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Models of HIV Infection at the Immune System Level Douglas E. Norton Villanova University Mellon Tri-Co Faculty Modeling Working Group Bryn Mawr College December 12, 2003 Villanova Summer Research Institute The National Health and Nutrition Examination Survey is a survey conducted by the National Center for Health Statistics (NCHS), Centers for Disease Control and Prevention. This survey has been designed to collect information about the health and diet of people in the United States. http://www.cdc.gov/nchs/nhanes.htm Our immediate academic goal: to formulate and explore some models of the immune system and its interaction with the human immunodeficiency virus. We present a small number of fundamental ideas – 6 in all – that participants can understand at a useful level and that we consider essentially sufficient to work intelligently with these models. 6 Pillars of Wisdom [William Fleischman, T. E. Lawrence with tolerance 1] I. We will be modeling dynamical systems. This may be a forbidding, technical sounding term; all it means is a system with quantities or properties we can measure that change over time. II. The models we will be examining are calculus-based. They are expressed formally as systems of differential equations, but there is a simple, intuitive way of thinking about them, interpreting the equations as rate of change expressions, that enables us to understand and then manipulate them without distorting their meaning. III. Although the formal solution and analysis of the models involve deep ideas from advanced mathematics, we can explore a big part of the landscape described by these models using only (high school) algebra and geometry. IV. Every model is a population model. V. Complex models are built from simple parts. They are based on useful analogies or metaphors. VI. Even models that are simple to the point of caricature can be profoundly useful tools in thinking about intricate biological systems. Be careful not to be dismissive of a model on account of its simplicity. After an introduction of the derivative as a limit of average rates of change, we use “rate of change” as our motivation for generating systems of differential equations to model biological phenomena. We move immediately to Euler’s Method as a computational tool, calling it simply The Recipe. The Recipe. STEP 1: Compute the value of the growth rate expression for the current time: dN/dt. STEP 2: Since dN/dt is approximately N/t, we can use the rate of change in step 1 to approximate the population change for a given time change. That is, for a given choice of t, we can use step 1 to estimate N over the time interval t. Specifically, N dN N ( t ) (t ), or N (growth rate) (t ). t dt STEP 3: Add the approximate change N to the population you began with to get the approximate population at the end of the time interval: (current N) + N (new N). STEP 4: Let the new value of N be the new choice for the current value of N and return to step 1. Repeat steps 1 through 3 until you reach the final time you want. dN birth rate death dt dN b N m N dt dN (b m) N dt dN r N dt rate 1 dN b m N dt per capita average average growth rate fecundity probabilit y of mortality b f ( N ) b0 kbN m g ( N ) m0 kmN b, m bo m0 N EQUILIBRIUM POINT b, m bo m0 N 1 dN (b0 kbN ) (m0 kmN ) N dt (b0 m0 ) (kmN kbN ) dN N [(b0 m0 ) (km kb) N ] dt ( b 0 m0 ) (km kb) N [ N] (km kb) dN dt k N [K N ] dN N r N 1 K dt 0 9.6 4.223483 1 18.3 3.564981 2 29 3.087903 3 47.2 2.571771 4 71.1 2.122624 5 119.1 1.522472 6 174.6 1.032724 7 257.3 0.460289 8 350.7 -0.10958 9 441 -0.6774 10 513.3 -1.21896 11 559.7 -1.67059 12 594.8 -2.13688 13 629.4 -2.87242 14 640.8 -3.27636 15 651.1 -3.84677 16 655.9 -4.27773 17 659.6 -4.80523 K N 1 Ae rt K N y ln( ) N 1 N N dN1 r N 1 1 2 1 12 K1 K1 dt dN2 r N 1 N2 N1 2 2 21 K K dt 2 2 EXAMPLE: Rabbits and sheep compete for a limited amount of grass. We assume logistic growth for each, that rabbits