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 .

Predation-induced dispersal toward fitness for predator invasion in predator-prey models

In this paper, we consider a predator-prey model with nonuniform predator dispersal, called predation-induced dispersal (PID), which represents predator motility depending on the maximal predation rate and the predator death rate in a spatially heterogeneous region. We study the local stability of the semitrivial steady state when predators are absent for models with PID and linear dispersal. We then investigate the local/global bifurcation from the semitrivial steady state of these models. Finally, we compare the results of the model with PID to the results of the model with linear dispersal. We conclude that the nonuniform dispersal of predators obeying PID increases fitness for predator invasion when rare; thus, predators with PID can invade a region with an increased probability even in cases wherein predators dispersed linearly cannot invade a certain region. Based on the results, we provide an ecological interpretation with the simulations.

A Local Analysis of a Mathematical Pattern for Interactions between the Human Immune System and a Pathogenic Agent

In the present study, we introduce a four-dimensional deterministic mathematical pattern in order to study the interactions between the human immune system and a virus. The model is based on a system with four first-order ordinary differential equations, and the main aim of the paper is to perform a mathematical analysis of the local behavior of the associated dynamical system using the tools of the qualitative theory of dynamical systems. Moreover, two types of patterns with controls were introduced; consequently, some very interesting theoretical conclusions with medical relevance were obtained.

Local and global stability of an HCV viral dynamics model with two routes of infection and adaptive immunity

The aim of this article is to formulate and study a mathematical model describing hepatitis C virus (HCV) infection dynamics. The model includes two essential modes of infection transmission, namely, virus-to-cell and cell-to-cell. The effect of therapy and adaptive immunity are incorporated in the suggested model. The adaptive immunity is represented by its two categories, namely, the humoral and cellular immune responses. Our article begins by establishing some mathematical results through proving the model's well-posedness in terms of existence, positivity and boundedness of solutions. We present all the steady states of the problem that depend on specific reproduction numbers. It moves then to the theoretical investigation of the local and global stability analysis of the free disease equilibrium and the four disease equilibria. The local and global stability analysis of the HCV mathematical model are established via the Routh-Hurwitz criteria and Lyapunov-LaSalle invariance principle, respectively. Finally, our article presents some numerical simulations to validate the analytical study of the global stability. Numerical simulations have shown the effect of the drug therapies on the system's dynamical behavior.

Intermittent instability is widespread in plankton communities

Chaotic dynamics appear to be prevalent in short-lived organisms including plankton and may limit long-term predictability. However, few studies have explored how dynamical stability varies through time, across space and at different taxonomic resolutions. Using plankton time series data from 17 lakes and 4 marine sites, we found seasonal patterns of local instability in many species, that short-term predictability was related to local instability, and that local instability occurred most often in the spring, associated with periods of high growth. Taxonomic aggregates were more stable and more predictable than finer groupings. Across sites, higher latitude locations had higher Lyapunov exponents and greater seasonality in local instability, but only at coarser taxonomic resolution. Overall, these results suggest that prediction accuracy, sensitivity to change and management efficacy may be greater at certain times of year and that prediction will be more feasible for taxonomic aggregates.

The effects of evolution on the stability of competing species

Based on evolutionary game theory and Darwinian evolution, we propose and study discrete-time competition models of two species where at least one species has an evolving trait that affects their intra-specific, but not their inter-specific competition coefficients. By using perturbation theory, and the theory of the limiting equations of non-autonomous discrete dynamical systems, we obtain global stability results. Our theoretical results indicate that evolution may promote and/or suppress the stability of the coexistence equilibrium depending on the environment. This relies crucially on the speed of evolution and on how the intra-specific competition coefficient depends on the evolving trait. In general, equilibrium destabilization occurs when α>2, when the speed of evolution is sufficiently slow. In this case, we conclude that evolution selects against complex dynamics. However, when evolution proceeds at a faster pace, destabilization can occur when α<2. In this case, if the competition coefficient is highly sensitive to changes in the trait v, destabilization and complex dynamics occur. Moreover, destabilization may lead to either a period-doubling bifurcation, as in the non-evolutionary Ricker equation, or to a Neimark-Sacker bifurcation.

Periodontal and Peri-Implant Microbiome Dysbiosis Is Associated With Alterations in the Microbial Community Structure and Local Stability

Periodontitis and peri-implantitis are common biofilm-mediated infectious diseases affecting teeth and dental implants and have been considered to be initiated with microbial dysbiosis. To further understand the essence of oral microbiome dysbiosis in terms of bacterial interactions, community structure, and microbial stability, we analyzed 64 plaque samples from 34 participants with teeth or implants under different health conditions using metagenomic sequencing. After taxonomical annotation, we computed the inter-species correlations, analyzed the bacterial community structure, and calculated the microbial stability in supra- and subgingival plaques from hosts with different health conditions. The results showed that when inflammation arose, the subgingival communities became less connective and competitive with fewer hub species. In contrast, the supragingival communities tended to be more connective and competitive with an increased number of hub species. Besides, periodontitis and peri-implantitis were associated with significantly increased microbial stability in subgingival microbiome. These findings indicated that the periodontal and peri-implant dysbiosis is associated with aberrant alterations in the bacterial correlations, community structures, and local stability. The highly connected hub species, as well as the major contributing species of negative correlations, should also be given more concern in future studies.

