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Immune Cell Polarization (M1/M2)

Macrophages switch phenotypes (M1 inflammatory to M2 reparative). Dysregulation of this switch impairs healing, leading to chronic wounds and persistent inflammation.
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The statement of the theorem

Let M1M_1 and M2M_2 be the concentrations of M1 (inflammatory) and M2 (reparative) macrophages. Their dynamic balance is modeled by a system of coupled ODEs dependent on local cytokine concentrations, CIL−4C_{IL-4} and CIFN−γC_{IFN-\gamma}:\nd[M1]dt=krec[Neutrophils]−kM1→M2[M1]CIL−4+kM2→M1[M2]CIFN−γ\frac{d[M_1]}{dt} = k_{rec} [\text{Neutrophils}] - k_{M1 \to M2} [M_1] C_{IL-4} + k_{M2 \to M1} [M_2] C_{IFN-\gamma} \nd[M2]dt=kM1→M2[M1]CIL−4−krec[Neutrophils]+kM2→M1[M2]CIFN−γ\frac{d[M_2]}{dt} = k_{M1 \to M2} [M_1] C_{IL-4} - k_{rec} [\text{Neutrophils}] + k_{M2 \to M1} [M_2] C_{IFN-\gamma} \n(Note: The signs are simplified to show the dependence on the polarizing cytokines.)