Transmission Dynamics Of Measles Research Articles

Unveiling measles transmission dynamics: Insights from a stochastic model with nonlinear incidence

AbstractIn this paper, taking into account the inevitable impact of environmental perturbations on disease transmission, we primarily investigate a stochastic model for measles infection with nonlinear incidence. The transmission rate in this model follows a logarithmic normal distribution influenced by an Ornstein–Uhlenbeck (OU) process. To analyze the dynamic properties of the stochastic model, our first step is to establish the existence and uniqueness of a global solution for the stochastic equations. Next, by constructing appropriate Lyapunov functions and utilizing the ergodicity of the OU process, we establish sufficient conditions for the existence of a stationary distribution, indicating the prevalence of the disease. Furthermore, we provide sufficient conditions for disease elimination. These conditions are derived by considering the interplay between the model parameters and the stochastic dynamics. Finally, we validate the theoretical conclusions through numerical simulations, which allow us to assess the practical implications of the established conditions and observe the dynamics of the stochastic model in action. By combining theoretical analysis and numerical simulations, we gain a comprehensive understanding of the stochastic model's behavior, contributing to the broader understanding of measles transmission dynamics and the development of effective control strategies.

A fractional modeling approach for the transmission dynamics of measles with double-dose vaccination

Measles, a member of the Paramyxoviridae family and the Morbillivirus genus, is an infectious disease caused by the measles virus that is extremely contagious and can be prevented through vaccination. When a person with the measles coughs or sneezes, the virus is disseminated by respiratory droplets. Normally, the appearance of measles symptoms takes 10–14 d following viral exposure. Conjunctivitis, a high temperature, a cough, a runny nose, and a distinctive rash are some of the symptoms. Despite the measles vaccination being available, it is still widespread worldwide. To eradicate measles, the Reproduction Number (i.e. R 0 < 1 ) must remain less than unity. This study examines a SEIVR compartmental model in the caputo sense using a double dose of vaccine to simulate the measles outbreak. The reproduction number R 0 and model properties are both thoroughly examined. Both the local and global stabilities of the proposed model are determined for R 0 less and greater than 1. To achieve the model’s global stability, the Lyapunov function is used while the existence and uniqueness of the proposed model are demonstrated In addition to the calculated and fitted biological parameters, the forward sensitivity indices for R 0 are also obtained. Simulations of the proposed fractional order (FO) caputo model are performed in order to analyse their graphical representations and the significance of FO derivatives to illustrate how our theoretical findings have an impact. The graphical results show that the measles outbreak is reduced by increasing vaccine dosage rates.

Stability and Threshold Dynamics in a Seasonal Mathematical Model for Measles Outbreaks with Double-Dose Vaccination

Measles is a highly contagious viral disease that can lead to serious complications, including death, particularly in young children. In this study, we developed a mathematical model that incorporates a seasonal transmission parameter to examine the measles transmission dynamics. We define the basic reproduction number (R0) and show its utility as a threshold parameter for global dynamics and the existence of periodic solutions. The model was applied to the measles outbreak that occurred in Pakistan from 2019 to 2021 and provided a good fit to the observed data. Our estimate of the basic reproduction number was found to be greater than one, indicating that the disease will persist in the population. The findings highlight the need to increase vaccination coverage and efficacy to mitigate the impact of the epidemic. The model also shows the long-term behavior of the disease, which becomes endemic and recurs annually. Our simulations demonstrate that a shorter incubation period accelerates the spread of the disease, while a higher vaccination coverage rate reduces its impact. The importance of the second dose of the measles vaccine is emphasized, and a higher vaccine efficacy rate can also help bring R0 below one. Our study provides valuable information for the development and implementation of effective measles control strategies. To prevent future outbreaks, increasing vaccination coverage among the population is the most effective way to reduce the transmission of measles.

