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This work seeks to describe intra-solution particle movement system. It makes use of data obtained from simulations of patients on efavirenz. A system of ordinary differential equations is used to model movement state at some particular concentration. The movement states’ description is found for the primary and secondary level. The primary system is found to be predominantly an unstable system while the secondary system is stable. This is derived from the state of dynamic eigenvalues associated with the system. The saturated solution-particle is projected to be stable both for the primary potential and secondary state. A volume conserving linear system has been suggested to describe the dynamical state of movement of a solution particle.

The intra-solution-particle movement system is described in this work. A system which consists of the primary and secondary movement states that is proposed in Nemaura (2015) is investigated. Differential Equations are used to describe dynamical systems. Most systems are described relative to time [

Multiple compartmental modelling finds its use in fields such as Physiologically Based Pharmacokinetics, Engineering and Mathematical Biology [

This work highlights possible intra-particle movement potential/states inferred to be present in the solution particle, at primary and secondary level and attempts to give mathematical form as in Nemaura (2015) [

The primary and secondary movement system of projected simulated data from patients who had been on efavirenz is used [

The primary system is a sub-system of the secondary and describes potential. It is projected to consist of four main movement entities that is convection, saturation, passive and advection. The form entity being the advective component [

A solution particle with concentration (x) is made up of four movement components (variables) at primary level C-convective, A-advective, P-Passive, and S-Saturation at primary level satisfying

where

We analyse the following system of differential equations in order to infer on the overall process occuring in describing state of movement in a solution particle (See

Subject to,

And

With

Solving the primary system above (1 - 5) in terms of advective components,

The equlibrium, steady state points are constant solutions,

which satisfy the nonlinear system of equations

points govern the behaviour of physical models. Thus we obtain

Local analysis is studied near each steady state point

where

The partial derivatives are evaluated at the equilibrium point

(I)Undisturbed potential

Eigenvalues of this system are given by,

(II) Disturbed (dissolving) potential

The dynamic eigenvalues follows from the characteristic equation of J given by,

which is similar,

Thus the eigenvalues for this system are,

And

where

(III) Saturated potential (

The product of advective primary interaction potential with its unique space (the relative uptake) results in a secondary advective movement system [

This is represented by the following constituent system (See

Subject to,

With

inducing movement interaction with four main movement entities,

Solving the secondary system above (13) - (15),

The equlibrium, steady state points are constant solutions,

satisfy the nonlinear system of equations

system is orbitally stable, which signifies that the solutions remain near the equilibrium point.

Local Analysis near Steady-State PointsConsidering,

Thus we obtain,

(space accessory state). It is important to note that the convective movement at secondary level is a stable state which is 0 (nullifying/stabilising accessory state). We study local analysis near each steady state point

where

The Jacobian matrix for the secondary system is given by,

The eigenvalues

The graphical representation of the advective components

The primary system and sub-system equations and estimated parameters are given (See Equations (21) - (27) and

Primary system equation,

Primary

Primary

Advective | Parameters | Estimate | Std Error | t value | |
---|---|---|---|---|---|

0.0034 0.8808 −0.0561 7.4315 0.0089 0.0598 | 0.0002 0.0874 0.0057 0.6922 0.0003 0.0015 | 14.010 10.076 −9.764 10.737 32.289 39.930 | |||

0.0457 1.4843 0.0452 0.2333(Fix) 0.0083 0.3241 | 0.0145 0.2279 0.0147 − 0.0020 0.0360 | 3.146 6.512 3.073 − 4.092 9.011 | 0.0032 | ||

−0.1852 0.3691 −0.2551 0.2333 0.0072 0.0254 | 0.0047 0.0131 0.0064 0.0133 0.0004 0.0007 | −39.74 28.19 −39.84 17.49 17.74 37.56 |

0.1803 0.3549 0.2393 0.2333(Fix) −0.0061 0.0242 | 0.0047 0.0170 0.0028 − 0.0006 0.0014 | 38.489 20.869 84.021 − −9.459 17.326 | |||
---|---|---|---|---|---|

1.6961 3.1504 3.8206 | 0.0036 0.002 0.0954 | 471.62 1599.33 40.04 | |||

−572.9 100.8 |

Primary

Primary

where

Primary

Primary

where

Furthermore,

The graphical representation of the advective components

The secondary system and sub-system equations and estimated parameters are given (See Equations (28) - (31) and

Secondary

Secondary

Secondary

Let

Advective | Parameters | Estimate | Std Error | t value | |
---|---|---|---|---|---|

−0.8281 6.9213 0.1345 0.0566 | 0.0059 0.1615 0.0007 0.0002 | −141.37 42.87 186.53 233.96 | |||

−0.2354 0.4450 0.2771 0.2354 9.199(Fix) −0.0100 0.0470 | 0.0108 0.0235 0.0221 0.0108 − 0.0005 0.0005 | −21.89 18.94 12.57 21.89 − −19.57 86.54 | |||

−0.2665 0.4271 0.2665 9.1990 0.0110 0.0351 | 0.0033 0.0084 0.0033 0.4912 0.0003 0.0004 | −81.76 51 81.76 18.73 33.19 87.31 | |||

0.1013 1.1959 −0.1013 9.199(Fix) −0.0052 0.0304 | 0.0026 0.0400 0.0026 − 0.0002 0.0009 | 38.40 29.90 −38.4 − −23.9 32.26 |

A nonautonomous linear system is developed that enables characterisation of the projected movement in a solution-particle [

The primary system is generally projected to be an unstable system relative to concentration. The saturated system is inferred to have stable potential. The reaction allowing state (non-saturated concentration) has unstable equilibrium potential at primary level. However, the secondary system is stable.

This work has managed to describe the possible state of kinetics of a solution-particle of efavirenz. It has shown that a saturated state can be equated to a stable state both at primary and secondary level. The constructed matrix in this case is traceless that is it has zero trace. The volume conserving systems have been considered in modelling of the physical systems and the theoretical framework is also developed in Lie Theory [

The author would like to thank the following; C. Nhachi, C. Masimirembwa, and G. Kadzirange, AIBST and The College of Health Sciences, University of Zimbabwe.

Nemaura, T. (2016) Modelling the Dynamical State of the Projected Primary and Secondary Intra-Solu- tion-Particle Movement System of Efavirenz In-Vivo. International Journal of Modern Nonlinear Theory and Application, 5, 235- 247. http://dx.doi.org/10.4236/ijmnta.2016.54021