The stochastic evolution of a rumor spreading model with two distinct spread inhibiting and attitude adjusting mechanisms in a homogeneous social network.

In this paper, we propose and analyze from a stability viewpoint a deterministic, ODE-based class of rumor spreading models with two distinct inhibiting and adjusting mechanisms, together with its corresponding stochastic counterpart. For the deterministic model, a threshold parameter R 0 defined ad hoc, called the basic influence number, is used to ascertain whether the rumors are prevailing or not. If R 0 < 1 , the rumor-free equilibrium is found to be locally asymptotically stable, while if R 0 > 1 it is shown that there is at least one additional rumor-prevailing equilibrium, which is necessarily locally asymptotically stable. For the stochastic model, we first show that there exists a unique global solution. Subsequently, we investigate the asymptotic behavior of the stochastic system around the equilibria of the deterministic system by constructing suitable Lyapunov functionals. Furthermore, numerical simulations are given to illustrate, support and enhance our theoretical analysis.

Transmission Dynamics and Control Mechanisms of Vector-Borne Diseases with Active and Passive Movements Between Urban and Satellite Cities.

A metapopulation model which explicitly integrates vector-borne and sexual transmission of an epidemic disease with passive and active movements between an urban city and a satellite city is formulated and analysed. The basic reproduction number of the disease is explicitly determined as a combination of sexual and vector-borne transmission parameters. The sensitivity analysis reveals that the disease is primarily transmitted via the vector-borne mode, rather than via sexual transmission, and that sexual transmission by itself may not initiate or sustain an outbreak. Also, increasing the population movements from one city to the other leads to an increase in the basic reproduction number of the later city but a decrease in the basic reproduction number of the former city. The influence of other significant parameters is also investigated via the analysis of suitable partial rank correlation coefficients. After gauging the effects of mobility, we explore the potential effects of optimal control strategies relying upon several distinct restrictions on population movement.

Effects of size refuge specificity on a predator-prey model.

Predation is a major cause of early-stage mortality for prey individuals, which are often forced to use refuges in order to reduce the risk of being consumed. The ability of certain genotypes in a prey population to reach a size refuge from predation may contribute significantly to the preservation of community diversity. We investigate how the specificity of this behavior affects the evolution of a given population by using a modified Lotka-Volterra model, in which the proportion of each genotype available for predation consists of two components: an intrinsic part and a combination from all genotypes present in the population. The trade-off of these components is characterized by a specificity parameter. From the viewpoint of population dynamics, we observe that the ability of the mutant to invade the resident population strongly depends on the values of this parameter. Finally, we describe the possible evolutionary outcomes, analytically and numerically.

The global stability of coexisting equilibria for three models of mutualism.

We analyze the dynamics of three models of mutualism, establishing the global stability of coexisting equilibria by means of Lyapunov's second method. This further establishes the usefulness of certain Lyapunov functionals of an abstract nature introduced in an earlier paper. As a consequence, it is seen that the use of higher order self-limiting terms cures the shortcomings of Lotka-Volterra mutualisms, preventing unbounded growth and promoting global stability.

An impulsively controlled pest management model with n predator species and a common prey.

This paper investigates the dynamics of a competitive single-prey n-predators model of integrated pest management, which is subject to periodic and impulsive controls, from the viewpoint of finding sufficient conditions for the extinction of prey and for prey and predator permanence. The per capita death rates of prey due to predation are given in abstract, unspecified forms, which encompass large classes of death rates arising from usual predator functional responses, both prey-dependent and predator-dependent. The stability and permanence conditions are then expressed as balance conditions between the cumulative death rate of prey in a period, due to predation from all predator species and to the use of control, and to the cumulative birth rate of prey in the same amount of time. These results are then specialized for the case of prey-dependent functional responses, their biological significance being also discussed.

On the impulsive controllability and bifurcation of a predator-pest model of IPM.

From a practical point of view, the most efficient strategy for pest control is to combine an array of techniques to control the wide variety of potential pests that may threaten crops in an approach known as integrated pest management (IPM). In this paper, we propose a predator-prey (pest) model of IPM in which pests are impulsively controlled by means of spraying pesticides (the chemical control) and releasing natural predators (the biological control). It is assumed that the biological and chemical control are used with the same periodicity, but not simultaneously. The functional response of the predator is allowed to be predator-dependent, in the form of a Beddington-DeAngelis functional response, rather than to have a perhaps more classical prey-only dependence. The local and global stability of the pest-eradication periodic solution, as well as the permanence of the system, are obtained under integral conditions which are shown to have biological significance. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations.