A reaction–diffusion delay model for spatial tumor virotherapy with optimal control

Fathalla A. Rihan; Fathalla A. Rihan

doi:10.3934/mbe.2026091

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 8: 2489-2515. doi: 10.3934/mbe.2026091

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A reaction–diffusion delay model for spatial tumor virotherapy with optimal control

Fathalla A. Rihan ^,

Department of Mathematical Sciences, College of Science, UAE University, Al-Ain, 15551, UAE

Received: 09 June 2026 Revised: 19 July 2026 Accepted: 10 August 2026 Published: 18 August 2026

Oncolytic virotherapy uses viruses that selectively infect and lyse tumor cells while promoting antitumor immunity. Because viral spread, tumor growth, and immune-cell migration are spatially heterogeneous, purely temporal models may miss important treatment dynamics. We develop a reaction–diffusion delay model for spatial tumor virotherapy with a virus-induced cytotoxic T-lymphocyte (CTL) response, thereby incorporating tumor growth, infection, lysis, immune-mediated killing, diffusion, and delayed immune activation. We prove positivity, local well-posedness, and global boundedness under explicit sufficient conditions, and analyze spatially homogeneous equilibria and modal stability. Then, we extend the model with a numerical optimal-control formulation for viral administration and immune stimulation, thus minimizing tumor burden, treatment cost, and excessive CTL proliferation. Simulations indicate that tumor–virus–immune dynamics can remain spatially heterogeneous and that the computed optimal control schedules improve tumor suppression for the chosen parameter set.
- oncolytic virotherapy,
- tumor–virus interaction,
- cytotoxic T lymphocytes,
- reaction–diffusion equations,
- delay differential equations,
- optimal control
Citation: Fathalla A. Rihan. A reaction–diffusion delay model for spatial tumor virotherapy with optimal control[J]. Mathematical Biosciences and Engineering, 2026, 23(8): 2489-2515. doi: 10.3934/mbe.2026091

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Abstract

Oncolytic virotherapy uses viruses that selectively infect and lyse tumor cells while promoting antitumor immunity. Because viral spread, tumor growth, and immune-cell migration are spatially heterogeneous, purely temporal models may miss important treatment dynamics. We develop a reaction–diffusion delay model for spatial tumor virotherapy with a virus-induced cytotoxic T-lymphocyte (CTL) response, thereby incorporating tumor growth, infection, lysis, immune-mediated killing, diffusion, and delayed immune activation. We prove positivity, local well-posedness, and global boundedness under explicit sufficient conditions, and analyze spatially homogeneous equilibria and modal stability. Then, we extend the model with a numerical optimal-control formulation for viral administration and immune stimulation, thus minimizing tumor burden, treatment cost, and excessive CTL proliferation. Simulations indicate that tumor–virus–immune dynamics can remain spatially heterogeneous and that the computed optimal control schedules improve tumor suppression for the chosen parameter set.

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