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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{R}}_{0}<1)$$\end{document}(i.e.R0<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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{ R}}_{0}<1)$$\end{document}(i.e.R0<1), and if greater than one \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\mathrm{i}.\mathrm{e}. \, \, {\mathrm{R}}_{0}>1)$$\end{document}(i.e.R0>1) epidemic occurs. While using the Routh-Hurwitz criteria, the equilibria are found to be locally asymptotically stable under the former condition on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{R}}_{0}$$\end{document}R0. The partial rank correlation coefficients (PRCCs), a global sensitivity analysis method is used to compute \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{R}}_{0}$$\end{document}R0 and measles prevalence \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left({\mathrm{I}}^{*}\right)$$\end{document}I∗ with respect to the estimated and fitted model parameters. We found that the transmission rate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\upbeta )$$\end{document}(β) 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.

www.nature.com/scientificreports/ and brain damage may occur due to measles 3 . There is no specific medicine for the treatment of infected people with measles. According to the complications of the patients, a specific treatment may be suggested. Patients may need complete bed rest, fluids, control of fever and pains, and antibiotics 9 . Now, the measles vaccine is available, which is effective and inexpensive, and it has been possible to remarkably reduce the number of people dying from measles through vaccination 5 . The measles, mumps, and rubella (MMR) vaccines efficacy is 95% for preventing measles if the first dose is given to children at 12 months of age, and the efficacy increase to 99% after the second dose is given to children at greater than 12 months of age 10 . Many countries and various public health organizations such as WHO, UNICEF, American Red Cross, Centers for Disease Control and Prevention (CDC), and United Nations Foundation have conducted tremendous efforts worldwide to fight against measles. These organizations launched the Measles and Rubella Initiative (MRI), a global partnership among these organizations to stop measles and rubella, in 2001 to reduce measles deaths globally by 90% by 2010 compared to 2000 estimates 11 . Presently, this partnership has taken the Measles and Rubella Strategic Framework 2021-2030 (MRSF 2021-2030) for a world free from measles and rubella 12 . However, despite being vaccine-preventable, measles is still a public health problem in many developing countries globally, especially in parts of Asia and Africa, because of low awareness, civil strife, vaccine hesitancy, lower immunization system and poor health infrastructures 13 .
In Bangladesh, one of the South East Asia Region (SEAR) countries, measles outbreaks occurred several times in different areas during [2000][2001][2002][2003][2004][2005][2006][2007][2008][2009][2010][2011][2012][2013][2014][2015][2016]. There were about 70,273 reported measles cases and 33,213 confirmed measles cases in Bangladesh during this period 14 . Although the Expanded Program of Immunization (EPI) was started in Bangladesh in 1979 to control and prevent measles 15 , the government adopted more initiatives like strengthening the surveillance system and introduction of the second dose of measles-containing vaccine (MCV2) in 2014 to eliminate measles from the country by 2018 16 . Consequently, measles cases were reduced up to 84% over the last decades. However, the number of estimated measles cases has been increasing since 2016 nationwide. Despite some existing challenges, Rohingya refugees is another challenge for the removal of measles from Bangladesh.
In recent decades, the research relating to measles in epidemiology has been one of the most important research fields to researchers. Many researchers have already proposed their ideas and accomplished their research mathematically, theoretically or experimentally, using different deterministic or compartmental models, to find the comparatively best ways for measles control and prevention, focusing on different areas of the world, for example, London 17 27 studied an SEIR deterministic epidemic model to investigate the impact of asymptomatic individuals at the latent period on measles dynamics. Adewale et al. 28 developed a mathematical model to ascertain the effect of distance between infected and non-infected persons in controlling the measles virus transmission. They observed that the number of infected individuals decreases due to increases in distance between infected and susceptible persons. Also, two studies highlighted the efficiency of vaccination in controlling and prevention of measles transmission 29,30 . Garba et al. 31 also studied a compartmental mathematical model to examine the effect of vaccination and treatment on measles dynamics. Beay 4 proposed a SIQR epidemic model and accomplished the numerical analysis of the model to explore the effect of treatment and quarantine on measles dynamics. The study demonstrated that the combined application of quarantine and treatment is more effective to control and prevent measles. It also observed that the measles spread reduces due to the treatment and quarantine of infected individuals.
In this study, we develop a novel compartmental measles model to simulate the prevalence of measles estimation in Bangladesh. We use the next-generation matrix method to determine the basic reproduction number of the system and found that this is an essential determinant for disease dynamics. To supplement and validate the analytic process, we use numerical techniques to solve the model equations and explore the epidemic trajectory for a range of possible parameters values and initial conditions. The local stability analyses of the disease-free and endemic equilibria are examined using the Routh-Hurwitz criteria. Following this, we perform a sensitivity analysis to investigate the model parameters that greatly influence measles prevalence. Finally, we investigate the impact of progression rate, transmission rate and double dose vaccinations on the dynamics of the measles outbreak.

