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 Students’ Proportional Reasoning on Rates, Ratios, & Proportions

This study aimed to develop and validate a proportional reasoning test (PRT) to measure students’ 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 (Cronbach’s 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 students’ 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, $\mathcal{R}_0 = 1$. 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