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University of Manitoba

1. Safi, Mohammad. Mathematical Analysis of The Role of Quarantine and Isolation in Epidemiology.

Degree: Mathematics, 2010, University of Manitoba

URL: http://hdl.handle.net/1993/4069

The quarantine of people suspected of being exposed to a disease, and the isolation of those with clinical symptoms of the disease, constitute what is probably the oldest infection control mechanism since the beginning of recorded human history. The thesis is based on using mathematical modelling and analysis to gain qualitative insight into the transmission dynamics of a disease that is controllable using quarantine and
isolation. A basic model, which takes the form of an autonomous deterministic system of non-linear differential equations with standard incidence, is formulated first of all.
Rigorous analysis of the basic model shows that its disease-free equilibrium is globally-asymptotically stable whenever a certain epidemiological threshold (denoted by Rc) is less than unity. The epidemiological implication of this result is that the disease will
be eliminated from the community if the use of quarantine and isolation could result in making Rc < 1. The model has a unique endemic equilibrium whenever Rc > 1. Using a Lyapunov function of Goh-Volterra type, it is shown that the unique endemic equilibrium is globally-asymptotically stable for a special case. The basic model is extended to
incorporate various epidemiological and biological aspects relating to the transmission dynamics and control of a communicable disease, such as the use of time delay to model the latency period, effect of periodicity (seasonality), the use of an imperfect vaccine and the use of multiple latent and infectious stages (coupled with gamma-distributed
average waiting times in these stages). One of the main mathematical findings of this thesis is that adding time delay, periodicity and multiple latent and infectious stages to the basic quarantine/isolation model does not alter the essential qualitative features
of the basic model (pertaining to the persistence or elimination of the disease). On the other hand, the use of an imperfect vaccine induces the phenomenon of backward bifurcation (a dynamical feature not present in the basic model), the consequence of which is that disease elimination becomes more difficult using quarantine and isolation (since, in this case, the epidemiological requirement Rc < 1 is, although necessary,
no longer sufficient for disease elimination). Numerous numerical simulations are carried out, using parameter values relevant to the 2003 SARS outbreaks in the Greater Toronto Area of Canada, to illustrate some of the theoretical findings as well as to evaluate the population-level impact of quarantine/isolation and an imperfect vaccine. In particular, threshold conditions for which the aforementioned control measures could
have a positive or negative population-level impact are determined.
*Advisors/Committee Members: Gumel, Abba (Mathematics), Shaun, Lui (Mathematics) (supervisor), Williams, Joseph (Mathematics).*

Subjects/Keywords: Isolation; quarantine; equilibria

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Safi, M. (2010). Mathematical Analysis of The Role of Quarantine and Isolation in Epidemiology. (Thesis). University of Manitoba. Retrieved from http://hdl.handle.net/1993/4069

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Safi, Mohammad. “Mathematical Analysis of The Role of Quarantine and Isolation in Epidemiology.” 2010. Thesis, University of Manitoba. Accessed September 22, 2019. http://hdl.handle.net/1993/4069.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Safi, Mohammad. “Mathematical Analysis of The Role of Quarantine and Isolation in Epidemiology.” 2010. Web. 22 Sep 2019.

Vancouver:

Safi M. Mathematical Analysis of The Role of Quarantine and Isolation in Epidemiology. [Internet] [Thesis]. University of Manitoba; 2010. [cited 2019 Sep 22]. Available from: http://hdl.handle.net/1993/4069.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Safi M. Mathematical Analysis of The Role of Quarantine and Isolation in Epidemiology. [Thesis]. University of Manitoba; 2010. Available from: http://hdl.handle.net/1993/4069

Not specified: Masters Thesis or Doctoral Dissertation

University of Manitoba

2. Sharomi, Oluwaseun Yusuf. Mathematical Analysis of Dynamics of Chlamydia trachomatis.

Degree: Mathematics, 2010, University of Manitoba

URL: http://hdl.handle.net/1993/4117

Chlamydia, caused by the bacterium Chlamydia trachomatis, is one of the most important sexually-transmitted infections globally. In addition to accounting for millions of cases every year, the disease causes numerous irreversible complications such as chronic pelvic pain, infertility in females and pelvic inflammatory disease. This thesis presents a number of mathematical models, of the form of deterministic systems of non-linear differential equations, for gaining qualitative insight into the transmission dynamics and control of Chlamydia within an infected host (in vivo) and in a population. The models designed address numerous important issues relating to the transmission dynamics of Chlamydia trachomatis, such as the roles of immune response, sex structure, time delay (in modelling the latency period) and risk structure (i.e., risk of acquiring or transmitting infection). The in-host model is shown to have a globally-asymptotically stable Chlamydia-free equilibrium whenever a certain biological threshold is less than unity. It has a unique Chlamydia-present equilibrium when the threshold exceeds unity. Unlike the in-host model, the two-group (males and females) population-level model undergoes a backward bifurcation, where a stable disease-free equilibrium co-exists with one or more stable endemic equilibria when the associated reproduction number is less than unity. This phenomenon, which is shown to be caused by the re-infection of recovered individuals, makes the effort to eliminate the disease from the population more difficult. Extending the two-group model to incorporate risk structure shows that the backward bifurcation phenomenon persists even when recovered individuals do not acquire re-infection. In other words, it is shown that stratifying the sexually-active population in terms of risk of acquiring or transmitting infection guarantees the presence of backward bifurcation in the transmission dynamics of Chlamydia in a population. Finally, it is shown (via numerical simulations) that a future Chlamydia vaccine that boosts cell-mediated immune response will be more effective in curtailing Chlamydia burden in vivo than a vaccine that enhances humoral immune response. The population-level impact of various targeted treatment strategies, in controlling the spread of Chlamydia in a population, are compared. In particular, it is shown that the use of treatment could have positive or negative population-level impact (depending on the sign of a certain epidemiological threshold).
*Advisors/Committee Members: Gumel, Abba (Mathematics) (supervisor), Williams, Joseph (Mathematics), .*