Methods and materials
Model description. We developed a compartmental transmission dynamics measles model between the following mutually exclusive compartments: susceptible individuals, S(t) ; those who have not yet infected with the disease but might become infected; first dose vaccinated individuals, V 1 (t) ; those who have received the first dose of vaccine; second dose vaccinated individuals, V 2 (t) ; those who have received the second dose of vaccine; Exposed individuals, E(t) ; representing those that are infected and have not yet developed active measles disease; Infected individuals, I(t) ; those who are infected and infectious; and Recovered individuals, R(t) ; those who were previously infected and successfully recovered. Individuals in the recovery class are neither infectious nor susceptible, including people in treatment, isolation, no longer contacting others or dead.
The total population size N(t) is assumed to be constant and well mixed: To ensure the population size constant, we replace all deaths as newborns in the susceptible compartment. It includes death through natural causes, which occurs in all states at the constant per-capita rate µ , and measlesrelated deaths, which occur at the constant per capita rate δ . Susceptible population (S) who receive the first dose of vaccine move to the vaccinated compartment at a rate η . The first dose of vaccinated population V 1 moves to the susceptible compartment at a rate ρ , and the rest of the population moves to the second dose of vaccinated www.nature.com/scientificreports/ population V 2 at a per-capita rate σ . The second dose of the vaccinated population also moves to the recovery compartment at a rate ω . Individuals in the S compartment may be infected with the measles virus at a rate = βSI , where β is the transmission rate between infected and susceptible population. Then infected individuals move to the exposed compartment E . A proportion of the exposed population progress to the infected compartment at a per-capita rate α . A proportion of the infected individuals move to the recovery compartment due to the treatment and natural recovery rate γ . The model flow diagram is presented in Fig. 1.
From those as mentioned above, the transmission dynamics of measles is given by the following compartmental system of nonlinear ordinary differential equations that describe the model: The initial conditions of the system (2)-(7) are of the form.
It can be easily shown that the solution of the system (2)-(7) subject to the initial conditions (8) exists and is nonnegative for all t ≥ 0.
Given the non-negative initial conditions of the system (2)-(7), it is direct to show that each state variable remains non-negative. Summing Eqs. (2)-(7), we find that the total population, N(t) satisfies in the absence of death due to measles or if there are no infected individuals (i.e. I = 0) 32 , then we have Integrating this equation, we find Given the constant population size and positivity of solutions, it naturally follows that each states S, V 1 , V 2 , E, I, R are bounded. www.nature.com/scientificreports/ Note that Eqs. (2)-(6) are independent of the recovery population; hence if we only wish to track measles incidence and prevalence, we can focus our attention on the following system: Given the positivity and boundedness of the system solutions, we find that the feasible region for Eqs. (9)- (13) given by.
where D is the positively invariant region for the system (9)- (13). Therefore, in this study, we consider Eqs. (9)-(13) in the set D.
Ethical approval. This study is based on aggregated measles surveillance data in Bangladesh provided by the World Health Organization. No confidential information was included because mathematical analyses were performed at the aggregate level.

Results
Existence of equilibria. Two types of equilibrium solutions appear in this system: the disease-free equilibrium, which is reached when the basic reproduction number is less than one, i.e. R 0 < 1 and the endemic equilibrium, which is reached when the basic reproduction number is greater than one, i.e. R 0 > 1 . We discuss these in order below.
Disease-free equilibrium point (X 0 ). In this section, we obtain the disease-free equilibrium point of the system (9)- (13) at which the epidemic is eliminated by applying E = I = 0 . Hence, the disease-free equilibrium point is given by That describes the state in which there is no infection in the community, and the total population N is constant at time t = 0.
Basic reproduction number (R 0 ). The basic reproduction number can be determined using the method of next-generation matrix 33 . The next-generation matrix is the production of matrices T and − −1 where the matrix T represents the rate of infection transmission in E and I compartments and the matrix describes all other transfer across the compartments. The matrices T and are given as The next-generation matrix is The basic reproduction number is the Eigen-value of the largest magnitude of the next-generation matrix (K) . Hence the basic reproduction number is obtained as , 0, 0 . www.nature.com/scientificreports/ Endemic equilibrium point (X * ). The endemic equilibrium point of the system (9)-(13) is discovered by applying S = V 1 = V 2 = E = I = 0 . Hence, the endemic equilibrium point is given by.
Stability analysis. To examining the stability of the equilibria of system (9)-(13), the following outcomes are proven: The disease-free equilibrium of the model is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1.

