called `epidemic` that will be activated only when there is someone inside the building. This reflex first computes the number of people inside the building (`T`), then the number of not infected people (`S`) and finally the number of infected ones (`I`).

If there is at least one people infected and one people not infected, the differential equations is integrated (according to the integration step value `h`) with the method Runge-Kutta 4 to compute the new value of infected people. We then sum the old value of `I_to1` with the number of people newly infected (this value is a float and not an integer). Finally, we cast this value as an integer, ask the corresponding number of not infected people to become infected, and decrement this integer value to `I\_to1`.

``species building {    ...    reflex epidemic when: nb_total > 0 {    T <- float(nb_total);    S <- float(nb_total - nb_infected);    I <- T - S;    float I0 <- I;    if (I > 0 and S > 0) {        solve SI method: #rk4 step_size: h;        I_to1 <- I_to1 + (I - I0);        int I_int <- min([int(S), int(I_to1)]);        I_to1 <- I_to1 - I_int;        ask (I_int among (people_in_building where (!each.is_infected))) {        is_infected <- true;        }    }    }    ...}``

## Complete Model​

``/*** Name: Differential Equation* Author: GAMA Team* Description: 7th part of the tutorial : Incremental Model* Tags: tutorial, chart, graph, 3d, light, multi-Level, equation*/model model7global {    int nb_people <- 500;    float agent_speed <- 5.0 #km / #h;    float step <- 1 #minutes;    float infection_distance <- 2.0 #m;    float proba_infection <- 0.05;    int nb_infected_init <- 5;    file roads_shapefile <- file("../includes/road.shp");    file buildings_shapefile <- file("../includes/building.shp");    geometry shape <- envelope(roads_shapefile);    graph road_network;    float staying_coeff update: 10.0 ^ (1 + min([abs(current_date.hour - 9), abs(current_date.hour - 12), abs(current_date.hour - 18)]));    float beta <- 0.01;    float h <- 0.1;    list<people_in_building> list_people_in_buildings update: (building accumulate each.people_in_building);    int nb_people_infected <- nb_infected_init update: (people + list_people_in_buildings) count (each.is_infected);    int nb_people_not_infected <- nb_people - nb_infected_init update: nb_people - nb_people_infected;    bool is_night <- true update: current_date.hour < 7 or current_date.hour > 20;    float infected_rate update: nb_people_infected / nb_people;    init {    create road from: roads_shapefile;    road_network <- as_edge_graph(road);    create building from: buildings_shapefile;    create people number: nb_people {        speed <- agent_speed;        building bd <- one_of(building);        location <- any_location_in(bd);    }    ask nb_infected_init among people {        is_infected <- true;    }    }    reflex end_simulation when: infected_rate = 1.0 {    do pause;    }}species people skills: [moving] {    bool is_infected <- false;    point target;    int staying_counter;    reflex move when: target != nil {    do goto target: target on: road_network;    if (location = target) {        target <- any_location_in(one_of(building));        target <- nil;        staying_counter <- 0;    }    }    reflex infect when: is_infected {    ask people at_distance infection_distance {        if flip(proba_infection) {        is_infected <- true;        }    }    }    aspect circle {    draw circle(5) color: is_infected ? #red : #green;    }    aspect sphere3D {    draw sphere(3) at: {location.x, location.y, location.z + 3} color: is_infected ? #red : #green;    }}species road {    geometry display_shape <- shape + 2.0;    aspect default {    draw display_shape color: #black depth: 3.0;    }}species building {    int nb_infected <- 0 update: self.people_in_building count each.is_infected;    int nb_total <- 0 update: length(self.people_in_building);    float height <- rnd(10 #m, 20 #m);    float I;    float S;    float T;    float t;    float I_to1;    species people_in_building parent: people schedules: [] { }    reflex let_people_leave {    ask self.people_in_building {        staying_counter <- staying_counter + 1;    }    release people_in_building where (flip((each).staying_counter / staying_coeff)) as: people in: world {        target <- any_location_in(one_of(building));    }    }    reflex let_people_enter {    capture (people inside self) where (each.target = nil) as: people_in_building;    }    equation SI {    diff(S, t) = (-beta * S * I / T);    diff(I, t) = (beta * S * I / T);    }    reflex epidemic when: nb_total > 0 {    T <- float(nb_total);    S <- float(nb_total - nb_infected);    I <- T - S;    float I0 <- I;    if (I > 0 and S > 0) {        solve SI method: #rk4 step_size: h;        I_to1 <- I_to1 + (I - I0);        int I_int <- min([int(S), int(I_to1)]);        I_to1 <- I_to1 - I_int;        ask (I_int among (people_in_building where (!each.is_infected))) {        is_infected <- true;        }    }    }    aspect default {    draw shape color: nb_total = 0 ? #gray : (float(nb_infected) / nb_total > 0.5 ? #red : #green) border: #black depth: height;    }}experiment main_experiment type: gui {    parameter "Infection distance" var: infection_distance;    parameter "Proba infection" var: proba_infection min: 0.0 max: 1.0;    parameter "Nb people infected at init" var: nb_infected_init;    output {    monitor "Current hour" value: current_date.hour;    monitor "Infected people rate" value: infected_rate;    display map_3D type: opengl {        light 1 color: (is_night ? 50 : 255);        image "../includes/soil.jpg";        species road;        species people aspect: sphere3D;        species building transparency: 0.5;    }    display chart refresh: every(10 #cycles) {        chart "Disease spreading" type: series {        data "susceptible" value: nb_people_not_infected color: #green marker: false;        data "infected" value: nb_people_infected color: #red marker: false;        }    }    }}``