TY - JOUR
T1 - Qualitative analysis of an age-structured SEIR epidemic model with treatment
AU - Safi, Mohammad A.
AU - Gumel, Abba B.
AU - Elbasha, Elamin H.
N1 - Funding Information:
ABG acknowledges, with thanks, the support in part of the Natural Science and Engineering Research Council (NSERC). The authors are grateful to the anonymous reviewers for their constructive comments.
PY - 2013
Y1 - 2013
N2 - A new age-structured model, which incorporates the use of treatment, is designed and qualitatively analysed. The model is, first of all, shown to be properly-posed mathematically by formulating it as an abstract Cauchy problem. For the case where the contact rate is separable (i.e., β(a,b)= β1(a) β2(b)), it is shown that the disease-free equilibrium of the model is locally- and globally-asymptotically stable whenever a certain epidemiological threshold, denoted by R0s, is less than unity. Furthermore, the model has a unique endemic equilibrium when the threshold exceeds unity (this equilibrium is shown to be locally-asymptotically stable if another condition holds). For the case where the natural death and contact rates are constant (i.e., independent of age), the unique endemic equilibrium of the resulting model is shown, using Lyapunov function theory and LaSalle's Invariance Principle, to be globally-asymptotically stable when it exists. Furthermore, for this reduced version of the model (with constant natural death and contact rates), it is shown that the use of treatment could offer positive or negative population-level impact, depending on the size of the parameter associated with the reduction of infectiousness of treated individuals.
AB - A new age-structured model, which incorporates the use of treatment, is designed and qualitatively analysed. The model is, first of all, shown to be properly-posed mathematically by formulating it as an abstract Cauchy problem. For the case where the contact rate is separable (i.e., β(a,b)= β1(a) β2(b)), it is shown that the disease-free equilibrium of the model is locally- and globally-asymptotically stable whenever a certain epidemiological threshold, denoted by R0s, is less than unity. Furthermore, the model has a unique endemic equilibrium when the threshold exceeds unity (this equilibrium is shown to be locally-asymptotically stable if another condition holds). For the case where the natural death and contact rates are constant (i.e., independent of age), the unique endemic equilibrium of the resulting model is shown, using Lyapunov function theory and LaSalle's Invariance Principle, to be globally-asymptotically stable when it exists. Furthermore, for this reduced version of the model (with constant natural death and contact rates), it is shown that the use of treatment could offer positive or negative population-level impact, depending on the size of the parameter associated with the reduction of infectiousness of treated individuals.
KW - Abstract Cauchy problem
KW - Age-structure
KW - C-semigroup
KW - Equilibria
KW - Stability
UR - http://www.scopus.com/inward/record.url?scp=84879059756&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84879059756&partnerID=8YFLogxK
U2 - 10.1016/j.amc.2013.03.126
DO - 10.1016/j.amc.2013.03.126
M3 - Article
AN - SCOPUS:84879059756
SN - 0096-3003
VL - 219
SP - 10627
EP - 10642
JO - Applied Mathematics and Computation
JF - Applied Mathematics and Computation
IS - 22
ER -