HIV and the Immune System Janet Cady

Introduction Modeling the relationship between the immune system and viruses Modeling the relationship between the immune system and viruses Two models: Two models: ▪ without immune system ▪ without immune system ▪ with immune system ▪ with immune system Why HIV is good at getting past the immune system Why HIV is good at getting past the immune system

Viruses Aren’t capable of reproducing on their own Aren’t capable of reproducing on their own Invade host cells and use cellular machinery to replicate their own DNA 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 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: R 0 = βca/dbf Virus spreads if R 0 >1

Nondimensionalization x=(d/c)X, y=(d/c)Y, v=(bf/ac)V, t=dT dx/dt=1-x- R 0 xv dy/dt=R 0 xv-αy εdv/dt= αy-v ε=d/b α=f/d

Steady States 0=(αy-v)/ ε v*= αy* 0=1-x- R 0 xv x*=1/(1+ R 0 v*) 0=R 0 xv-αy x*=1/R 0 x*=1/R 0 y*=1/ α(1- 1/R 0 ) v*=1- 1/R 0

Immune System B Cells: made in bone B Cells: made in bone ▪ Produce antibodies ▪ Produce antibodies ▪ Kill free viruses ▪ Kill free viruses T Cells: made in Thymus gland T Cells: made in Thymus gland ▪ Helper T cells- alert cytotoxic cells ▪ Helper T cells- alert cytotoxic cells ▪ Cytotoxic killer cells- kill infected 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- R 0 xv εdv/dt= αy-v dy/dt=R 0 xv-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)) R 0

AIDS HIV-human immunodeficiency virus HIV-human immunodeficiency virus ▪ attacks CD4+ cells ▪ attacks CD4+ cells AIDS-acquired immunodeficiency syndrome AIDS-acquired immunodeficiency syndrome ▪ advanced stage of HIV ▪ advanced stage of HIV ▪ fewer than 200 CD4+ cells/mm 3 blood ▪ fewer than 200 CD4+ cells/mm 3 blood Opportunistic infections Opportunistic infections

References Knorr, A.L., Srivastava,R. (2004) Evaluation of HIV-1 kinetic models using quantitative discrimination analysis. Bioinformatics. 21(8) Knorr, A.L., Srivastava,R. (2004) Evaluation of HIV-1 kinetic models using quantitative discrimination analysis. Bioinformatics. 21(8) Britton, N.F. (2003) Essential Mathematical Biology. Springer-Verlag, London Britton, N.F. (2003) Essential Mathematical Biology. Springer-Verlag, London