Investigating mathematical models of immuno-interactions with early-stage cancer under an agent-based modelling perspective

Table 12 Transition rates calculations from the mathematical equations for case 3

Agent	Transition	Mathematical equation	Transition rate
Effector Cell	Reproduce	$\frac{p 1 I E}{g 1 + I} \times (p 1 - \frac{q 1 S}{q 2 + S})$	$\frac{p 1 \times T o t a l I L_2}{g 1 + T o t a l I L_2} \times (p 1 - \frac{q 1 \times T o t a l T G F B e t a}{q 2 + T o t a l T G F B e t a})$
	Die	μ ₁ E	mu 1
	ProduceIL2	$\frac{p 3 T E}{(g 4 + T) (1 + a l p h a S)}$	$\frac{p 3 . T o t a l T u m o u r}{(g 4 + T o t a l T u m o u r) (1 + a l p h a . T o t a l T G F)}$
	KillTumour	$\frac{a_{a} . T E}{g 2 + T}$	$\frac{a a \times T o t a l T u m o u r \times T o t a l E f f e c t o r}{g 2 + T o t a l T u m o u r}$
Tumour Cell	Reproduce	$(a T (1 - \frac{T}{1000000000}))$	$(T o t a l T u m o u r . a (1 - \frac{T o t a l T u m o u r}{1000000000}))$
	Die	$(a T (1 - \frac{T}{1000000000}))$	$(T o t a l T u m o u r . a (1 - \frac{T o t a l T u m o u r}{1000000000}))$
	DieKilledByEffector	$\frac{a_{a} . T E}{g 2 + T}$	message from effector
	ProduceTGF	$\frac{p 4 T^{2}}{t e t a^{2} + T^{2}}$	$\frac{p 4 T u m o u r C e l l s}{t e t a^{2} + T u m o u r C e l l s^{2}}$
	EffectorRecruitment	$\frac{c T}{1 + γ S}$	$\frac{c}{1 + g a m m a . T o t a l t G F}$
IL-2	Loss	μ ₂ I	mu 2
TGF-β	Loss	μ ₃ S	mu 3
	Stimulates TumourGrowth	$\frac{p 2 T}{g 3 + S}$	$\frac{p 2 . T o t a l T G F}{g 3 + T o t a l T G F}$

ISSN: 1471-2105