Proof
We consider the Jacobian of the system (9)-(13) which is given by which, at the infection-free equilibrium point, X 0 , reduces to Now we have to provide that all the eigenvalues of J X 0 are negative. As the 3rd column indicates only the diagonal terms which form the one negative eigenvalue, −(ω + µ) the other eigenvalues can be derived from the sub-matrix, J 1 (X 0 ) formed by excluding the 3rd row and column of J X 0 . Which gives This matrix can be written in block form as The characteristic equation of the two-by-two block matrix J 1 (X 0 ) is Now we can apply the Routh-Hurwitz criteria for stability to matrices A 1 and A 4 directly and independently. We require that the trace is negative and the determinant is positive for each matrix. Now for A 1 matrix and (15) α((α+µ)(γ+δ+µ))(ρ+σ+µ) , Hence, the disease-free equilibrium X 0 is locally asymptotically stable for R 0 < 1 . If either R 0 > 1 , at least one of the roots of the characteristic equation has a positive real part and X 0 is unstable.

Lemma 2
The endemic equilibrium X * is locally asymptotically stable if R 0 > 1.
Hence, by the Routh-Hurwitz stability criterion, the endemic equilibrium point X * is locally asymptotically stable for R 0 > 1.
In order to verify the nature of the disease-free and endemic equilibrium analysis, we used Monte Carlo simulation 34 to prove the conditions by calculating the real part of the eigenvalues of the Jacobian matrix of disease-free and endemic equilibriums. The simulation outcomes are presented in Figs. 2 and 3. Figure 2 represented that the disease-free equilibrium is locally stable as all the five eigenvalues (real part) are negative (i.e. 1 , 2 , 3 , 4 , and 5 < 0) . Whilst Fig. 3 represented that the endemic equilibrium is unstable as two eigenvalues (real part) is positive (i.e. 4 , 5 > 0).
Using the parameter values in Table 1, we have the following values and the nature of their signs in Table 2.   www.nature.com/scientificreports/ In the sensitivity indices of R 0 the most sensitive parameter is the transmission rate (β) of measles. Another significant parameter is the first dose of vaccination rate (η) . The least sensitive parameter is the measles related death rate (δ) . Hence, increasing (or decreasing) the transmission rate (β) of measles by 100% increases (or decreases) the basic reproduction number R 0 by 100%. Similarly, increasing (or decreasing) the measles-related death rate (δ) by 100% decreases (or increases) R 0 by 16.9%.
Numerical simulation. In this section, we carry out detailed numerical simulations to support the analytic results and to assess the impact of model parameters, including progression rate, transmission rate and double dose vaccination. For illustration, we have chosen baseline parameter values consistent with measles infection and transmission (see Table 1). We found two equilibrium points following the analytical results: the diseasefree equilibrium (X 0 ) and an endemic equilibrium (X * ) . We used different initial conditions for the exposed and  www.nature.com/scientificreports/ infected population and found that if the basic reproduction number less than one (i.e. R 0 < 1) , then the diseasefree equilibrium is locally asymptotically stable. Furthermore, if R 0 > 1 then measles persists in the population. Figure 7 illustrates the stability of the disease-free equilibrium (i.e. whenR 0 < 1) by depicting system trajectories through the EvsI plane originating from different initial conditions. In this case, measles disease dies out. Figure 8 shows the stability of the endemic equilibrium (i.e.whenR 0 > 1) , and in this case, measles disease persists in the population. Figures 9 and 10 show that the effect of progression rate and transmission rate on measles prevalence. From these figures, we observed that the burden of the measles prevalence increase if the progression and transmission rates increase, which means that those have a positive correlation with a measles outbreak. Figures 11 and  12 show that the increases in first and second dose vaccination rates reduce the measles prevalence and reduce the risk of an outbreak.

Summary and conclusion
This paper has developed and analyzed a compartmental transmission dynamics measles model with double dose vaccination in Bangladesh. We have determined an analytic expression for the basic reproduction number using the next-generation matrix and found that the disease-free equilibrium is locally asymptotically stable if the basic reproduction number is less than one. We have also found that measles disease persists in the community if the basic reproduction number is greater than one. Sensitivity analysis has also been performed to explore the impact of model parameters and findings showed that the spread of the measles disease largely depends on the transmission rate. Therefore, effort should be made to minimize unnecessary transmission with measles infected individuals. However, if we treat early measles infected individuals, it will also reduce transmission from infected person to uninfected person. This study has also highlighted the significance of vaccination in controlling and preventing the spread of measles in the community of Bangladesh. Vaccination in a population is the best way to control an outbreak of measles. Numerical analysis has revealed that vaccination has a negative impact on the prevalence of measles. This finding indicates that the improvement in vaccination dose rate decreases the spread of measles. Therefore, to attain a high level of herd immunity for the disease, mass vaccination exercise should be encouraged to cover most of the population to prevent an outbreak of measles in Bangladesh.  www.nature.com/scientificreports/

Data availability
The datasets produced during the study are available from the corresponding author on reasonable request.