Subjects/Keywords: Chlamydia trachomatis; Mathematical epidemiology; Persistence theory; Permanence theory; Mathematical biology; Lyapunov functions; Equilibria; Reproduction number

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Sharomi, O. Y. (2010). Mathematical Analysis of Dynamics of Chlamydia trachomatis. (Thesis). University of Manitoba. Retrieved from http://hdl.handle.net/1993/4117

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Sharomi, Oluwaseun Yusuf. “Mathematical Analysis of Dynamics of Chlamydia trachomatis.” 2010. Thesis, University of Manitoba. Accessed September 22, 2019. http://hdl.handle.net/1993/4117.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Sharomi, Oluwaseun Yusuf. “Mathematical Analysis of Dynamics of Chlamydia trachomatis.” 2010. Web. 22 Sep 2019.

Vancouver:

Sharomi OY. Mathematical Analysis of Dynamics of Chlamydia trachomatis. [Internet] [Thesis]. University of Manitoba; 2010. [cited 2019 Sep 22]. Available from: http://hdl.handle.net/1993/4117.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Sharomi OY. Mathematical Analysis of Dynamics of Chlamydia trachomatis. [Thesis]. University of Manitoba; 2010. Available from: http://hdl.handle.net/1993/4117

Not specified: Masters Thesis or Doctoral Dissertation

University of Manitoba

3. Podder, Chandra Nath. Mathematics of HSV-2 Dynamics.

Degree: Mathematics, 2010, University of Manitoba

URL: http://hdl.handle.net/1993/4082

The thesis is based on using dynamical systems theories and techniques to study the qualitative dynamics of herpes simplex virus type 2 (HSV-2), a sexually-transmitted disease of major public health significance.
A deterministic model for the interaction of the virus with the immune system in the body of an infected individual (in vivo) is designed first
of all. It is shown, using Lyapunov function and LaSalle's Invariance Principle, that
the virus-free equilibrium of the model is globally-asymptotically stable whenever a
certain biological threshold, known as the reproduction number, is less than unity. Furthermore, the model has at least one virus-present equilibrium when the threshold quantity exceeds unity. Using persistence theory, it is shown that the virus will always be present in vivo whenever the reproduction threshold exceeds unity. The analyses (theoretical and numerical) of this model show that a future HSV-2 vaccine that enhances cell-mediated immune response will be effective in
curtailling HSV-2 burden in vivo.
A new single-group model for the spread of HSV-2 in
a homogenously-mixed sexually-active population is also designed. The disease-free equilibrium of the model is globally-asymptotically stable when its associated reproduction number is less
than unity. The model has a unique endemic equilibrium, which is shown to be
globally-stable for a special case, when the reproduction number exceeds unity.
The model is extended to incorporate an imperfect vaccine with some therapeutic benefits.
Using centre manifold theory, it is shown that the resulting vaccination model undergoes a vaccine-induced backward bifurcation (the epidemiological
importance of the phenomenon of backward bifurcation is that the
classical requirement of having the reproduction threshold less than unity is, although necessary, no longer sufficient for disease elimination. In such a case, disease elimination depends upon the initial sizes of the
sub-populations of the model). Furthermore, it is shown that the use of such an
imperfect vaccine could lead to a positive or detrimental population-level impact (depending on the sign of a certain threshold quantity).
The model is extended to incorporate the effect of variability in HSV-2 susceptibility due to gender differences. The resulting two-group (sex-structured) model is shown to have essentially the
same qualitative dynamics as the single-group model. Furthermore, it is shown that adding periodicity to the corresponding autonomous two-group model does not alter the dynamics of the autonomous two-group model (with respect to the elimination of the disease). The model is used to evaluate the impact of various anti-HSV control strategies.
Finally, the two-group model is further extended to address the effect of risk structure (i.e., risk of acquiring or transmitting HSV-2). Unlike the two-group model described above, it
is shown that the risk-structured model undergoes backward
bifurcation under certain conditions (the backward bifurcation property can be…
*Advisors/Committee Members: Gumel, Abba (Mathematics) (supervisor), Lui, Shaun (Mathematics), Williams, Joseph (Mathematics), Shamseddine, Khodr (Physics and Astronomy), Castillo-Chavez, Carlos (Arizona State University) (examiningcommittee).*

Subjects/Keywords: Epidemiology; Mathematical Modeling; Disease-free Equilibrium; Local Stability; Global Stability; Basic Reproduction Number; Endemic Equilibrium; Lyapunov Function; Centre Manifold Theory; Backward Bifurcation

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Podder, C. N. (2010). Mathematics of HSV-2 Dynamics. (Thesis). University of Manitoba. Retrieved from http://hdl.handle.net/1993/4082

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Podder, Chandra Nath. “Mathematics of HSV-2 Dynamics.” 2010. Thesis, University of Manitoba. Accessed September 22, 2019. http://hdl.handle.net/1993/4082.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Podder, Chandra Nath. “Mathematics of HSV-2 Dynamics.” 2010. Web. 22 Sep 2019.

Vancouver:

Podder CN. Mathematics of HSV-2 Dynamics. [Internet] [Thesis]. University of Manitoba; 2010. [cited 2019 Sep 22]. Available from: http://hdl.handle.net/1993/4082.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Podder CN. Mathematics of HSV-2 Dynamics. [Thesis]. University of Manitoba; 2010. Available from: http://hdl.handle.net/1993/4082

Not specified: Masters Thesis or Doctoral Dissertation