Study Design: The mathematical model is constructed based on contemporary research data condensed from the clinical literature on HAART of AIDS. A system of coupled non-linear deterministic differential equations are used to characterize the patho-physiodynamics of HAART during the post chronic phase. The mathematical analysis of the model equations and the computer simulations are performed with regard to HAART protocols with constant continuous intravenous and transdermal drug infusions. A syncytium term with stoichiometric coefficient is introduced into the model to account for the formation of large multinucleated gp120 bearing CD4+ T cells that are observed in some AIDS patients. By assigning a zero value to the stoichiometric coefficient, the role of syncytium is abrogated.

Place and Duration of Study: This research was done at Fayetteville State University, North Carolina USA, and is sponsored by the FSU Mini-Grant Award and the HBCU Graduate STEM Grant. The research was conducted during the Spring of 2012.

Methodology: The deterministic nonlinear dynamics of HIV-1 AIDS equations are analyzed using the techniques of dynamical system theory, principles of linearized stability, non-linear system theory, and other relevant mathematical techniques. The clinically desirable equilibrium states, their local existence, and global stability are analyzed. Investigative computer simulations are performed illustrating some physiological outcomes.

Results: The therapeutic outcomes are presented in the form of theoretical criteria which are obtained from mathematical analysis of the model equations. In particular, the critical parameters which govern the dynamics of HIV-1 virions during HAART are clearly identified. Some clinical implications of HAART are elucidated in the computer simulations using hypothetical physiological parametric configurations.

Conclusion: This research has demonstrated the existence of plausible criteria under which HIV-1 virions can be annihilated using HAART. The latent HIV-1 virion reservoirs are implicated in unsuccessful scenarios of HAART. It has also been demonstrated that the rate constants associated with activation of lymphocytes by cytokine interleukin-2 (IL-2) play a significant role in determining the efficacious outcomes of HAART.

]]>Study Design: The model is based on contemporary published patho-physiological data on acute and clinical chronic phase HIV-1 induced AIDS. These data are meticulously condensed into a clinically plausible four-compartmental mathematical model that incorporates the dynamics and interactions between non-HIV-1 infected CD4+ T lymphocytes. HIV-1 infected lymphocytes, free HIV-1 virions in the blood plasma, and HIV-1 specific cytotoxic CD8+ T lymphocytes. The relevant stoichiometric interaction rate constants, apoptotic rate constants, rate constants for viral recruitment from latent reservoirs, and other relevant parameters are clearly exhibited in the mathematical model. The role of CD4+ T cell-induced syncytia is explicitly incorporated into the HIV-1 virion dynamical equation.

Place and Duration of Study: This research was done at Fayetteville State University, North Carolina USA and is sponsored by the FSU Mini-Grant Award and the HBCU Graduate STEM Grant. The research was conducted during the Spring of 2012.

Methodology: The deterministic nonlinear HIV-1 AIDS patho-physio-dynamical equations are analyzed using the techniques of dynamical system theory, principles of linearized stability, Hartman-Grobman theory and other relevant mathematical techniques. The clinically desirable equilibrium states are and their local existence and global stability are analyzed. Investigative computer simulations are performed illustrating some physiological outcomes.

Results: Mathematical criteria are derived under which the clinically desired outcomes can occur. These criteria are presented in terms of theorems. Investigative computer simulations are presented which elucidate a number of physiological scenarios of primary HIV-1 infection, involving the annihilation and persistence of HIV-1 in the absence of AIDS Pharmacotherapy.

Conclusion: This research has demonstrated the existence of plausible criteria under which an HIV-1 sero-positive person can be maintained at an asymptomatic chronic state indefinitely. Some of the criteria are configured in terms of clinically measurable and biological quantifiable parameters which have been verified by the computer simulations.

]]>- Aims: To construct a clinically plausible mathematical model of the patho-physiological dynamics of HIV-1 induced AIDS during the acute and chronic phases which incorporates the interactions between uninfected CD4+ T cells, HIV-1 infected CD4+ T cells, HIV-1 virions in the blood plasma, and specific cytotoxic CD8+ T cells. In particular, the model describes quantitatively the time evolution of AIDS in the patient during the acute phase and the asymptomatic chronic clinical latency phase and elucidates the effect of latent HIV-1 reservoirs on the prognosis of AIDS. The major objective is to derive mathematical criteria depicting the necessary and sufficient conditions under which the HIV-1 virions can be maintained definitely at the subclinical viral blood plasma level such that the HIV-1 seriopositive person does not develop full-blown AIDS.
- Study design: The model is based on contemporary published patho-physiological data on acute and clinical chronic phase HIV-1 induced AIDS. These data are meticulously condensed into a clinically plausible four compartmental mathematical model that incorporates the dynamics and interactions between non-HIV-1 infected CD4+ T lymphocytes. HIV-1 infected lymphocytes, free HIV-1 virions in the blood plasma, and HIV-1 specific cytotoxic CD8+ T lymphocytes. The relevant stoichiometric interaction rate constants, apoptotic rate constants, rate constants for viral recruitment from latent reservoirs, and other relevant parameters are clearly exhibited in the mathematical model.
- Place and Duration of Study: This research was done at Fayetteville State University, North Carolina USA, and is sponsored by the FSU Mini-Grant Award and the HBCU Graduate STEM Grant. The research was done during the Spring of 2012.
- Methodology: The deterministic nonlinear HIV-1 AIDS patho-physio-dynamical equations are analyzed using the techniques of dynamical system theory, principles of linearized stability, Hartman-Grobman theory, and other relevant mathematical techniques. The clinically desirable equilibrium states are and their local existence and global stability are analyzed. Investigative computer simulations are performed illustrating some physiological outcomes.
- Results: Mathematical criteria are derived under which the clinically desired outcomes can occur. Investigative computer simulations are presented which elucidate a number of physiological scenarios of primary HIV-1 infection, involving the annihilation, and persistence of HIV-1 in the absence of AIDS Pharmacotherapy.
- Conclusion: Mathematical modeling can be a useful technique in the derivation of prognostic criteria and quantitative analysis of AIDS during the acute and chronic phases.

This project is devoted to developing Liapunov direct method for fractional differential equations and systems. The method (constructing a system related scalar function) enables investigators to analyze the qualitative behavior of solutions of a differential equation without actually solving it. We are able to convert some fractional differential equations (semi-linear) to integral equations with singular kernels and construct Liapunov functionals for the integral equations to deduce conditions for boundedness and stability of solutions. Extending such a method to fully nonlinear equations presents a significant challenge to investigators and will be a major area of research for many years to come.

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tralizers and Lie centralizers on triangular rings and nest algebras are also

presented. ]]>