Mathematical models have been of great importance when attempting to analyze problems that are faced by society. With the widespread outbreak of HIV, many models have been established to explain some of the characteristics of the HIV infection. In this research, we use a three dimensional model to study the interactions of uninfected CD4+ T cells, actively infected CD4+ T cells, and viral particles.
For the model described by a three-dimensional ODE system, a bounded positively invariant set Γ in R3+ is found. We show that if N < Nc, that is if the number of viral particles produced by a single actively infected T cell is less than or equal to a critical number Nc, the system has only one uninfected equilibrium point P0 on the boundary of Γ. If N > Nc, that is if the number of viral particles produced by an actively infected CD4+ T cell is greater than the critical value, then there are two equilibrium points within the feasible region Γ; namely P0 and an endemically infected equilibrium point in the interior of Γ.
The local and global stability of each equilibrium point is then studied. If N < Nc, the local stability of the uninfected state is confirmed by showing the eigenvalues of the Jacobian matrix have negative real parts. A Lyapunov function is found to show the global stability of the uninfected case. Thus, we show that if N < Nc then all solutions starting inside Γ will converge to P0. Biologically this means that the HIV infection dies out in time and no disease persists. If N > Nc, P0 becomes unstable. The local stability of the endemically infected equilibrium point P1 is established using the Routh-Hurwitz conditions and the global stability of P1 is then evaluated numerically. The mathematical results show that when N > Nc the solutions always converge to an endemically infected steady state. Thus the disease persists.
","abstract_html":"<p>Mathematical models have been of great importance when attempting to analyze problems that are faced by society. With the widespread outbreak of HIV, many models have been established to explain some of the characteristics of the HIV infection. In this research, we use a three dimensional model to study the interactions of uninfected CD4<sup>+</sup> T cells, actively infected CD4<sup>+</sup> T cells, and viral particles.</p> <p>For the model described by a three-dimensional ODE system, a bounded positively invariant set Γ in R<sup>3</sup><sub>+</sub> is found. We show that if N < N<sub><em>c</em></sub>, that is if the number of viral particles produced by a single actively infected T cell is less than or equal to a critical number N<sub><em>c</em></sub>, the system has only one uninfected equilibrium point P<sub>0 </sub>on the boundary of Γ. If N > N<sub><em>c</em></sub>, that is if the number of viral particles produced by an actively infected CD4<sup>+</sup> T cell is greater than the critical value, then there are two equilibrium points within the feasible region Γ; namely P<sub>0</sub> and an endemically infected equilibrium point in the interior of Γ.</p> <p>The local and global stability of each equilibrium point is then studied. If N < N<sub>c</sub>, the local stability of the uninfected state is confirmed by showing the eigenvalues of the Jacobian matrix have negative real parts. A Lyapunov function is found to show the global stability of the uninfected case. Thus, we show that if N < N<sub><em>c</em></sub> then all solutions starting inside Γ will converge to P<sub>0</sub>. Biologically this means that the HIV infection dies out in time and no disease persists. If N > N<sub><em>c</em></sub>, P<sub>0</sub> becomes unstable. The local stability of the endemically infected equilibrium point P<sub>1</sub> is established using the Routh-Hurwitz conditions and the global stability of P<sub>1</sub> is then evaluated numerically. The mathematical results show that when N > N<sub><em>c</em></sub> the solutions always converge to an endemically infected steady state. Thus the disease persists.</p>","abstract_has_math":false,"creators":["Hart, Samuel Bee"],"institution":null,"degree_name":"Master of Science","degree_level":"Thesis (restricted to Georgia Southern)","degree_discipline":"Department of Mathematics","degree_department":null,"school":null,"contributors":["Donald Fausett","Matthew Schuette"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2003,"date_issued":"2003-01-01T08:00:00Z","date_published":"2003-01-01T08:00:00Z","updated_at":"2026-07-24T02:28:33Z","subjects":["ETD","Mathematical models","HIV","ODE system","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://digitalcommons.georgiasouthern.edu/etd_legacy/440","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Donald Fausett","Matthew Schuette"]},{"key":"dc:creator","label":"Author","values":["Hart, Samuel Bee"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["1970-01-01T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Department of Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis (restricted to Georgia Southern)"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["ETD","Mathematical models","HIV","ODE system","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.georgiasouthern.edu/etd_legacy/440"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Mathematical models have been of great importance when attempting to analyze problems that are faced by society. With the widespread outbreak of HIV, many models have been established to explain some of the characteristics of the HIV infection. In this research, we use a three dimensional model to study the interactions of uninfected CD4+ T cells, actively infected CD4+ T cells, and viral particles.
