Global dynamics of discrete mathematical models of tuberculosis

In this paper, we develop discrete models of Tuberculosis (TB). This includes SEI endogenous and exogenous models without treatment. These models are then extended to a SEIT model with treatment. We develop two types of net reproduction numbers, one is the traditional R 0 which is based on the disease-free equilibrium, and a new net reproduction number R 0 ( E ∗ ) based on the endemic equilibrium. It is shown that the disease-free equilibrium is globally asymptotically stable if R 0 ≤   1 and unstable if R 0 > 1 . Moreover, the endemic equilibrium is locally asymptotically stable if R 0 ( E ∗ ) < 1 < R 0 .

Mem-fractive properties of mushrooms

Memristors close the loop forI-Vcharacteristics of the traditional, passive, semi-conductor devices. A memristor is a physical realisation of the material implication and thus is a universal logical element. Memristors are getting particular interest in the field of bioelectronics. Electrical properties of living substrates are not binary and there is nearly a continuous transitions from being non-memristive to mem-fractive (exhibiting a combination of passive memory) to ideally memristive. In laboratory experiments we show that living oyster mushroomsPleurotus ostreatusexhibit mem-fractive properties. We offer a piece-wise polynomial approximation of theI-Vbehaviour of the oyster mushrooms. We also report spiking activity, oscillations in conduced current of the oyster mushrooms.

Bifurcation analysis of a model of tuberculosis epidemic with treatment of wider population suggesting a possible role in the seasonality of this disease

In this paper, a novel epidemiological model describing the evolution of tuberculosis in a human population is proposed. This model is of the form SEIR, where S stands for Susceptible people, E for Exposed (infected but not yet infectious) people, I for Infectious people, and R for Recovered people. The main characteristic of this model inspired from the disease biology and some realistic hypothesis is that the treatment is administered not only to infectious but also to exposed people. Moreover, this model is characterized by an open structure, as it considers the transfer of infected or infectious people to other regions more conducive to their care and accepts treatment for exposed or infectious patients coming from other regions without care facilities. Stability and bifurcation of the solutions of this model are investigated. It is found that saddle-focus bifurcation occurs when the contact parameter β passes through some critical values. The model undergoes a Hopf bifurcation when the quality of treatment r is considered as a bifurcation parameter. It is shown also that the system exhibits saddle-node bifurcation, which is a transcritical bifurcation between equilibrium points. Numerical simulations are done to illustrate these theoretical results. Amazingly, the Hopf bifurcation suggests an unexpected and never suggested explanation of seasonality of such a disease, linked to the quality of treatment.

Some elements for a history of the dynamical systems theory

Writing a history of a scientific theory is always difficult because it requires to focus on some key contributors and to "reconstruct" some supposed influences. In the 1970s, a new way of performing science under the name "chaos" emerged, combining the mathematics from the nonlinear dynamical systems theory and numerical simulations. To provide a direct testimony of how contributors can be influenced by other scientists or works, we here collected some writings about the early times of a few contributors to chaos theory. The purpose is to exhibit the diversity in the paths and to bring some elements-which were never published-illustrating the atmosphere of this period. Some peculiarities of chaos theory are also discussed.