Stochastic model of measles transmission dynamics with double dose vaccination

Abstract

In this paper we developed a stochastic model of measles transmission dynamics with double dose vaccination. The total population in this model was sub-divided in to five compartments, namely Susceptible , Infected Vaccinated first dose Vaccinated second dose and Recovered First the model was developed by deterministic approach and then transformed into stochastic one, which is known to play a significant role by providing additional degree of realism compared to the deterministic approach. The analysis of the model was done in both approaches. The qualitative behavior of the model, like conditions for positivity of solutions, invariant region of the solution, the existence of equilibrium points of the model and their stability, and also sensitivity analysis of the model were analyzed. We showed that in both deterministic and stochastic cases if the basic reproduction number is less than 1 or greater than 1 the disease free equilibrium point is stable or unstable respectively, so that the disease dies out or persists within the population. Numerical simulations were carried out using MATLAB to support our analytical solutions. These simulations show that how double dose vaccination affect the dynamics of human population.

Mathematical analysis of a measles transmission dynamics model in Bangladesh with double dose vaccination

“…( 2 )–( 7 ), we find that the total population, satisfies in the absence of death due to measles or if there are no infected individuals (i.e. I = 0) 32 , then we have …”

Mathematical analysis of a COVID-19 model with double dose vaccination in Bangladesh

“…Based on different considerations, a significant number of mathematical models for COVID-19 transmission dynamics have become omnipresent. A series of mathematical models have been analyzed to delineate the transmission dynamic pattern of infectious diseases [3] , [39] , [40] , [41] . We considered the total population size among nine mutually exclusive compartments: susceptible individuals, S(t), are uninfected people with the disease but a chance to be infected; the first dose vaccinated individuals, V 1 (t), though still have to chance to be infected; second doses vaccinated individuals, V 2 (t), those who have completed the both dose vaccination within on time; exposed individuals, E(t), are those who are affected with a disease but have not yet developed respiratory illness; mild individuals, M(t), are indicated those who are asymptomatic; critical individuals, C(t), are express the COVID-19 symptoms clearly; non-hospitalized individuals, NH(t), are not in serious health crisis; hospitalized individuals, H(t), are those who are in critical health and respiratory crisis; and recovered individuals, R(t), are recovered against the disease who were infected.…”

“…Since the middle of the 20th century, the transmission dynamic of infectious diseases outbreak can be analyzed with the help of deterministic and stochastic epidemiology models. Those models represent real-world phenomena and predict the severity of the infectious disease using mathematical concepts [3] . This research is motivated by the ongoing vaccination to protect the transmission of the novel coronavirus among human beings.…”

“…According to the high and low basic reproduction number, a multi-wave solution to the model of coronavirus was demonstrated by Shayak et al, 2021 [19] . A few numbers of researchers are considered the effect of vaccines in their proposed model to represent more specific transmission dynamics of the disease [3] , [12] , [13] , [14] , [15] . They performed SIR and SEIR mathematical models with a two-phase vaccination process.…”

Analysis of a stochastic coronavirus (COVID-19) Lévy jump model with protective measures

“…[12] studied a stochastic SIRD epidemic model of Ebola and explored the existence of a unique stationary distribution. Respecting measles transmission dynamics, [13] developed a stochastic SIR-type model incorporating double dose vaccination and studied the stability of disease-free and endemic equilibria. For other diseases modelling using SDEs, we refer, for instance, to [9,14,15,16] and references cited therin.…”