In this paper, two novel extensions of Banach and Meir-Keeler fixed point theorems for $L$-fuzzy set-valued map defined on a complete metric space are presented by using new general contractive conditions under suitable assumptions. In the case when the $L$-fuzzy set-valued map is reduced to its crisp counterparts, our obtained concepts herein generalize several significant fixed point theorems of point-to-point and point-to-set valued mappings in the comparable literature. We provide non-trivial examples to validate the hypotheses of our results.
Banana Xanthomonas Wilt (BXW) is an infectious disease caused by Xanthomonas campestris pv. musacearum. The model incorporates a new class of protected banana plants into banana plant population. This new class are the susceptible banana plant that are treated with fertilizers. The basic reproduction number, Ro , is obtained using next generation matrix. The model analysis is done and equilibrium points are analysed to establish the local and global stability of disease-free and endemic equilibrium solution. It is shown that if the basic reproduction number, Ro≤1, then banana xanthomonas wilt is cleared from banana plantation and is globally asymptotically stable and if Ro>1, the endemic equilibrium point is globally asymptotically stable and the disease persists in banana plant population. The impact of parameters in BXW model is investigated using sensitivity analysis. Numerical simulations are performed to justify the analytical findings.
Choosing a response surface design to fit certain kinds of models is a difficult task. Extensive research comprising a collection of efficient second-order response surface designs from which a researcher may choose to best fit his/her needs has been conducted, which are based solely on a widely-accepted assumption of a completely randomized error structure of statistically-designed experiments. However, this assumption is not feasible in industrial experiments, which are often split-plot in nature and for which randomization of some factors have to be restricted due to certain constraints. The performance of such experimental designs depends strongly on the relative magnitude (d) of the whole-plot and sub-plot error variances. This work focuses on reduced second-order models having one, two, or all of their quadratic and/or interaction terms removed from the full models of some chosen candidate split-plot central composite designs (CCDs). It investigates the effects of model reduction on efficiency of these designs by computing the relative D-efficiencies for the formulated reduced models with respect to their corresponding full designs and assessing the efficiency losses under specific values of d. The study revealed a significant loss of D-efficiency in these designs, which depend strongly on the removed term(s) and increases, across all values of d, as the number of whole-plot factors increases.
In the present paper, we have focused on the antiviral treatment of avian influenza to predict the situation of the disease and analyzed the stability of the model at the equilibrium points (disease-free and endemic). In this concern, we have applied the SITR model based on the well-known SIR model to calculate the basic reproduction number and final size relation. Important parameters, such as susceptible, infective, treatment, and removal (SITR) rate under the compartmental method have been studied theoretically. The analytical results highlight that the model results are locally and globally stable at disease-free equilibrium if the basic reproduction number is less than one and it is locally and globally stable at endemic equilibrium if the basic reproduction number is greater than one. The numerical simulations of the developed model (SITR) are performed graphically with the help of the Range-Kutta method and we have also observed that each compartment has much affected by the infection and death rate, whereas re-susceptive has no significant effect on compartments.
This given paper can be considered as a continuation of previous work, doing on cancer models and their control by a set-valued method, in the context of viability theory. We analyze a class models of ordinary differential equations, taking into account the possibility to directly acting on tumor. However we can augment the class by a simple ordinary differential equation of the tumor control term, and join it to the other variables state. This will allow to exploit results to generalize the approach.
