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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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Presentation Transcript
introduction
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
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
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
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
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
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
Model II-with immune system

dV/dT=aY-bV

dX/dT=c-dX-βXV

dY/dT= βXV-fY-γYZ

dZ/dT=g-hZ

nondimensionalization1
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 states1
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

slide11
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
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.