Modeling of infectious diseases is essential to comprehend dynamic behavior for the transmission of an epidemic. This research study consists of a newly proposed mathematical system for transmission dynamics of the measles epidemic. The measles system is based upon mass action principle wherein human population is divided into five mutually disjoint compartments: susceptible S(t)-vaccinated V (t)-exposed E(t)-infectious I (t)-recovered R(t). Using real measles cases reported from January 2019 to October 2019 in Pakistan, the system has been validated. Two unique equilibria called measles-free and endemic (measles-present) are shown to be locally asymptotically stable for basic reproductive number R 0 < 1 and R 0 > 1, respectively. While using Lyapunov functions, the equilibria are found to be globally asymptotically stable under the former conditions on R 0 . However, backward bifurcation shows coexistence of stable endemic equilibrium with a stable measles-free equilibrium for R 0 < 1. A strategy for measles control based on herd immunity is presented. The forward sensitivity indices for R 0 are also computed with respect to the estimated and fitted biological parameters. Finally, numerical simulations exhibit dynamical behavior of the measles system under influence of its parameters which further suggest improvement in both the vaccine efficacy and its coverage rate for substantial reduction in the measles epidemic.
For investigators working on criminal covert networks, identification of key actor(s) in the network is a major objective. Taking out key nodes will decrease the ability of the criminal network to function normally. Traditionally, the node centrality measurements have relied solely on the number of edges incident to nodes but not on the weights of those edges. However, in some generalizations for centrality measures for weighted networks, the focus shifts solely to the weights of the links and they don't account for the number of ties which was the central idea in the original centrality measures. Hence, answering which nodes are most central in a network with weighted relations depends on what imporantce is given the weights of the incident edges in comparison to the number of those edges. Opsahl et al. propose a generalized method for controlling the relative importance between the number of incident ties (nodal degree) versus the total weight of those ties (nodal strength). Research in TNA has largely focused on un-weighted ties, whereas richer and more sophisticated models of covert networks are needed to give precise and more realistic knowledge about such networks. Moreover, the existing implementation of node centrality algorithms in TNA tools don't support networks having weighted/values relations among nodes. New implementations of node centrality algorithms for weighted networks based on the generalized approach have been developed in CrimeFighter Assistant tool and are evaluated with known network dataset of the 9/11 incident.
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