Texts: How to do (5) ? ? ? 🍓🍇🧸🌟🩵
2.1 An Ordinary Differential Equation: human immunodeficiency virus (HIV)
Here's a simple ordinary differential equation (ODE) model describing the dynamics of HIV infection involving uninfected target cells (T), infected cells (I), and plasma virus (V). This model is known as the Basic Model and is often referred to as the Mass Action Model. A HIV dynamic model [ref] can be written as:
- Target cell dynamics:
dU/dt = X - tVU - pU (1)
- Infected cell dynamics:
dI/dt = tVU - 8U (2)
- Virus dynamics:
dV/dt = N8U - cV (3)
where Ut is the concentration of uninfected target cells at time t, It is the concentration of infected cells at time t, Vt is the concentration of plasma virus at time t, A is the production rate of target cells, t is the time-varying rate of virus infecting uninfected target cells, δ is the death rate of uninfected target cells, η is the death rate of target cells, N is the number of virions produced from each infected cell, and c is the clearance rate of virus. Please note that this is a very basic model and does not capture all the complexities of HIV dynamics, immune responses, and antiretroviral therapy effects. More advanced models incorporate additional factors like immune response, different stages of infected cells, and drug treatment effects. The functions U, I, and V are state variables, and A, t, δ, η, N, and c are unknown dynamic parameters.
(1) Simulation Data generated by ODE with the following parameters [Example Code: www or pdf]:
The initial values: U0, I0, V0 = 600, 30, 10
The true values of parameters: A, t, δ, η, N, c = 36, 0.108, 0.5, 10^3, 3
The time-varying parameter: t = 9 * 10^-5 - 0.9 cos(rt/1000)
2. Consider CD4+ T cell counts T = U + I. Confirm that
dT/dt = X - e^T - 1
dV/dt = αo + αTt + αTt - cVt N&X N8p N&
where:
a) Find α and N in terms of δ and η:
α = p - 8δ
N = p - 8η
3) Observations models for the ODE are:
(4) y1 = Tt + e1i
(5) y2i = Vt + 2i
where i, 2i are independent and follow normal distributions with mean zero and variances σ1^2 = 20, σ2^2 = 30, σ3^2 = 40, and σ4^2 = 100, σ5^2 = 150, σ6^2 = 200, respectively. Note that the simulated data U, I, V were generated by numerically solving equations 1, 2, and 3, and T = U + I. [Matlab code: randn Normally distributed random numbers X ~ N(μ, σ^2), N(a, b) ~ Fa(X + b) ~ N(a + b, σ^2)]
4) Apply smoothing methods (DeNoise processes) to obtain the estimators Vt, Vt, Tt, and Tt of Vt, Vt, Tt, and T(t), respectively.
5) Assume that the parameters δ and η are known as the above parameters and estimate parameters α and N by the least squares (LS) method.
6) Applications to clinical data of HIV. Here are the data about Patient 1 CD4+ T cell data and Viral load, and Patient 2 CD4+ T cell and Viral load can be downloaded. Research articles related to HIV/AIDS often include data on CD4+ T cell counts and other experimental findings. Websites like PubMed can help you search for relevant publications.