For the model described by a three-dimensional ODE system, a bounded positively invariant set Γ in R3+ is found. We show that if N < Nc, that is if the number of viral particles produced by a single actively infected T cell is less than or equal to a critical number Nc, the system has only one uninfected equilibrium point P0 on the boundary of Γ. If N > Nc, that is if the number of viral particles produced by an actively infected CD4+ T cell is greater than the critical value, then there are two equilibrium points within the feasible region Γ; namely P0 and an endemically infected equilibrium point in the interior of Γ.
The local and global stability of each equilibrium point is then studied. If N < Nc, the local stability of the uninfected state is confirmed by showing the eigenvalues of the Jacobian matrix have negative real parts. A Lyapunov function is found to show the global stability of the uninfected case. Thus, we show that if N < Nc then all solutions starting inside Γ will converge to P0. Biologically this means that the HIV infection dies out in time and no disease persists. If N > Nc, P0 becomes unstable. The local stability of the endemically infected equilibrium point P1 is established using the Routh-Hurwitz conditions and the global stability of P1 is then evaluated numerically. The mathematical results show that when N > Nc the solutions always converge to an endemically infected steady state. Thus the disease persists.
"]},{"key":"dc:title","label":"Title","values":["A Three Dimensional Model of Virus Dynamics in HIV Patients"]}]}],"canonical_facts":{"dc:contributor":["Donald Fausett","Matthew Schuette"],"dc:creator":["Hart, Samuel Bee"],"dc:date.available":["1970-01-01T08:00:00Z"],"dc:description.abstract":["Mathematical models have been of great importance when attempting to analyze problems that are faced by society. With the widespread outbreak of HIV, many models have been established to explain some of the characteristics of the HIV infection. In this research, we use a three dimensional model to study the interactions of uninfected CD4+ T cells, actively infected CD4+ T cells, and viral particles.
For the model described by a three-dimensional ODE system, a bounded positively invariant set Γ in R3+ is found. We show that if N < Nc, that is if the number of viral particles produced by a single actively infected T cell is less than or equal to a critical number Nc, the system has only one uninfected equilibrium point P0 on the boundary of Γ. If N > Nc, that is if the number of viral particles produced by an actively infected CD4+ T cell is greater than the critical value, then there are two equilibrium points within the feasible region Γ; namely P0 and an endemically infected equilibrium point in the interior of Γ.
The local and global stability of each equilibrium point is then studied. If N < Nc, the local stability of the uninfected state is confirmed by showing the eigenvalues of the Jacobian matrix have negative real parts. A Lyapunov function is found to show the global stability of the uninfected case. Thus, we show that if N < Nc then all solutions starting inside Γ will converge to P0. Biologically this means that the HIV infection dies out in time and no disease persists. If N > Nc, P0 becomes unstable. The local stability of the endemically infected equilibrium point P1 is established using the Routh-Hurwitz conditions and the global stability of P1 is then evaluated numerically. The mathematical results show that when N > Nc the solutions always converge to an endemically infected steady state. Thus the disease persists.
"],"dc:identifier":["https://digitalcommons.georgiasouthern.edu/etd_legacy/440"],"dc:subject":["ETD","Mathematical models","HIV","ODE system","Mathematics"],"dc:title":["A Three Dimensional Model of Virus Dynamics in HIV Patients"],"thesis:degree_discipline":["Department of Mathematics"],"thesis:degree_level":["Thesis (restricted to Georgia Southern)"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T02:28:33Z"}