Modelling, Analysis, and Simulation of Measles Disease Transmission Dynamics

Measles is one of the top communicable diseases, which is still responsible for 2.6 million deaths every year. Due to this reason, the paper focuses on measles transmission dynamics concerning the impact of indirect contact rate (transmitted from the host of the virus to the healthy individual) and improving the SEVIR model into the SVIRP model. From the model, we first estimated the disease-free equilibrium, calculated the effective reproduction numberREff, and established the stability analysis. The Castillo–Chavez stability criterion is used to demonstrate the global stability of the disease-free equilibrium point, while the linearization method is used to justify its local stability analysis and gives a result REff&lt;1. The stability analysis of endemic equilibrium point is explained by defining a Lyapunov function, and its global stability exists whenREff&gt; 1. To identify the effect of parameters on the transmission dynamics, we performed sensitivity index and numerical simulation. From the result, we obtained that the indirect contact rate has the highest impact in maximizing the transmission dynamics of measles. Also, we found that working on prevention and treatment strategies brings a significant contribution in reducing the disease effect in the community.

Theoretical analysis of a measles model with nonlinear incidence functions

Measles is a highly contagious respiratory disease of global public health concern. A deterministic mathematical model for the transmission dynamics of measles in a population with Crowley–Martin incidence function to account for the inhibitory effect due to susceptible and infected individuals and vaccination is formulated and analyzed using standard dynamical systems methods. The basic reproduction number is computed. By constructing a suitable Lyapunov function, the disease-free equilibrium is shown to be globally asymptotically stable. Using the Center Manifold theory, the model exhibits a forward bifurcation, which implies that the endemic equilibrium is also globally asymptotically stable. To determine the optimal choice of intervention measures to mitigate the spread of the disease, an optimal control problem is formulated (by introducing a set of three time-dependent control variables representing the first and second vaccine doses, and the palliative treatment) and analyzed using Pontryagin’s Maximum Principle. To account for the scarcity of measles vaccines during a major outbreak or other causes such as the COVID-19 pandemic, a Holling type-II incidence function is introduced at the model simulation stage. The control strategies have a positive population level impact on the evolution of the disease dynamics. Graphical results reveal that when the mass-action incidence function is used, the number of individuals who received first and second vaccine dose is smaller compared to the numbers when the Crowley–Martin incidence-type function is used. Inhibitory effect of susceptibles tends to have the same effect on the population level as the Crowley–Martin incidence function, while the control profiles when inhibitory effect of the infectives is considered have similar effect as when the mass-action incidence is used, or when there is limitation in the availability of measles vaccines. Missing out the second measles vaccine dose has a negative impact on the initial disease prevalence.

The impact of supplementary immunization activities on measles transmission dynamics and implications for measles elimination goals: A mathematical modelling study

BackgroundMeasles has re-emerged globally due to the accumulation of susceptible individuals and immunity gap, which causes challenges in eliminating measles. Routine vaccination and supplementary immunization activities (SIAs) have greatly improved measles control, but the impact of SIAs on the measles transmission dynamics remains unclear as the vaccine-induced immunity wanes. MethodsWe developed a comprehensive measles transmission dynamics model by taking into account population demographics, age-specific contact patterns, seasonality, routine vaccination, SIAs, and the waning vaccine-induced immunity. The model was calibrated by the monthly age-specific cases data from 2005 to 2018 in Jiangsu Province, China, and validated by the dynamic sero-prevalence data. We aimed to investigate the short-term and long-term impact of three-time SIAs during 2009–2012 (9.68 million and 4.25 million children aged 8 months-14 years in March 2009 and September 2010, respectively, and 140,000 children aged 8 months-6 years in March 2012) on the measles disease burden and explored whether additional SIAs could accelerate the measles elimination. ResultsWe estimated that the cumulative numbers of measles cases from March 2009 to December 2012 (in the short run) and to December 2018 (in the long run) after three-time SIAs (base case) were 6,699 (95% confidence interval [CI]: 2,928–10,469), and 22,411 (15,146–29,675), which averted 45.0% (42.9%-47.0%) and 34.3% (30.7%-37.9%) of 12,226 (4,916–19,537) and 34,274 (21,350–47,199) cases without SIAs, respectively. The fraction of susceptibles for children aged 8–23 months and 2–14 years decreased from 8.3% and 10.8% in March 2009 to 5.8% and 5.8% in April 2012, respectively. However, the fraction of susceptibles aged 15–49 years and above 50 years increased gradually to about 15% in 2018 irrespective of SIAs due to the waning immunity. The measles elimination goal would be reached in 2028, and administrating additional one-off SIAs in September 2022 to children aged 8–23 months, or young adolescents aged 15–19 years could accelerate the elimination one year earlier. ConclusionsSIAs have greatly reduced the measles incidence and the fraction of susceptibles, but the benefit may wane over time. Under the current interventions, Jiangsu province would reach the measles elimination goal in 2028. Additional SIAs may accelerate the measles elimination one year earlier.

