Mathematical Modelling of Host – Pathogen – Drug Interaction Dynamics and Immune Modulation in Viral Infection

Paul Kipkurgat Choge; Titus Kiplimo Rotich; Wesley Cheruiyot Koech

doi:doi:10.11648/j.ijtam.20251104.11

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Mathematical Modelling of Host – Pathogen – Drug Interaction Dynamics and Immune Modulation in Viral Infection

Paul Kipkurgat Choge

, Titus Kiplimo Rotich^*

, Wesley Cheruiyot Koech

Published in International Journal of Theoretical and Applied Mathematics (Volume 11, Issue 4)

Received: 8 October 2025 Accepted: 25 October 2025 Published: 3 December 2025

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Abstract

Viral infections target the same immune system cells which fight against the infection, significantly curtailing the capacity of the body to fight associated diseases. Once the virus successfully infects a cell, it uses the same highly proliferating immune host cells to multiply, and enjoy host immunity. With the absence of HIV treatment, available chemotherapy is used to boost immune system, inhibit infection, and disrupt the assembling of viral materials during reproduction. This is achieved by use of Highly Active Anti-Retroviral Therapy (HAART) which contains immune proliferation boosters, reverse transcriptase inhibitors (RTI) and Protease Inhibitors (PI) components. In this paper, a host-pathogen-drug interaction triangle mathematical model is formulated to depict the effect of chemotherapeutic control on reproductive ratio. A SEIR paradigm was used and modified to show differentiated adaptive immune T and B cells, and a viral load compartment. Reproductive ratio R₀ was computed using the next generation matrix, together with its elasticity to control parameters. It was found that in absence of any controls, the reproductive ratio R₀ = 0.659 and as C_E increases, this value reduces with elasticity of E_CE = -1.298 at the critical drug concentration at effect site of C_E(t) = 0.72 of the dose. Simulation revealed that the most sensitive component of HAART is the PI at elasticity of E_π = -1.573, followed by the drug potency to directly kill viral materials ω, closesly followed by the drug potency to kill infected immune cells ψ and lastly by the RTI’s ability to prevent infection η. In conclusion, correct composition of HAART and consist dosing to maintain therapeutic window concentration reduces the viral load, boosts the immune system to normalcy, and generates a pool of memory cells, ready for immediate attack in the subsequent re-infection. This restores the health of People Living with HIV/AIDS (PLWHA), reduces the force of infection of susceptible cells and consequently reduce disease incidence rate across the population.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 11, Issue 4)
DOI	10.11648/j.ijtam.20251104.11
Page(s)	55-64
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Pharmacokinetics, Therapeutic Window, Chemotherapeutic Potency, Compartmental Modelling, Reproductive Ratio, Next Generation Matrix, Parameter Sensitivity, Protease Inhibitors

References

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APA Style

Choge, P. K., Rotich, T. K., Koech, W. C. (2025). Mathematical Modelling of Host – Pathogen – Drug Interaction Dynamics and Immune Modulation in Viral Infection. International Journal of Theoretical and Applied Mathematics, 11(4), 55-64. https://doi.org/10.11648/j.ijtam.20251104.11

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ACS Style

Choge, P. K.; Rotich, T. K.; Koech, W. C. Mathematical Modelling of Host – Pathogen – Drug Interaction Dynamics and Immune Modulation in Viral Infection. Int. J. Theor. Appl. Math. 2025, 11(4), 55-64. doi: 10.11648/j.ijtam.20251104.11

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AMA Style

Choge PK, Rotich TK, Koech WC. Mathematical Modelling of Host – Pathogen – Drug Interaction Dynamics and Immune Modulation in Viral Infection. Int J Theor Appl Math. 2025;11(4):55-64. doi: 10.11648/j.ijtam.20251104.11

