### AIMS Mathematics

2020, Issue 6: 7191-7213. doi: 10.3934/math.2020460
Research article

# A mathematical study of effects of delays arising from the interaction of anti-drug antibody and therapeutic protein in the immune response system

• Received: 29 March 2020 Accepted: 11 August 2020 Published: 11 September 2020
• MSC : 34A34, 92B05

• Immunogenicity is the ability of substances to evoke an immune response such as a therapeutic protein drug that is considered as a foreign object in the human body. The rise of the immune response results in the production of Anti-Drug Antibody (ADA) that requires a certain period to be activated since it is influenced by the number of injected doses of the drug. The entry of ADA from the depot into the plasma also requires a certain period since the ADA must pass through a series of compartments, hence rises a delay. Both processes are considered as a natural process where the system experiences delay with different delay periods. Immunogenicity on therapeutic protein pharmacokinetics is modelled as a nonlinear delay differential system. From the formulated model, one positive equilibrium solution is obtained under some conditions. Qualitative analysis gives a pair of critical delays in terms of a time delay of the accumulation of protein drug injection and a time required by the ADA to enter the plasma and binding the drug in the plasma. Numerical simulations show that the critical delays result in the appearance of oscillatory behavior in the system. For the system to remain stable, the entering process of ADA into the plasma is delayed in accordance with the obtained critical delay. It is intended such that the injected therapeutic protein drugs provide an optimal effect.

Citation: Kasbawati, Mariani, Nur Erawaty, Naimah Aris. A mathematical study of effects of delays arising from the interaction of anti-drug antibody and therapeutic protein in the immune response system[J]. AIMS Mathematics, 2020, 5(6): 7191-7213. doi: 10.3934/math.2020460

### Related Papers:

• Immunogenicity is the ability of substances to evoke an immune response such as a therapeutic protein drug that is considered as a foreign object in the human body. The rise of the immune response results in the production of Anti-Drug Antibody (ADA) that requires a certain period to be activated since it is influenced by the number of injected doses of the drug. The entry of ADA from the depot into the plasma also requires a certain period since the ADA must pass through a series of compartments, hence rises a delay. Both processes are considered as a natural process where the system experiences delay with different delay periods. Immunogenicity on therapeutic protein pharmacokinetics is modelled as a nonlinear delay differential system. From the formulated model, one positive equilibrium solution is obtained under some conditions. Qualitative analysis gives a pair of critical delays in terms of a time delay of the accumulation of protein drug injection and a time required by the ADA to enter the plasma and binding the drug in the plasma. Numerical simulations show that the critical delays result in the appearance of oscillatory behavior in the system. For the system to remain stable, the entering process of ADA into the plasma is delayed in accordance with the obtained critical delay. It is intended such that the injected therapeutic protein drugs provide an optimal effect.

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沈阳化工大学材料科学与工程学院 沈阳 110142

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