Fractional Dynamics of a Measles Epidemic Model

In this work, we replaced the integer derivative with Caputo derivative to model the transmission dynamics of measles in an epidemic situation. We began by recalling some results on the local and global stability of the measles-free equilibrium point as well as the local stability of the endemic equilibrium point. We computed the basic reproduction number of the fractional model and found that is it equal to the one in the integer model when the fractional order ν = 1. We then performed a sensitivity analysis using the global method. Indeed, we computed the partial rank correlation coefficient (PRCC) between each model parameter and the basic reproduction number R0 as well as each variable state. We then demonstrated that the fractional model admits a unique solution and that it is globally stable using the Ulam–Hyers stability criterion. Simulations using the Adams-type predictor–corrector iterative scheme were conducted to validate our theoretical results and to see the impact of the variation of the fractional order on the quantitative disease dynamics.

Mathematical model of measles transmission dynamics using real data from Nigeria

Measles is a highly contagious and life-threatening disease caused by a virus called morbillivirus, despite the availability of a safe and cost-effective vaccine, it remains a leading cause of death, especially in children. Measles spreads easily from person to person via infected people's coughs and sneezes. It can also be transmitted through direct contact with the mouth or contaminated surfaces. To have a better knowledge of measles epidemiology in Nigeria, we develop a deterministic mathematical model to study the transmission dynamics of the disease in the population. The boundary of the model solution is performed, both equilibrium points are calculated, and the basic reproduction number is determined. We have proved that when , the disease-free equilibrium point is both locally and globally stable. When , the endemic equilibrium point exists and is stable if it satisfies Routh–Hurwitz criteria. We demonstrate the model's effectiveness by using a real-life application of the disease spread in Nigeria. We fit the proposed model using available data from Nigeria Center for Disease Control (NCDC) from January to December 2020 to obtain the best fit, this help us to determine the accuracy of the proposed model's representation to the real-world data. We investigate the impact of vaccination rate and hospitalization of infected individuals on the dynamics of measles in the population. The result shows that the combined control strategies reduce the peak of infection faster than the single control strategy.

Modeling transmission dynamics of measles in Nepal and its control with monitored vaccination program.

Measles is one of the highly contagious human viral diseases. Despite the availability of vaccines, measles outbreak frequently occurs in many places, including Nepal, partly due to the lack of compliance with vaccination. In this study, we develop a novel transmission dynamics model to evaluate the effects of monitored vaccination programs to control and eliminate measles. We use our model, parameterized with the data from the measles outbreak in Nepal, to calculate the vaccinated reproduction number, $ R_v $, of measles in Nepal. We perform model analyses to establish the global asymptotic stability of the disease-free equilibrium point for $ R_v < 1 $ and the uniform persistence of the disease for $ R_v > 1 $. Moreover, we perform model simulations to identify monitored vaccination strategies for the successful control of measles in Nepal. Our model predicts that the monitored vaccination programs can help control the potential resurgence of the disease.