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@article{10.11648/j.ijtam.20251104.11,
  author = {Paul Kipkurgat Choge and Titus Kiplimo Rotich and Wesley Cheruiyot Koech},
  title = {Mathematical Modelling of Host – Pathogen – Drug Interaction Dynamics and Immune Modulation in Viral Infection
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {11},
  number = {4},
  pages = {55-64},
  doi = {10.11648/j.ijtam.20251104.11},
  url = {https://doi.org/10.11648/j.ijtam.20251104.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20251104.11},
  abstract = {Viral infections target the same immune system cells which fight against the infection, significantly curtailing the capacity of the body to fight associated diseases. Once the virus successfully infects a cell, it uses the same highly proliferating immune host cells to multiply, and enjoy host immunity. With the absence of HIV treatment, available chemotherapy is used to boost immune system, inhibit infection, and disrupt the assembling of viral materials during reproduction. This is achieved by use of Highly Active Anti-Retroviral Therapy (HAART) which contains immune proliferation boosters, reverse transcriptase inhibitors (RTI) and Protease Inhibitors (PI) components. In this paper, a host-pathogen-drug interaction triangle mathematical model is formulated to depict the effect of chemotherapeutic control on reproductive ratio. A SEIR paradigm was used and modified to show differentiated adaptive immune T and B cells, and a viral load compartment. Reproductive ratio R0 was computed using the next generation matrix, together with its elasticity to control parameters. It was found that in absence of any controls, the reproductive ratio R0 = 0.659 and as CE increases, this value reduces with elasticity of ECE = -1.298 at the critical drug concentration at effect site of CE(t) = 0.72 of the dose. Simulation revealed that the most sensitive component of HAART is the PI at elasticity of Eπ = -1.573, followed by the drug potency to directly kill viral materials ω, closesly followed by the drug potency to kill infected immune cells ψ and lastly by the RTI’s ability to prevent infection η. In conclusion, correct composition of HAART and consist dosing to maintain therapeutic window concentration reduces the viral load, boosts the immune system to normalcy, and generates a pool of memory cells, ready for immediate attack in the subsequent re-infection. This restores the health of People Living with HIV/AIDS (PLWHA), reduces the force of infection of susceptible cells and consequently reduce disease incidence rate across the population.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Mathematical Modelling of Host – Pathogen – Drug Interaction Dynamics and Immune Modulation in Viral Infection

AU  - Paul Kipkurgat Choge
AU  - Titus Kiplimo Rotich
AU  - Wesley Cheruiyot Koech
Y1  - 2025/12/03
PY  - 2025
N1  - https://doi.org/10.11648/j.ijtam.20251104.11
DO  - 10.11648/j.ijtam.20251104.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 55
EP  - 64
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20251104.11
AB  - Viral infections target the same immune system cells which fight against the infection, significantly curtailing the capacity of the body to fight associated diseases. Once the virus successfully infects a cell, it uses the same highly proliferating immune host cells to multiply, and enjoy host immunity. With the absence of HIV treatment, available chemotherapy is used to boost immune system, inhibit infection, and disrupt the assembling of viral materials during reproduction. This is achieved by use of Highly Active Anti-Retroviral Therapy (HAART) which contains immune proliferation boosters, reverse transcriptase inhibitors (RTI) and Protease Inhibitors (PI) components. In this paper, a host-pathogen-drug interaction triangle mathematical model is formulated to depict the effect of chemotherapeutic control on reproductive ratio. A SEIR paradigm was used and modified to show differentiated adaptive immune T and B cells, and a viral load compartment. Reproductive ratio R0 was computed using the next generation matrix, together with its elasticity to control parameters. It was found that in absence of any controls, the reproductive ratio R0 = 0.659 and as CE increases, this value reduces with elasticity of ECE = -1.298 at the critical drug concentration at effect site of CE(t) = 0.72 of the dose. Simulation revealed that the most sensitive component of HAART is the PI at elasticity of Eπ = -1.573, followed by the drug potency to directly kill viral materials ω, closesly followed by the drug potency to kill infected immune cells ψ and lastly by the RTI’s ability to prevent infection η. In conclusion, correct composition of HAART and consist dosing to maintain therapeutic window concentration reduces the viral load, boosts the immune system to normalcy, and generates a pool of memory cells, ready for immediate attack in the subsequent re-infection. This restores the health of People Living with HIV/AIDS (PLWHA), reduces the force of infection of susceptible cells and consequently reduce disease incidence rate across the population.

VL  - 11
IS  - 4
ER  -

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Author Information

Paul Kipkurgat Choge

Department of Mathematics, Physics and Computing, Moi University, Eldoret, Kenya

Contact Email

http://orcid.org/0009-0009-6430-7881
Titus Kiplimo Rotich

Department of Mathematics, Physics and Computing, Moi University, Eldoret, Kenya

Contact Email

http://orcid.org/0009-0003-6091-5616
Wesley Cheruiyot Koech

Department of Mathematics, Physics and Computing, Moi University, Eldoret, Kenya

Contact Email

http://orcid.org/0000-0002-7667-5624

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APA Style

Choge, P. K., Rotich, T. K., Koech, W. C. (2025). Mathematical Modelling of Host – Pathogen – Drug Interaction Dynamics and Immune Modulation in Viral Infection. International Journal of Theoretical and Applied Mathematics, 11(4), 55-64. https://doi.org/10.11648/j.ijtam.20251104.11

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ACS Style

Choge, P. K.; Rotich, T. K.; Koech, W. C. Mathematical Modelling of Host – Pathogen – Drug Interaction Dynamics and Immune Modulation in Viral Infection. Int. J. Theor. Appl. Math. 2025, 11(4), 55-64. doi: 10.11648/j.ijtam.20251104.11

