/*
* Self-tolerance and Autoimmunity in a Regulatory T Cell Model
*
* Model Status
*
* This CellML model represents system 4 (equations 15a-15d) in
* the original publication. The model runs in both COR and OpenCell
* to replicate the published results (figure 2) with R0=0.8. The
* units have been checked and they are consistent.
*
* Model Structure
*
* ABSTRACT: The class of immunosuppressive lymphocytes known as
* regulatory T cells (Tregs) has been identified as a key component
* in preventing autoimmune diseases. Although Tregs have been
* incorporated previously in mathematical models of autoimmunity,
* we take a novel approach which emphasizes the importance of
* professional antigen presenting cells (pAPCs). We examine three
* possible mechanisms of Treg action (each in isolation) through
* ordinary differential equation (ODE) models. The immune response
* against a particular autoantigen is suppressed both by Tregs
* specific for that antigen and by Tregs of arbitrary specificities,
* through their action on either maturing or already mature pAPCs
* or on autoreactive effector T cells. In this deterministic approach,
* we find that qualitative long-term behaviour is predicted by
* the basic reproductive ratio R(0) for each system. When R(0)
* is less tHAN 1, only the trivial equilibrium exists and is stable;
* when R(0) is greater than 1, this equilibrium loses its stability
* and a stable non-trivial equilibrium appears. We interpret the
* absence of self-damaging populations at the trivial equilibrium
* to imply a state of self-tolerance, and their presence at the
* non-trivial equilibrium to imply a state of chronic autoimmunity.
* Irrespective of mechanism, our model predicts that Tregs specific
* for the autoantigen in question play no role in the system's
* qualitative long-term behaviour, but have quantitative effects
* that could potentially reduce an autoimmune response to sub-clinical
* levels. Our results also suggest an important role for Tregs
* of arbitrary specificities in modulating the qualitative outcome.
* A stochastic treatment of the same model demonstrates that the
* probability of developing a chronic autoimmune response increases
* with the initial exposure to self antigen or autoreactive effector
* T cells. The three different mechanisms we consider, while leading
* to a number of similar predictions, also exhibit key differences
* in both transient dynamics (ODE approach) and the probability
* of chronic autoimmunity (stochastic approach).
*
* model diagram
*
* [[Image file: alexander_2010.png]]
*
* Flow chart illustrating interactions among populations and flow
* in/out of compartments in System 1. Populations to be modelled
* explicitly are in black boxes; those that are considered only
* implicitly or as intermediaries are in grey boxes. Movement
* in or out of a compartment is indicated by a black arrow; activating
* influences are indicated with green arrows; suppressive influences
* are indicated with red arrows; and other interactions or effects
* are indicated with blue arrows.
*
* The original paper reference is cited below:
*
* Self-tolerance and Autoimmunity in a Regulatory T Cell Model,
* Alexander HK and Wahl LM, 2010, Bulletin of Mathematical Biology.
* PubMed ID: 20195912
*/
import nsrunit;
unit conversion on;
unit day=86400 second^1;
unit first_order_rate_constant=1.1574074E-5 second^(-1);
math main {
realDomain time day;
time.min=0;
extern time.max;
extern time.delta;
real A(time) dimensionless;
when(time=time.min) A=1.0;
real v_tilday first_order_rate_constant;
v_tilday=0.0025;
real f dimensionless;
f=1e-4;
real muA first_order_rate_constant;
muA=0.25;
real b4 dimensionless;
b4=1.00;
real sigma4 dimensionless;
sigma4=1.2e-5;
real G(time) dimensionless;
when(time=time.min) G=1e8;
real R(time) dimensionless;
when(time=time.min) R=0.0;
real pi4 first_order_rate_constant;
pi4=0.016;
real beta first_order_rate_constant;
beta=200.0;
real muR first_order_rate_constant;
muR=0.25;
real E(time) dimensionless;
when(time=time.min) E=0.0;
real lambdaE first_order_rate_constant;
lambdaE=1000.0;
real muE first_order_rate_constant;
muE=0.25;
real gamma first_order_rate_constant;
gamma=2000.0;
real muG first_order_rate_constant;
muG=5.0;
//
//
A:time=(f*v_tilday*G/(1+b4+sigma4*R)-muA*A);
//
R:time=((pi4*E+beta)*A-muR*R);
//
E:time=(lambdaE*A-muE*E);
//
G:time=(gamma*E-(v_tilday*G+muG*G));
}