Table 2 Equations for model V2, which differs from model V1 in having an additional assumption of lymphocyte exhaustion process. Parameter definitions and values: HSC population, H = 0.5; homeostasis probability of HSC differentiation into a myeloid cell, aM = 0.2; myeloid death rate, μM = 0.025; lymphocyte proliferation rate, pL =0.2; lymphocyte death rate, μL = 0. 4; pathogen proliferation rate, pP values vary. Pathogen killing rate by myeloid cells, κM = 0.6; pathogen killing rate by lymphocytes, κL = 1; HSC skew regulator, α = 0.8; rate of APC stimulation of lymphocytes, β = 0.2. Exhaustion effect increases in presence of pathogen. Rate of increase in exhaustion effect, γ1 = 0.005; degree to which pathogen affects onset of exhaustion, γ2 = 100; rate of dissipation of exhaustion effect, μexh = 0.002. See Additional file 1 for a concise summary and explanation of the equations and parameters for all models presented in this article
Variable/function | Equation | Initial value |
|---|---|---|
Myeloid cells (M) | \( \frac{dM}{dt}={f}_1(P)\cdotp {a}_MH-{\mu}_MM \) | M(t = o) = 4 |
Lymphocytes (L) | \( \frac{dL}{dt}=\left(1-{f}_1\cdotp {a}_M\right)H+{f}_2(M)\cdotp {p}_LL-\left(1+ exh\right)\cdotp {\mu}_LL \) | L(t = o) = 2 |
Pathogen (P) | \( \frac{dP}{dt}={p}_PP\left(1-\frac{1}{P_{\infty }}\right)-{\kappa}_MM\frac{P}{k+P}-{\kappa}_LL\frac{P}{k+P} \) | P(t = o) = 3 |
HSC differentiation skew | \( {f}_1(P)=\frac{1+\frac{\alpha }{a_M}P}{1+P} \) | f1(t = o) = 1 |
APC stimulation of lymphocytes | f2(M) = 1 + β(M − M0) | f2(t = o) = 1 |
Exhaustion function dynamics | \( \frac{dexh}{dt}=\frac{\gamma_1}{1+{e}^{-{\gamma}_2\left(P-1\right)}}-{\mu}_{exh} exh \) | exh(t = 0) = 0 |