Measles epidemic in Southern Vietnam: an age-stratified spatio-temporal model for infectious disease counts.

Measles resurged in Vietnam between 2018 and 2020, especially in the Southern region. The proportion of children with measles infection showed quite some variation at the provincial level. We applied a spatio-temporal endemic-epidemic modelling framework for age-stratified infectious disease counts using measles surveillance data collected in Southern Vietnam between 1 January 2018 and 30 June 2020. We found that disease transmission within age groups was greatest in young children aged 0-4 years whereas a relatively high between-group transmission was observed in older age groups (5-14 years, 15-24 years and 25+ years groups). At the provincial level, spatial transmission followed an age-dependent distance decay with measles spread mainly depending on local and neighbouring transmission. Our study helped to clarify the measles transmission dynamics in a more detailed fashion with respect to age strata, time and space. Findings from this study may help determine proper strategies in measles outbreak control including promotion of age-targeted intervention programmes in specific areas.

Modeling the Transmission Dynamics of Measles in the Presence of Treatment as Control Strategy

We present in this research work, mathematical modeling of the transmission dynamics of measles using treatment as a control measure. We determined the Disease Free Equilibrium (DFE) point of the model after which we obtained the Basic Reproduction Number ( R0 ) of the model using the next generation approach. The model Endemic Equilibrium (EE) point was also determined after which we performed Local Stability Analysis(LAS) of the Disease Free Equilibrium point and result shows that the Disease Free Equilibrium point of the model would be stable if ( R0 &lt;1). Global Stability Analysis (GAS) result shows that, ( R0 ≤ 1) remains the necessary and sufficient condition for the infection to go into extinction from a population. We carried out Sensitivity Analysis of the model using the Basic Reproduction Number and we discovered that ( δ , μ, ν , θ ) are sensitive parameters that should be targeted towards control intervention strategy as an increase in these values can reduce the value of ( R0 ) to a value less than unity and such can reduce the spread of measles in a population. Model simulation was carried out using mat lab software to support our analytical results.

Modelling the relative benefits of using the measles vaccine outside cold chain for outbreak response

IntroductionRapid outbreak response vaccination is a strategy for measles control and elimination. Measles vaccines must be stored and transported within a specified temperature range, but this can present significant challenges when targeting remote populations. Measles vaccine licensure for delivery outside cold chain (OCC) could provide more vaccine transport/storage space without ice packs, and a solution to shorten response times. However, due to vaccine safety and wastage considerations, the OCC strategy will require other operational changes, potentially including the use of 1-dose (monodose) instead of 10-dose vials, requiring larger transport/storage equipment currently achieved with 10-dose vials. These trade-offs require quantitative comparisons of vaccine delivery options to evaluate their relative benefits. MethodsWe developed a modelling framework combining elements of the vaccine supply chain - cold chain, vial, team, and transport equipment types - with a measles transmission dynamics model to compare vaccine delivery options. We compared 10 strategies resulting from combinations of the vaccine supply elements and grouped into three main classes: OCC, partial cold chain (PCC), and full cold chain (FCC). For each strategy, we explored a campaign with 20 teams sequentially targeting 5 locations with 100,000 individuals each. We characterised the time needed to freeze ice packs and complete the campaign (campaign duration), vaccination coverage, and cases averted, assuming a fixed pre-deployment delay before campaign commencement. We performed sensitivity analyses of the pre-deployment delay, population sizes, and two team allocation schemes. ResultsThe OCC, PCC, and FCC strategies achieve campaign durations of 50, 51, and 52 days, respectively. Nine of the ten strategies can achieve a vaccination coverage of 80%, and OCC averts the most cases. DiscussionThe OCC strategy, therefore, presents improved operational and epidemiological outcomes relative to current practice and the other options considered.

