Annals of Communications in Mathematics

Abstract:In our study, we propose a vector-host epidemic model with fixed-time pulse interventions guided by population size of susceptible humans and susceptible vectors. This approach aims to reflect the situation where public health interventions, namely, vaccination campaigns, mass treatments, insecticide spraying, or sterile mosquito releases, are generally implemented periodically rather than continuously. A saturated incidence function is used to fit the situation where the number of infective cases is getting larger with the limited medical facilities. The model incorporates a general rule of implementation of control measures such us vaccination, treatment and mosquitoes elimination. We theoretically investigate the dynamic behavior with the appearance of (k + m)T - periodic solutions. Firstly, we derive the existence and stability of the order-1 (k + 1)T - disease-free periodic solution with nonnegative integer k and the order-m (1 + m)T -disease-free periodic solution with m being any positive integer. Furthermore, we study the existence and stability of the disease-free (1 + 2)T -periodic solution. In addition, numerical simulations are performed to support the theoretical results. Our results indicate that there exists a critical monitoring period or vaccination rate such that the intervention strategy can successfully control the disease.

Abstract:This paper presents the notion of ideal leras topological spaces as an extension of leras topological spaces. The connections between these two classes of spaces are examined in detail. Different forms of closed sets arising in ideal leras topological spaces are introduced and their interrelations are discussed. Several characterizations and fundamental properties associated with these sets are obtained. Examples are included to clarify the introduce concepts and the corresponding results.

Abstract:This paper studies \( B \)-algebras through the Boolean-centre-valued operators \( x^\nabla = 1 \to x \) and \( x^\Delta = 0 \leftarrow x \). The operator \( {}^\Delta \) defines a centre-valued negation satisfying \( (x \vee y)^\Delta = x^\Delta \wedge y^\Delta \) for all \( x,y \in A \), together with the inequality \( x^\Delta \vee y^\Delta \leq (x \wedge y)^\Delta \). Equality in the latter holds on the Boolean centre \( B(A) \); an explicit example confirms that restricting to \( B(A) \) is necessary. The Boolean centre, equipped with the restricted operations, satisfies the equations of one standard axiomatisation of Nelson algebras [14]. An involutive distributive lattice \( \mathcal{N}(A) = A \times A \) is constructed together with an injective map \( \iota(x) = (x,x^\Delta) \). The embedding preserves joins globally and preserves meets on \( B(A) \). Prime-spectrum and prime-filter representation results are obtained for \( B \)-algebras. Under an injectivity condition on the pseudo-supplement map, an ultrafilter representation follows. The bi-implication \( x \leftrightarrow y = (x \Rightarrow y) \wedge (y \Rightarrow x) \) characterises equality and satisfies a transitivity inequality.

Abstract: The purpose of this paper is to extend the study of generalized closed sets in leras topological spaces to the ideal setting. Using the leras local function and leras \( \star \)-closure operator associated with an ideal, we introduce and investigate two new classes of sets, called \( ideal\ leras\ generalized\ closed\ sets \) (\( I_{lrg} \)-closed sets) and \( ideal\ leras\ generalized\ open\ sets \) (\( I_{lrg} \)-open sets), in ideal leras topological spaces. We show that every \( lrg \)-closed set is \( I_{lrg} \)-closed, and that every \( I^\star \)-closed set is \( I_{lrg} \)-closed, while the converses fail in general, thereby placing the new classes correctly within the existing hierarchy of sets in leras and ideal leras topological spaces. Several characterizations of \( I_{lrg} \)-closed sets are obtained, including formulations in terms of the leras \( \star \)-closure operator, the leras closures of singletons, and the non-existence of nonempty \( Ir \)-closed sets in certain associated difference sets. Additional properties, such as finite unions of \( I_{lrg} \)-closed sets being \( I_{lrg} \)-closed and sufficient conditions under which \( I_{lrg} \)-closed sets coincide with \( lrg \)-closed sets, are also established. Appropriate examples are constructed throughout to illustrate the developed concepts and to demonstrate that the implications obtained cannot, in general, be reversed.

