9 research outputs found
Complex bifurcation structures in SIRWS and SIRWJS models of pertussis with asymmetric partition of immunity period
Complex bifurcation structures in SIRWS and SIRWJS models of pertussis with asymmetric partition of immunity perio
Dynamics of an SIRWS model with waning of immunity and varying immune boosting period
SIRS models capture transmission dynamics of infectious diseases for which
immunity is not lifelong. Extending these models by a W compartment for
individuals with waning immunity, the boosting of the immune system upon
repeated exposure may be incorporated. Previous analyses assumed identical
waning rates from R to W and from W to S. This implicitly assumes equal length
for the period of full immunity and of waned immunity. We relax this
restriction, and allow an asymmetric partitioning of the total immune period.
Stability switches of the endemic equilibrium are investigated with a
combination of analytic and numerical tools. Then, continuation methods are
applied to track bifurcations along the equilibrium branch. We find rich
dynamics: Hopf bifurcations, endemic double bubbles, and regions of
bistability. Our results highlight that the length of the period in which
waning immunity can be boosted is a crucial parameter significantly influencing
long term epidemiological dynamics
Construction and Validation of a Test for Measuring Studentsâ Proportional Reasoning on Rates, Ratios, & Proportions
This study aimed to develop and validate a proportional reasoning test (PRT) to measure studentsâ proportional reasoning skills on rates, ratios, and proportions in mathematics. The test items were carefully designed to reflect the aspects of proportional reasoning identified in the existing literature and aligned them to the instructional objectives as stated in the Kenya secondary mathematics curriculum. The test underwent different developmental processes to establish content-related validity before it was piloted. The pilot was conducted on a sample of 45 form three students from one school in Kenya. The results showed an acceptable internal consistency level (Cronbachâs α= .83). The item analysis revealed that all the items had a moderate difficulty level ranging from .39 to .50. The discrimination index for most items ranged from .22 to .44. The findings suggest that the PRT is a valid and reliable instrument that can be used to measure the domain-specific skills on proportional reasoning. More specifically, the instrument can be used to measure the impact of a formative assessment intervention on studentsâ achievement on proportional reasoning skills
Bifurcation analysis of waning-boosting epidemiological models with repeat infections and varying immunity periods
We consider the SIRWJS epidemiological model that includes the waning and
boosting of immunity via secondary infections. We carry out combined analytical
and numerical investigations of the dynamics. The formulae describing the
existence and stability of equilibria are derived. Combining this analysis with
numerical continuation techniques, we construct global bifurcation diagrams
with respect to several epidemiological parameters. The bifurcation analysis
reveals a very rich structure of possible global dynamics. We show that
backward bifurcation is possible at the critical value of the basic
reproduction number, . Furthermore, we find stability
switches and Hopf bifurcations from steady states forming multiple endemic
bubbles, and saddle-node bifurcations of periodic orbits. Regions of
bistability are also found, where either two stable steady states, or a stable
steady state and a stable periodic orbit coexist. This work provides an insight
to the rich and complicated infectious disease dynamics that can emerge from
the waning and boosting of immunity