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

Although the availability of the measles vaccine, it is still epidemic in many countries globally, including Bangladesh. Eradication of measles needs to keep the basic reproduction number less than one (mathrm{i}.mathrm{e}. , , {mathrm{R}}_{0}<1). This paper investigates a modified (SVEIR) measles compartmental model with double dose vaccination in Bangladesh to simulate the measles prevalence. We perform a dynamical analysis of the resulting system and find that the model contains two equilibrium points: a disease-free equilibrium and an endemic equilibrium. The disease will be died out if the basic reproduction number is less than one (mathrm{i}.mathrm{e}. , , {mathrm{ R}}_{0}<1), and if greater than one (mathrm{i}.mathrm{e}. , , {mathrm{R}}_{0}>1) epidemic occurs. While using the Routh-Hurwitz criteria, the equilibria are found to be locally asymptotically stable under the former condition on {mathrm{R}}_{0}. The partial rank correlation coefficients (PRCCs), a global sensitivity analysis method is used to compute {mathrm{R}}_{0} and measles prevalence left({mathrm{I}}^{*}right) with respect to the estimated and fitted model parameters. We found that the transmission rate (upbeta ) had the most significant influence on measles prevalence. Numerical simulations were carried out to commissions our analytical outcomes. These findings show that how progression rate, transmission rate and double dose vaccination rate affect the dynamics of measles prevalence. The information that we generate from this study may help government and public health professionals in making strategies to deal with the omissions of a measles outbreak and thus control and prevent an epidemic in Bangladesh.

Modeling of measles epidemic with optimized fractional order under Caputo differential operator

Abstract Memory is an important characteristic of an epidemic. One of such memory dependent and highly contagious viral diseases is measles that is also responsible for more than 140,000 deaths in 2018 in various regions of Asia and Africa. In order to better understand the transmission dynamics of measles, we have developed a new epidemiological model while considering both integer and fractional order operators and presented comparison. The Caputo fractional model has a unique solution with the positively invariant region. On the basis of basic reproduction number R 0 , stability analysis is discussed and sensitivity of parameters is investigated using PRCC global technique. Not only parameters but fractional order χ is also optimized via nonlinear least-squares approach with availability of statistical data obtained from WHO. Various simulations in terms of time series plots, 3D meshes and contours are carried out to observe effects of parameters on dynamics of the epidemic wherein it is said to be persistent for χ → 0 demonstrating the role being played by Caputo fractional derivative towards measles dynamics.

Solusi numerik model matematika SHEIR pada penyebaran penyakit campak dengan vaksinasi Dan kekebalan kelompok

In this paper, mathematical model of measles transmission dynamics considering vaccination and herd immunity is discussed. The solution of the model is investigated using euler, atangana, dan nonstandard finite difference method. After comparing the solutions of the model, we observe that the solutions obtained by using euler and atangana method diverge for certain step. On the other hand, the solutions obtained by using nonstandard finite difference always converge.

A comparative epidemiological stability analysis of predictor corrector type non-standard finite difference scheme for the transmissibility of measles

In this article, a predictor-corrector type non-standard finite difference scheme has been formulated to avoid unnatural chaos, and numerical instabilities depend on a long time step, lead to a better strategy for overcoming the transmission of infectious diseases. The construction, development, and analysis of our proposed numerical scheme is done for the SEIR epidemiological model regarding the transmission dynamics of measles. Based on the mathematical study of the proposed scheme, the stability analyses are performed in detail. Further, the behavior of the scheme is accessed by the evaluation of the Eigenvalues of the Jacobian at a steady state. MATLAB algorithm is used for numerical simulations, and results demonstrated that the NSFD-PC scheme double refines the solution and gives physically realistic solutions even for large step sizes. The effectiveness of the scheme has been investigated by comparison with renowned numerical methods in literature like the RK4 and Euler method of a predictor-corrector type. It has been found that the presented scheme is dynamically compatible with the continuous system, unconditionally convergent, and satisfies the positivity of the state variables involved in the system of the SEIR model. Conclusively, the proposed numerical scheme preserves all essential control measuring features of the corresponding dynamical system and reduces additional cost when examined over long periods. Therefore, it is highly recommended.

