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HIV and the Immune System. Janet Cady. Introduction. Modeling the relationship between the immune system and viruses Two models: ▪ without immune system ▪ with immune system Why HIV is good at getting past the immune system. Viruses. Aren’t capable of reproducing on their own
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HIV and the Immune System Janet Cady
Introduction • Modeling the relationship between the immune system and viruses • Two models: ▪ without immune system ▪ with immune system • Why HIV is good at getting past the immune system
Viruses • Aren’t capable of reproducing on their own • Invade host cells and use cellular machinery to replicate their own DNA • When new viruses are mature, burst out of cell, results in death of cell
Model I- no immune response dV/dT=aY-bV dX/dT=c-dX-βXV dY/dT= βXV-fY Basic reproductive ratio: R0= βca/dbf Virus spreads if R0>1
Nondimensionalization x=(d/c)X, y=(d/c)Y, v=(bf/ac)V, t=dT dx/dt=1-x- R0xv dy/dt=R0xv-αy εdv/dt= αy-v ε=d/b α=f/d
Steady States 0=(αy-v)/ ε v*= αy* 0=1-x- R0xv x*=1/(1+ R0v*) 0=R0xv-αy x*=1/R0 x*=1/R0 y*=1/ α(1- 1/R0) v*=1- 1/R0
Immune System • B Cells: made in bone ▪ Produce antibodies ▪ Kill free viruses • T Cells: made in Thymus gland ▪ Helper T cells- alert cytotoxic cells ▪ Cytotoxic killer cells- kill infected cells
Model II-with immune system dV/dT=aY-bV dX/dT=c-dX-βXV dY/dT= βXV-fY-γYZ dZ/dT=g-hZ
Nondimensionalization z=hZ/g dx/dt=1-x- R0xv εdv/dt= αy-v dy/dt=R0xv-αy-kyz dz/dt=λ(1-z) k= γg/dh λ=h/d
Steady States x*=1/R’0 y*=1/(α+k)(1- 1/R’0) v*= α /(α+k)(1- 1/R’0) z*=1 R’0=(α /(α+k)) R0
AIDS • HIV-human immunodeficiency virus ▪ attacks CD4+ cells • AIDS-acquired immunodeficiency syndrome ▪ advanced stage of HIV ▪ fewer than 200 CD4+ cells/mm3 blood • Opportunistic infections
References • Knorr, A.L., Srivastava,R. (2004) Evaluation of HIV-1 kinetic models using quantitative discrimination analysis. Bioinformatics. 21(8) 1668-1677. • Britton, N.F. (2003) Essential Mathematical Biology. Springer-Verlag, London. 197-201.