1. HIV dynamics in sequence space Shiwu Zhang Based on [Kamp2002from, Kamp2002co-evolution]

2. Issues on HIV dynamics HIV infection in patient –HIV development stages. [Santos2001, Hershberg2000, AAMAS-HIVreport] –Factors influence (mutation rate, antigenic diversity…) –Distribution of HIV latency period [Kamp2002from, Kamp2002co-evolution] HIV epidemic –Spreading on social network [Dezso2002, Satorras2001]

3. Background: Percolation theory –Occupation probability (Susceptibility) –Clusters –Spanning probability (transmissibility) –Percolation threshold (P c )

4. Background: Sequence space Viral genome & immune receptor: length l Viral mutation: change one bit (1,2, … ) Constructing a sequence space, size= l Viral mutation means random walk in space Dimension: l

5. Model Site status –Susceptible S(t) Site can harbor a virus –Infected  v (t) Site is infected by virus –Recovered R(t) After immune response (immune memory) Viral genome is not arbitrary (D 0 ) Immunological presence (  0 )

6. Model (2) Rules: –Random select site If the site harbor immune receptor –Mutate with certain probability –If mutate and the mutant match an infected site then set the infected site to recovered If the site is infected –Mutate with certain probability –If a new strain is generated and corresponds to a susceptible site, the site become infected –For HIV, another rule Viral strain has probability  is (t) to meet an receptor infect it with probability p

7. Result Simulation result could capture HIV population dynamics from clinical latency stage to onset on AIDS, but fail to reflect initial immune response Initial distribution  0 is important factor to affect result Increasing probability p will shorten waiting time Distribution of HIV incubation period distribution from simulation fits in well with that from real data

8. Summary Characteristics: –Sequence space –Percolation theory –Accounting for important interactions HIV mutation Immune cells stimulation Immune system’s global ability:memory Shortage: –Omitting physical space –Using strain denote population (without strain size distribution) –Don’t account for initial response

9. Related Papers C. Kamp, S. Bornholdt (2002). From HIV infection to AIDS: A dynamically induced percolation transition?, Proc. R. Soc. London B (2002), accepted for publication. http://arxiv.org/abs/cond- mat/0201482 C. Kamp, S. Bornholdt (2002). Co-evolution of quasispecies: B-cell mutation rates maximize viral error catastrophes, Phys. Rev. Lett. 88, 068104. http://www.tp.umu.se/~kim/Network/Marek/sem.pdf D. Stauffer, A. Aharony (1992). Introduction to Percolation Theory, (Taylor and Francis, London). H. Mannion et al. (2000). A Monte Carlo Approach to Population Dynamics of Cell in an HIV Immune Response Model. Theory in Bioscience: 119(94) U. Hershberg et al.(2001). HIV time hierarchy: Winning the war while losing all the battles. Physica A:289 (1-2). http://arxiv.org/abs/nlin.AO/0006023