An Accurate Predictor-Corrector-Type Nonstandard Finite Difference Scheme for an SEIR Epidemic Model

The present work deals with the construction, development, and analysis of a viable normalized predictor-corrector-type nonstandard finite difference scheme for the SEIR model concerning the transmission dynamics of measles. The proposed numerical scheme double refines the solution and gives realistic results even for large step sizes, thus making it economical when integrating over long time periods. Moreover, it is dynamically consistent with a continuous system and unconditionally convergent and preserves the positive behavior of the state variables involved in the system. Simulations are performed to guarantee the results, and its effectiveness is compared with well-known numerical methods such as Runge–Kutta (RK) and Euler method of a predictor-corrector type.

Modelling the Periodic Outbreak of Measles in Mainland China

In mainland China, measles infection reached the lowest level in 2012 but resurged again after that with a seasonally fluctuating pattern. To investigate the phenomenon of periodic outbreak and identify the crucial parameters that play in the transmission dynamics of measles, we formulate a mathematical model incorporating periodic transmission rate and asymptomatic infection with waning immunity. We define the basic reproduction number as the threshold value to govern whether measles infection dies out or not. Fitting the reported measles cases from 2013 to 2016 to our proposed model, we estimate the basic reproduction number R0 with immunization to be 1.0077. From numerical simulations, we conclude asymptomatic infection does not cause much new infections and the key parameters affecting the transmission of measles are vaccination rate, transmission rate, and recovery rate, which suggests the public to enhance vaccination and protection measures to reduce effective contacts between susceptible and infective individuals and treat infected individuals timely. To minimize the number of infected individuals at a minimal cost, we formulate an optimal control system to design optimal control strategies. Numerical simulations show the effectiveness of optimal control strategies and recommend us to implement the control strategies as soon as possible. In particular, enhancing vaccination is especially effective in lowering the initial outbreak and making disease recurrence less likely.

Control Policy Mix in Measles Transmission Dynamics Using Vaccination, Therapy, and Treatment

This paper considers a deterministic model for the dynamics of measles transmission in a population divided into six classes with respect to the disease states: susceptible, vaccinated, exposed, infected, treated, and recovered. First, we investigate the dynamical properties of the SVEITR model such as its equilibrium points, their stability, and parameter sensitivity by applying constant controls. Criteria for determining the stability of disease-free and endemic equilibrium points are provided in terms of basic reproduction number. The model is then extended by incorporating vaccination, therapy, and treatment rates as time-dependent control variables representing the level of coverages. Application of Pontryagin’s maximum principle provides the necessary conditions that must be satisfied for the existence of optimal controls aiming at minimization of the number of exposed and infected individuals simultaneously with the control effort. Numerical simulations that were carried out using the backward sweep method and Runge–Kutta scheme suggest that optimal controls under moderate and high scenarios can effectively reduce the cases of measles. In particular, the moderate scenario that utilizes the existing coverage level of 86% for MCV1 and 69% for MCV2 can degrade the cost functional by 47% of the low scenario. Meanwhile, high scenario that takes the 2020 target of 96% as coverage only makes a slight difference in reducing the number of exposed and infected individuals.

A Fractional Order Model for the Transmission Dynamics of Measles with Vaccination

In this paper, the fractional order model was adopted to describe the dynamics of measles and to establish how the virus that causes measles is transmitted as well as how to mitigate the conditions that cause the spread. We showed the existence of the equilibrium states. The threshold parameter of the model was evaluated in terms of parameters in the model using the next generation matrix approach. We provided the conditions for the stability of the disease free and the endemic equilibrium points. Also, the stability of the various equilibrium points was studied. Numerical simulations of the model are presented graphically using Adam-Bashforth Method and the results were interpreted. The result also shows that the use of vaccination is the best way to prevent measles outbreak.