RESUMEN
Life style of people almost in every country has been changed with arrival of corona virus. Under the drastic influence of the virus, mathematicians, statisticians, epidemiologists, microbiologists, environmentalists, health providers, and government officials started searching for strategies including mathematical modeling, lock-down, face masks, isolation, quarantine, and social distancing. With quarantine and isolation being the most effective tools, we have formulated a new nonlinear deterministic model based upon ordinary differential equations containing six compartments (susceptible S ( t ) , exposed E ( t ) , quarantined Q ( t ) , infected I ( t ) , isolated J ( t ) and recovered R ( t ) ). The model is found to have positively invariant region whereas equilibrium points of the model are investigated for their local stability with respect to the basic reproductive number R 0 . The computed value of R 0 = 1.31 proves endemic level of the epidemic. Using nonlinear least-squares method and real prevalence of COVID-19 cases in Pakistan, best parameters are obtained and their sensitivity is analyzed. Various simulations are presented to appreciate quarantined and isolated strategies if applied sensibly.
RESUMEN
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.