N.I.M.R.O.D.  

HIV : Activated T-cells model

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Model used in : Prague M., Commenges D., Drylewicz J. and Thiebaut R., Treatment Monitoring of HIV-Infected Patients based on Mechanistic Model. Biometrics. 68(3): 902911 (sept. 2012).

-----> Download here : VIH.zip

Mathematical model (ode.f90)

VIH.jpg

\[ \frac{dQ}{dt}= \lambda + \rho T - \alpha Q- \mu_{Q} Q, \]

\[ \frac{dT}{dt}= \alpha Q -\gamma TV - \rho T - \mu_{T}T, \]

\[ \frac{dT^{*}}{dt}= \gamma TV - \mu _{T^{*}} T^{*}, \]

\[ \frac{dV}{dt} = \pi T^{*} - \mu _{V}V. \]

with system in equilibrium at initial points :

\[ Q(0) = \frac{\lambda \gamma \pi +\rho \mu_{T^*}\mu _{v}}{\gamma \pi(\alpha+\mu_{Q})}, \]

\[ T(0) = \frac{\mu_{T^*} \mu _{V}}{ \gamma \pi}, \]

\[ T*(0) = \frac{\lambda \gamma \pi \alpha-\rho \mu _{V}\mu_{T^*}\mu_{Q}-\alpha \mu _{V}\mu_{T^*}\mu_{T}-\mu _{V}\mu_{T^*}\mu_{Q}\mu_{T}}{\gamma \pi \mu_{T^*}(\alpha+\mu_{Q})}, \]

\[ V(0) = \frac{\lambda \gamma \pi \alpha-\rho \mu _{V}\mu_{T^*}\mu_{Q}-\alpha \mu _{V}\mu_{T^*}\mu_{T}-\mu _{V}\mu_{T^*}\mu_{Q}\mu_{T}}{\gamma \mu_{T^*}\mu _{V}(\alpha+\mu_{Q})}. \]

Observational model (observationModelSpe.f90)

\[ Y^i_{1}(t_{ij})=log_{10}(V(X^i(t_{ij})))+\epsilon_{ij1}, \qquad \epsilon_{ij1}\sim \mathcal{N}\left(0,\sigma^2_{VL}\right). \]

\[ Y^i_{2}(t_{ij})=(Q(X^i(t_{ij}))+T(X^i(t_{ij}))+T*(X^i(t_{ij})))^{0.25}+\epsilon_{ij2}, \qquad \epsilon_{ij2}\sim \mathcal{N}\left(0,\sigma^2_{CD4}\right). \]

Statistical model (parameterTransformation.f90)

All parameters are observed in log transformation. We consider the effect of one covariate : treatment with HAART.

\[ \tilde{\gamma}=log(\gamma)=\tilde{\gamma}_0+ \beta I_{HAART=yes}. \]

Data (inAndOutUser.f90 / pk.txt)

Data must be in this shape : "id" "time" "VL" "CD4" "time of treatment beginning" "Viral load censorship indicator" (See the vih.txt file) The input file used in this example is constituted of simulated data. Values for simulation are :

\[\tilde{\lambda} = 2.3\]

\[\tilde{\mu}_{T*} = 0.15 \]

\[\tilde{\mu}_Q = -9.0 \]

\[\tilde{\alpha} = -3.0 \]

\[\tilde{\rho} =-4.0 \]

\[\tilde{\mu}_T = -2.5\]

\[\tilde{\gamma} =-4.1\]

\[\tilde{\pi} =3.5 \]

\[\tilde{\mu}_V = 3.0 \]

\[\tilde{\beta} = -0.75 \]

\[\sigma_{\lambda}= 0.3\]

\[\sigma_{\mu_{T*}}=0.5\]

\[\sigma_{VL}= 0.8\]

\[\sigma_{CD4}=0.4\]

Priors (Input.txt)

Priors are choosen in accordance with the litterature with rather small standard deviation for V0 to avoid the flip-flop paradox.

\[\tilde{\lambda} \sim \mathcal{N}(2.55;1.90)\]

\[\tilde{\mu}_{T*} \sim \mathcal{N}(-0.05;1.0)\]

\[\tilde{\mu}_Q \sim \mathcal{N}(-9.0;1.0)\]

\[\tilde{\alpha} \sim \mathcal{N}(-4.0;2.0)\]

\[\tilde{\rho} \sim \mathcal{N}(-4.34;1.38)\]

\[\tilde{\mu}_T \sim \mathcal{N}(-2.59;0.34)\]

\[\tilde{\gamma} \sim \mathcal{N}(-5.76;4.02)\]

\[\tilde{\pi} \sim \mathcal{N}(4.04;2.66)\]

\[\tilde{\mu}_V \sim \mathcal{N}(2.90;0.68)\]

\[\beta\sim no prior\]

\[\sigma_{\lambda}\sim no prior\]

\[\sigma_{\mu_{T*}}\sim no prior\]

\[\sigma_{VL}\sim no prior\]

\[\sigma_{CD4}\sim no prior\]

Results