Abstract:Let \( G \) be a connected graph. A non-empty set \( T \subseteq V(G) \) is a 2-movable total dominating set of \( G \) if \( T \) is a total dominating set and for every pair \( x,y \in T \), \( T \setminus \{x,y\} \) is a total dominating set in \( G \), or there exist \( u,v \in V(G) \setminus T \) such that \( u \) and \( v \) are adjacent to \( x \) and \( y \), respectively, and \( (T \setminus \{x,y\}) \cup \{u,v\} \) is a total dominating set in \( G \). The 2-movable total domination number of \( G \), denoted by \( \gamma_{mt}^{2}(G) \), is the minimum cardinality of a 2-movable total dominating set of \( G \). A 2-movable total dominating set with cardinality equal to \( \gamma_{mt}^{2}(G) \) is called a \( \gamma_{mt}^{2} \)-set of \( G \). This paper presents the 2-movable total domination in the join and corona of graphs.

Abstract: Let \( G = (V(G),E(G)) \) be a connected graph and let \( \alpha = (V_0,V_1,V_2) \) be a CvRDF on \( G \). A function \( \alpha \) is an outer-connected convex Roman dominating function (OCCvRDF) on \( G \) provided that for every \( v \in V_0 \), there exists \( u \in V_2 \) such that \( uv \in E(G) \), \( V_1 \cup V_2 \) is a convex set on \( G \), and \( \langle V_0 \rangle \) is a connected subgraph. The weight of OCCvRDF \( \alpha \) is denoted by \( \widetilde{\omega}_G^{cCvR}(\alpha) \) and is defined as \( \widetilde{\omega}_G^{cCvR}(\alpha) = \sum_{z \in V(G)} \alpha(z) = |V_1| + 2|V_2| \). The outer-connected convex Roman domination number of graph \( G \) is denoted by \( \widetilde{\gamma}_{cCvR}(G) \) and is defined as the minimum weight of an OCCvRDF \( \alpha \) on \( G \), which can be written as \( \widetilde{\gamma}_{cCvR}(G) = \min\{\widetilde{\omega}_G^{cCvR}(\alpha) : \alpha \text{ is an OCCvRDF on } G\} \). Each OCCvRDF \( \alpha \) on \( G \) satisfying the equation \( \widetilde{\omega}_G^{cCvR}(\alpha) = \widetilde{\gamma}_{cCvR}(G) \) is called a \( \widetilde{\gamma}_{cCvR} \)-function on \( G \). In this paper, we introduce a new study of convex Roman domination in graphs, called the outer-connected convex Roman dominating function, and give some of its graph-theoretic properties.

Abstract:A connected dominating set \( C \) in \( G \) is a 2-movable connected dominating set of \( G \) if for every pair \( x,y \in C \), \( C \setminus \{x,y\} \) is a connected dominating set in \( G \), or there exists \( u,v \in V(G) \setminus C \) such that \( u \) and \( v \) are adjacent to \( x \) and \( y \), respectively, and \( (C \setminus \{x,y\}) \cup \{u,v\} \) is a connected dominating set of \( G \). The minimum cardinality of a 2-movable connected dominating set of \( G \), denoted by \( \gamma_{mc}^{2}(G) \) is the 2-movable connected domination number of \( G \). A 2-movable connected dominating set with cardinality \( \gamma_{mc}^{2}(G) \) is called a minimum 2-movable connected dominating set or a \( \gamma_{mc}^{2} \)-set of \( G \). In this paper, the researcher obtained the 2-movable connected domination number of the Complete Graph \( K_n \), Join Graph \( G + H \), and Generalized Wheel Graph \( W_{m,n} \).

Abstract:In this short paper, we establish a generalization of an inequality involving the Rieman zeta function. The inequality was established by Laforgia and Natalini in 2006 and subsequently improved by other researchers. The main tools used to prove our results are the logarithmic convexity property of the Riemann zeta function and monotonicity properties of certain functions containing the Riemann zeta function.