reproduce rapidly, and that the sheep can crowd out the rabbits. Then some unrealistic coefficients (just to make this example look friendlier) and these assumptions could give the following model: dx/dt = x(3 – x – 2y) dy/dt = y(2 – x – y), where x(t) = size of the rabbit population and y(t) = size of the sheep population. This is an example of what the N1 – N2 system may look like, algebraically speaking, if multiplication and constant renaming are utilized. We can explore this model with a natural extension of our recipe to find equilibrium points and possible extinction of one population. EXAMPLE (a different but related example): Three populations: grass, sheep, and wolves. Without the sheep, the grass grows logistically (?!). Sheep eat grass. Wolves eat sheep. Let x, y, and z represent the sizes of the wolf, sheep, and grass populations, respectively. Then the following system is a possible model: dx/dt = –x + xy dy/dt = –y + 2yz – xz dz/dt = 2z – z2 – yz. ONE MORE EXAMPLE: predator-prey with child-care (!). x1 = young prey; x2 = adult prey; y = predators. The young are protected from predators. Interpret the terms in this system: dx1/dt = ax2 – bx1 – cx1 dx2/dt = bx1 – dx2 – ex2y dy/dt = – fy + gx2y. Dissolved Oxygen Biodegradation kinetics and reaeration may be integrated into a single model to predict the effect of organic pollution on streams. The deficit equation in these natural systems must consider the oxygen consumption due to biodegradation and the oxygen replenishment due to reaeration. The differential equation that describes these conditions is: Integration of this summation yields the general equation for the deficit in a stream (sag-curve equation or Streeter-Phelps equation): REM: The Register of Ecological Models (REM) is a metadatabase for existing mathematical models in ecology. ECOBAS: The ECOBAS project provides a system for documentation of mathematical descriptions of ecological processes. The Classical S-I-R Model Model the spread of a disease: N = S + I + R = total population, where S = the number of “susceptibles,” I = number of “infecteds,” and R = number of “recovereds” (immune). First version: do not include births or deaths. Then the only changes are from S to I (susceptibles becoming infected) and from I to R (from infected to recovered) To become infected, a susceptible must come in contact with an infected, we use an analogy with the Law of Mass Action in chemical kinetics: the rate at which this type of change occurs (S to I) is proportional to the product of the sizes of the two populations involved. For recovery, the rate is proportional to the number of infecteds. Then with constants = “transmission coefficient” and = “recovery rate coefficient,” we have: dS/dt = –SI dI/dt = SI – I dR/dt = I. A First Modification = the extra per capita mortality rate due to the disease (decreases I) A = rate of additions (increases S) b = per capita mortality rate (decreases all three) = per capita rate of loss of immunity (turns R’s back into S’s) a = per capita birth rate, independent of whether S, I, or R. (If there are no additions other than by birth, then A = a(S + I + R).) Then the “new and improved” model would be: dS/dt = A – bS – SI + R dI/dt = SI – (b + + I dR/dt = I – ( + b)R. Epidemiological Models Immune System Level Our Beginning Model In direct analogy with human population models, we model the immune system players in HIV infection. Rather than susceptible, infected, or recovered persons, our dynamic variables now represent: T T* T** V : : : : healthy T-cells (cells per ml) latently infected T-cells (cells per ml) actively infected T-cells (cells per ml) viral particles (free virus) (particles per ml). T T * T * * dT k1VT s T T rT 1 dt Tmax dT * k1VT T T * k 2T * dt dT * * k 2T * bT * * dt dV N bT * * k1VT V V dt T T*T** dT s TT rT 1 (k1V k3M*)T dt Tmax dT* (k1VTk3M*T) T*T*k2T* dt dT** k2T*bT** dt dV NbT**MM*k1VTVV dt dM M(EM M) k4VM dt dM* k4VMM*M* dt T T*T** dT s TT rT 1 (k1V k3M*)T dt Tmax dT* (k1VTk3M*T) T*T*k2T* dt dT** k2T*bT** dt dV NbT**MM*k1VTVV dt dM M(EM M) k4VM dt dM* k4VMM*M* dt Dependent Variables Values T = uninfected CD4+ T cell population 1000 mm-3 A = CD8+ T cell population 500 mm-3 Ts* = Latently infected CD4+ T cell population (slow-replicating 0 virus) Tf* = Latently infected CD4+ T cell population (fast-replicating 0 virus) Ts** = Actively infected CD4+ T cell population (slow-replicating 0 virus) Tf** = Latently infected CD4+ T cell population (fast-replicating 0 virus) Vs = Slow replicating infectious HIV population .001 mm-3 Vf = Fast replication infectious HIV population 0 M = Uninfected macrophage population 30 mm-3 Ms* = Infected macrophage population (slow-replicating) 0 Mf* = Infected macrophage population (fast-replicating) 0 dT TV ST T T rT ( K11 Vs K12V f )T ( K41 M s * K42 M f *)T dt CT V dA AV S A A A rA dt CA V dTs * K11Vs T T Ts * K2 Ts * K41 M s * T dt dTf * K12V f T T Tf * K2 Tf * K42 M f * T dt dTs ** K2 Ts * bs Ts ** K5Ts ** A dt dTf ** K2 Tf * bf Tf ** K5Tf ** A dt dVs N s bs Ts ** wII Ms M s * (1 y ) II Mf M f * VV K11Vs T K31Vs M dt dV f N f bf Tf ** (1 w) II Ms M s * y ) II Mf M f * VV K12V f T K32V f M dt dM S M M M ( K31Vs K32V f ) M dt dM s * K31Vs M M * M s * K6 M s * A dt dM f * K32V f M M * M f * K6 M f * A dt Parameters ST = source term for uninfected T4 cells SA = source term for killer cells rT = maximal proliferation of CD4+ T cell population rA= maximal proliferation of CD8+ T cell population CT = half-saturation constant of proliferation process (helper) CA = half-saturation constant of proliferation process (killer) K11= rate T4 cell becomes infected by Vs K12 = rate T4 cell becomes infected by Vf K2 = rate T* converts to actively infected K31 = rate macrophage becomes infected by Vs K32 = rate macrophage becomes infected by Vf K41 = rate Ms* infects T4 cells K42 = rate Mf* infects T4 cells K5 = rate CD8+ cells kill T** cells K6 = rate CD8+ cells kill M* T = death rate of uninfected T4 cells A = death rate of CD8+ cells v = death rate of virus M = death rate of uninfected macrophage M* = death rate of infected macrophage bs = death rate of infected Ts** cells bf = death rate of infected Tf** cells EM = equilibrium for macrophage Ns = number of free virus produced by Ts** cells Nf = number of free virus produced by Tf** cells Ms = rate of free virus produced by infected Ms* Mf = rate of free virus produced by infected Mf* y = fast replicating mutate to slow replicating w = slow replicating mutate to fast replicating Working values 10 d-1 mm-3 15 d-1 mm-3 0.02 d-1 0.03 d-1 100 mm-3 100 mm-3 1.8 x 10-5 mm3 d-1 2 x 10-4 mm3 d-1 0.003 mm3 d-1 8 x 10-6 mm3 d-1 5 x 10-6 mm3 d-1 1 x 10-7 mm3 d-1 1 x 10-7 mm3 d-1 7.4 x 10-4 mm3 d-1 7.4 x 10-4 mm3 d-1 0.02 d-1 0.02 d-1 0.4 d-1 0.005 d-1 0.005 d-1 0.24 d-1 0.3 d-1 30 mm-3 1000 1000 300 d-1 300 d-1 0.95 0.8 T = healthy T-cells (initial value = 1000 to 2000 cells/ml) T* = all infected T-cells (initial value = 0) Tf* = T-cell infected with fast replicating HIV (initial value = 0) Ts* = T-cell infected with slow replicating HIV (initial value = T) V = both strands of HIV (initial value = 0.001 virions/ml) Vf = fast replicating HIV (initial value = 0) Vs = slow replicating HIV (initial value = V) M = macrophage uninfected by HIV (initial value = 85 or 100) M* = macrophages infected with either strand of HIV (initial value = 0) Mf* = macrophage infected with fast replicating virus Ms* = macrophage infected with slow replicating virus Tb = tuberculosis cell (initial value = 1 mm-3) sT = source of healthy T-cells = 10 (cells/ml/day) sM = source of uninfected macrophages = T = natural healthy T-cell death rate = 0.02 (1/day) b = lysis rate of an infected T-cell = 0.24 (1/day) v = natural death rate of HIV particles = 2.4 (1/day) M = natural death rate of macrophages = 0.005 or 0.003 (1/day) Tb = natural death rate of Tb cells k1 = rate at which a fast replicating virus infects a healthy T-cell = 