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AMA Style

Choge PK, Rotich TK, Koech WC. Mathematical Modelling of Host – Pathogen – Drug Interaction Dynamics and Immune Modulation in Viral Infection. Int J Theor Appl Math. 2025;11(4):55-64. doi: 10.11648/j.ijtam.20251104.11

Copy | Download

@article{10.11648/j.ijtam.20251104.11,
  author = {Paul Kipkurgat Choge and Titus Kiplimo Rotich and Wesley Cheruiyot Koech},
  title = {Mathematical Modelling of Host – Pathogen – Drug Interaction Dynamics and Immune Modulation in Viral Infection
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {11},
  number = {4},
  pages = {55-64},
  doi = {10.11648/j.ijtam.20251104.11},
  url = {https://doi.org/10.11648/j.ijtam.20251104.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20251104.11},
  abstract = {Viral infections target the same immune system cells which fight against the infection, significantly curtailing the capacity of the body to fight associated diseases. Once the virus successfully infects a cell, it uses the same highly proliferating immune host cells to multiply, and enjoy host immunity. With the absence of HIV treatment, available chemotherapy is used to boost immune system, inhibit infection, and disrupt the assembling of viral materials during reproduction. This is achieved by use of Highly Active Anti-Retroviral Therapy (HAART) which contains immune proliferation boosters, reverse transcriptase inhibitors (RTI) and Protease Inhibitors (PI) components. In this paper, a host-pathogen-drug interaction triangle mathematical model is formulated to depict the effect of chemotherapeutic control on reproductive ratio. A SEIR paradigm was used and modified to show differentiated adaptive immune T and B cells, and a viral load compartment. Reproductive ratio R0 was computed using the next generation matrix, together with its elasticity to control parameters. It was found that in absence of any controls, the reproductive ratio R0 = 0.659 and as CE increases, this value reduces with elasticity of ECE = -1.298 at the critical drug concentration at effect site of CE(t) = 0.72 of the dose. Simulation revealed that the most sensitive component of HAART is the PI at elasticity of Eπ = -1.573, followed by the drug potency to directly kill viral materials ω, closesly followed by the drug potency to kill infected immune cells ψ and lastly by the RTI’s ability to prevent infection η. In conclusion, correct composition of HAART and consist dosing to maintain therapeutic window concentration reduces the viral load, boosts the immune system to normalcy, and generates a pool of memory cells, ready for immediate attack in the subsequent re-infection. This restores the health of People Living with HIV/AIDS (PLWHA), reduces the force of infection of susceptible cells and consequently reduce disease incidence rate across the population.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Mathematical Modelling of Host – Pathogen – Drug Interaction Dynamics and Immune Modulation in Viral Infection

AU  - Paul Kipkurgat Choge
AU  - Titus Kiplimo Rotich
AU  - Wesley Cheruiyot Koech
Y1  - 2025/12/03
PY  - 2025
N1  - https://doi.org/10.11648/j.ijtam.20251104.11
DO  - 10.11648/j.ijtam.20251104.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 55
EP  - 64
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20251104.11
AB  - Viral infections target the same immune system cells which fight against the infection, significantly curtailing the capacity of the body to fight associated diseases. Once the virus successfully infects a cell, it uses the same highly proliferating immune host cells to multiply, and enjoy host immunity. With the absence of HIV treatment, available chemotherapy is used to boost immune system, inhibit infection, and disrupt the assembling of viral materials during reproduction. This is achieved by use of Highly Active Anti-Retroviral Therapy (HAART) which contains immune proliferation boosters, reverse transcriptase inhibitors (RTI) and Protease Inhibitors (PI) components. In this paper, a host-pathogen-drug interaction triangle mathematical model is formulated to depict the effect of chemotherapeutic control on reproductive ratio. A SEIR paradigm was used and modified to show differentiated adaptive immune T and B cells, and a viral load compartment. Reproductive ratio R0 was computed using the next generation matrix, together with its elasticity to control parameters. It was found that in absence of any controls, the reproductive ratio R0 = 0.659 and as CE increases, this value reduces with elasticity of ECE = -1.298 at the critical drug concentration at effect site of CE(t) = 0.72 of the dose. Simulation revealed that the most sensitive component of HAART is the PI at elasticity of Eπ = -1.573, followed by the drug potency to directly kill viral materials ω, closesly followed by the drug potency to kill infected immune cells ψ and lastly by the RTI’s ability to prevent infection η. In conclusion, correct composition of HAART and consist dosing to maintain therapeutic window concentration reduces the viral load, boosts the immune system to normalcy, and generates a pool of memory cells, ready for immediate attack in the subsequent re-infection. This restores the health of People Living with HIV/AIDS (PLWHA), reduces the force of infection of susceptible cells and consequently reduce disease incidence rate across the population.

VL  - 11
IS  - 4
ER  -

Copy | Download