Evolution in pecunia

The paper models evolution in pecunia-in the realm of finance. Financial markets are explored as evolving biological systems. Diverse investment strategies compete for the market capital invested in long-lived dividend-paying assets. Some strategies survive and some become extinct. The basis of our paper is that dividends are not exogenous but increase with the wealth invested in an asset, as is the case in a production economy. This might create a positive feedback loop in which more investment in some asset leads to higher dividends which in turn lead to higher investments. Nevertheless, we are able to identify a unique evolutionary stable investment strategy. The problem is studied in a framework combining stochastic dynamics and evolutionary game theory. The model proposed employs only objectively observable market data, in contrast with traditional settings relying upon unobservable investors' characteristics (utilities and beliefs). Our method is analytical and based on mathematical reasoning. A numerical illustration of the main result is provided.

Dynamics of a diffusive vaccination model with therapeutic impact and non-linear incidence in epidemiology

In this paper, we study a more general diffusive spatially dependent vaccination model for infectious disease. In our diffusive vaccination model, we consider both therapeutic impact and nonlinear incidence rate. Also, in this model, the number of compartments of susceptible, vaccinated and infectious individuals are considered to be functions of both time and location, where the set of locations (equivalently, spatial habitats) is a subset of Rn with a smooth boundary. Both local and global stability of the model are studied. Our study shows that if the threshold level R0≤1, the disease-free equilibrium E0 is globally asymptotically stable. On the other hand, if R0>1 then there exists a unique stable disease equilibrium E∗. The existence of solutions of the model and uniform persistence results are studied. Finally, using finite difference scheme, we present a number of numerical examples to verify our analytical results. Our results indicate that the global dynamics of the model are completely determined by the threshold value R0.

Modeling and optimal control analysis of transmission dynamics of COVID-19: The case of Ethiopia

A mathematical model to estimate transmission dynamics of COVID-19 is developed. A real data of confirmed cases for Ethiopia is used for parameter estimation via model fitting. Results showed that, the diseases free and endemic equilibrium points are found to be locally and globally asymptotically stable for R_o < 1 and R_o > 1 respectively. The basic reproduction number is R_o = 1.5085. Optimal control analysis also showed that, combination of optimal preventive strategies such as public health education, personal protective measures and treatment of hospitalized cases are effective to significantly decrease the number of COVID-19 cases in different compartments of the model.

On global dynamics of COVID-19 by using SQIR type model under non-linear saturated incidence rate

This paper investigates a new mathematical SQIR model for COVID-19 by means of four dimensions; susceptible, quarantine, infected and recovered (SQIR) via Non-linear Saturated Incidence Rate. First of all the model is formulated in the form of differential equations. Disease-free, endemic equilibriums and Basic Reproduction Number are found for the said model. Local Stability is analyzed through Jacobean Matrix while Lyapunov Function is constructed for the study of Global Stability of the Model. Using nonstandard finite difference method, numerical results are simulated. By Simulation, we mean how protection, exposure, death and cure rates affect the Susceptible, Quarantined, Infected and recovered population with the passage of time.

Optimization and Stability of Heat Engines: The Role of Entropy Evolution

Local stability of maximum power and maximum compromise (Omega) operation regimes dynamic evolution for a low-dissipation heat engine is analyzed. The thermodynamic behavior of trajectories to the stationary state, after perturbing the operation regime, display a trade-off between stability, entropy production, efficiency and power output. This allows considering stability and optimization as connected pieces of a single phenomenon. Trajectories inside the basin of attraction display the smallest entropy drops. Additionally, it was found that time constraints, related with irreversible and endoreversible behaviors, influence the thermodynamic evolution of relaxation trajectories. The behavior of the evolution in terms of the symmetries of the model and the applied thermal gradients was analyzed.

Stability analysis of pine wilt disease model by periodic use of insecticides

This work is related to qualitative behaviour of an epidemic model of pine wilt disease. More precisely, we proved that the reproductive number has sharp threshold properties. It has been shown that how vector population can be reduced by the periodic use of insecticides. Numerical simulations show that epidemic level of infected vectors becomes independent of saturation level by including the transmission through mating.

Dynamical behaviour of a two-predator model with prey refuge

A three-component model consisting on one-prey and two-predator populations is considered with a Holling type II response function incorporating a constant proportion of prey refuge. We also consider the competition among predators for their food (prey) and shelter. The essential mathematical features of the model have been analyzed thoroughly in terms of stability and bifurcations arising in some selected situations. Threshold values for some parameters indicating the feasibility and stability conditions of some equilibria are determined. The range of significant parameters under which the system admits different types of bifurcations is investigated. Numerical illustrations are performed in order to validate the applicability of the model under consideration.

Anterior-posterior and medial-lateral control of sway in infants during sitting acquisition does not become adult-like

We examined (1) how sitting postural control in infants develops in the anterior-posterior (A/P) and medial-lateral (M/L) directions of sway, and (2) whether this control is already adult-like during the late phase of infant's sitting acquisition. COP data were acquired from 14 healthy infants (from the onset of sitting until independent sitting) and 21 healthy adults while sitting on a force platform. Attractor dimensionality (CoD: correlation dimension), attractor predictability (LyE: largest Lyapunov exponent), and sway variability (RMS: root-mean square) were calculated from the COP data to evaluate postural control. In the A/P direction, sitting was mastered by the infants by decreasing the active degrees of freedom of the postural system (decreased CoD), using a more predictable and (locally) stable sway (decreased LyE), and increasing sway variability (increased RMS). Control of sitting became practically simple, stable and exploratory with infant development. This may support the hypothesis that the sitting posture serves as the foundation for the development of other motor skills, as reaching. In the M/L direction, only sway variability decreased with development, possibly due to changes in the infant's body dimensions. Taken together, these findings indicate that early in development the focus is more in the A/P than the M/L direction. Adults' postural control was found more adaptable than the infants in both directions, involving more active degrees of freedom and less predictable sway patterns. Identifying the factors that make the dynamics of the postural system adult-like requires further research.