0.000024 (1/cells*day/ml) k3 = rate at which fast replicating virus is engulfed by macrophage = 0.000002 k4 = rate at which infected macrophage “infects” healthy T-cell = k5 = rate at which a healthy T-cell kills a Tb cell = 0.5 (ml/day) k6 = rate at which a macrophage kills a Tb cell = k7= = rate at which a Tb cell kills a healthy T-cell = k8 = rate at which a Tb cell kills a T-cell infected by fast-replicating HIV particles = k11 = rate at which a slow replicating virus infects a healthy T-cell k13 = rate at which a slow replicating virus is engulfed by a macrophage = rT = coefficient for immune response initiated by emergence of foreign invaders = 0.02 rTb = coefficient for immune response inititated by appearance of Tb = 1 K = carrying capacity of Tb = 1000 z = effect of AZT on burst size of virion particles N = burst size of virion particles from infected T-cells = = burst size of virion particles from infected macrophages = from 100-1000 = percent fast replicating viruses coming from an infected macrophage 1 - = percent of slow replicating viruses coming from an infected macrophage C = something = 1000 per cubic mm 1. V Tb dT * sT T T rT T k1V f T k 4 M T k 7 Tb T k11Vs T dt C V Tb = source / natural death / immune response growth / infection / “infection” / death by TB /infection 2. dT * f dt k1V f T T T f* b T f* k 4 M *T k 8Tb T f* = source / natural death / burst / source by macrophage / death by TB * 3. dTs k11V s T T Ts* b Ts* 1 k 4 M *T k 8Tb Ts* dt = source / natural death / burst / source by macrophage / death by TB 4. dV f zN b T * k1V f T vV f k 3 MV f zM * M * zN k 8Tb T * dt = source / death by T-cell / natural death / engulfed / source by macrophage / burst size 5. dVs 1 zM * M * k11Vs T k13 MV s dt = source / death by T-cell / engulfed 6. dM s M k MV r 2 MV r 1 MT k MV M M 3 f M M b 13 s dt = source / natural death / death by HIV / stimulation / recruitment / death by HIV * dM f 7. k 3 MV f M * M *f dt = source / natural death * 8. dM s k MV * M * 13 s s M dt = source / natural death 9. dTb r T ( T ) T T k T k M Tb b b Tb b b 5 6 dt = source / natural death / death by immune system 10. V V f Vs * * * T T T 11. f s * * * M M M 12. f s Goals • Accurately implement the current models • Modify existing equations to make them more mathematically accurate and biologically realistic • Create equations to model the viral load, number of HIV strains, and the immune response • Model the effects of the number of viral strains on the progression of the virus Original System of Equations • dTp/dt = CLTL(t) – CPTP(t) • dTlp/dt = CLTlL(t) – CPTlP(t) • dTL/dt = CPTP(t) – CLTL(t) – kTL(t) + ųaTaL(t) • dTlL/dt = pkTL(t) – CLTlL(t) + CPTlP(t) – ųlTlL(t) – slTlL(t) + siTiL(t) • dTaL/dt = rkTL(t) – ųaTaL(t) • dTiL/dt = qkTL(t) – ųiTiL(t) + slTlL(t) – siTiL(t) Modifications • dTp/dt = CLTL(t) – CPTP(t) + s*(1-(Tp(t)+Tlp (t)+TL (t)+TlL (t)+TaL (t)+TiL (t))/Smax) ųu*Tp(t) • dTL/dt = CPTP(t) – CLTL(t) – kTL(t) + ųaTaL(t) – ųu* TL(t) • dV/dt = bTil(t) - cV(t) - KR(t) • dS/dt = un*(q*k* TL(t) + Sl * TlL(t)) • dR/dt = [g* V(t) * R(t) * (1- R(t) / Rmax)]/ floor S(t) Future Modifications • dTL/dt = CPTP(t) – CLTL(t) – kV(t)TL(t) + ųaTaL(t) – muU*Tp(t) • dTaL/dt = rkV(t)TL(t) – ųaTaL(t) • dTlL/dt = pkV(t)TL(t) – CLTlL(t) + CPTlP(t) – ųlTlL(t) – slTlL(t) + siTiL(t) • dTiL/dt = qkV(t)TL(t) – ųiTiL(t) + slTlL(t) – siTiL(t) • dS/dt = un*(q*k*V(t)*Tl(t) + Sl * Tll(t)) Uninfected blood CD4+ cells over 10 years Before After Incorrect display of uninfected T cells • The cell count does not get low enough to induce AIDS Uninfected CD4+ cells in blood Uninfected CD4+ cells in lymph Latently infected CD4+ cells in blood over 10 years Before After Uninfected CD4+ cells in lymph over 10 years Before After Latently (red), abortively (green), and actively (yellow) infected CD4+ cells in the lymph over 10 years Before After Viral Load over 1 year (in powers of 10) Viral Load over 10 years (in powers of 10) Number of Virus Strains over 10 years