A Compartmental Approach to Modeling the Measles Disease: A Fractional Order Optimal Control Model

Abstract

Measles is the most infectious disease with a high basic reproduction number ($R_0$). For measles, it is reported that $R_0$ lies between 12 and 18 in an endemic situation. In this paper, a fractional order mathematical model for measles disease is proposed to identify the dynamics of disease transmission following a declining memory process. In the proposed model, a fractional order differential operator is used to justify the effect and success rate of vaccination. The total population of the model is subdivided into five sub-compartments: susceptible (S), exposed (E), infected (I), vaccinated (V), and recovered (R). Here, we consider the first dose of measles vaccination and convert the model to a controlled system. Finally, we transform the control-induced model to an optimal control model using control theory. Both models are analyzed to find the stability of the system, the basic reproduction number, the optimal control input, and the adjoint equations with the boundary conditions. Also, the numerical simulation of the model is presented along with using the analytical findings. We also verify the effective role of the fractional order parameter alpha on the model dynamics and changes in the dynamical behavior of the model with $R_0=1$.

1. Introduction

Measles is an infectious disease that causes respiratory illness. Cough, coryza, and conjunctivitis are major symptoms of measles. Also, the maculopapular rash is observed after two weeks of infection, and it spreads rapidly all over the body. After four days of infection, the patients are considered contagious to before four days after the rash appears. The morbillivirus in the Paramyxoviridae family is known as the measles virus. It is a single-stranded RNA virus with one serotype. The measles virus can only be found in humans.

The infection rate of measles through person-to-person transmission is very high, and its attack rate is over 90%. It is the first infectious disease that humans can ever face in their childhood [1]. The basic symptom is a red rash, and sometimes it leads to serious fetal complications like pneumonia, diarrhea, and encephalitis. Some children suffer from blindness, deafness, or vision impairment. But measles infection creates lifelong immunity from future attacks [2]. In the wild, more than 30 to 40 million children suffer from measles, and over 1 million face complications like pneumonia, diarrhea, and malnutrition [3]. It has been reported that measles is the major cause of child death worldwide in spite of control measures [4].

The only efficient and safe method of protecting against measles infection is vaccination [5]. Without immunization, pregnant women and small children are at significant risk. As per WHO, the measles vaccine has been found to last for at least 20 years and is typically maintained for life in the majority of people. The efficacy of the measles vaccination is around 85% for children aged nine to eleven months, and it rises to 95% following a second dose given after a year [6].

Except for vaccination, there is no specific medicine for the treatment of measles. Complete bed rest and an intake of plenty of healthy fluids helps to reduce the fever and pains, as well as antibiotic use [7,8].

Since the previous decade, mathematical modeling has been used as one of the most essential tools for understanding the dynamics of human or animal disease [9]. For measles disease dynamics, Momoh et al. [10] proposed and analyzed an SEIR measles model to look at how the latent period affected the dynamics of the disease. In 2014, Adewale et al. [11] explained the impact of distance between infected and uninfected people. Ochoche and Gwervina [12] explored the SIR model of measles with two stages of vaccination. They conducted research to show that a full course of immunization can eradicate the illness. Edward [13] has researched the mathematical model for control dynamics in measles infection. Smith [14] conducted a quantitative analysis of the effectiveness of vaccination in avoiding measles. Raymond researched the stochastic measles model with vaccination impact [15]. Garba et al. [16] studied the role of vaccines on measles infection. Christopher et al. [17] studied the dynamics of the measles epidemic under the influence of vaccination combinations. The control and transmission dynamics of measles were used by Mitku et al. [18] to study the SEIR mathematical model of the disease. A stochastic differential equation model with double-dose vaccination was developed by Tilahun in 2020 [8]. In order to estimate the frequency of measles in Bangladesh, Kuddus [19] studied a compartmental mathematical model of the measles disease. In [20], a generalized SVEIR epidemic model with a general nonlinear incidence rate was proposed as a candidate model for measles virus dynamics.

The above-mentioned models are based on ordinary differential equations. But the deterministic technique for ordinary differential equations has several limitations. Thus, our motivation is to explore the fractional order differential problem (FODE) in measles disease dynamics.

Optimal control theory is useful for managing a disease in cost-effective way. In [7], an optimal control of measles was explored with vaccination using a mathematical model of integer order. Here, we formulated fractional optimal control (FOC) to combat the measles infection.

In this study, we derived a fractional order differential equation model of the dynamics of measles transmission with vaccines in optimal control approach. Five compartments were used in this model to separate the entire population: vulnerable, exposed, infected, immunized, and recovered. The model was put forth using the fractional method, which is recognized to be important since it adds more realism than the deterministic approach. We are aware that in nonlinear mathematical modeling, fractional order differential equations (FODE) are crucial [21,22,23,24].

To determine the efficacy of immunization, multiple analyses of the suggested model were conducted. Positiveness, existence, basic reproduction number, and model stability analysis made up the qualitative study of the model. Additionally, we looked at the model’s optimum control theory method. MATLAB software version R2018a was utilized to carry out numerical solutions.

The arrangement of the article is as follows: A fractional order mathematical model of measles in the presence of vaccination is proposed and the uniqueness and positivity of solutions are examined in Section 2. The model analysis (equilibria and stability) and the basic reproduction number are determined in Section 3. The fractional optimal control problem is formulated and analyzed in Section 4. Numerical results of the analytical discoveries are presented in Section 5. Finally, the discussion in Section 6 and the conclusions in Section 7 complete the paper.

2. The Basic Assumptions and the Mathematical Model

Here, we consider S, V, E, I, and R as the concentrations of the susceptible population, vaccination populations, exposed population but not infections, infected population, and recovered population, with some basic assumptions. The schematic diagram is represented in Figure 1. To construct the mathematical model, we consider the following assumptions:

$\left(A1\right)$

p is the probability of a susceptible population, and $\lambda$ is the birth rate of populations. $\mu$ is the death rate of populations, and $\nu$ is the vaccination rate from S,

$\left(A2\right)$

$\beta$ is the disease transmission rate,

$\left(A3\right)$

$\rho$ is the rate of infection,

$\left(A4\right)$

$\theta$ is the decay rate for the first doses of the vaccine to the susceptible population, and b is the rate at which the vaccinated population moves to the recover class. $\eta$ is the death rate for the infected class due to infection, and $\omega$ is the natural recovery rate of the infected population. $\varphi$ is the rate at which the first dose of the vaccine is administered. Thus, our model based on (A1)–(A4) is constructed.

The above assumptions, (A1)–(A4), lead us to formulate the model equations as follows:

$$\begin{array}{ccc}\hfill D^{\alpha }S& =& p\lambda -\beta SI+\theta V-(\nu +\mu )S,\hfill \\ \hfill D^{\alpha }V& =& (1-p)\lambda +\nu S-(\theta +b+\mu )V,\hfill \\ \hfill D^{\alpha }E& =& \beta SI-(\rho +\mu )E,\hfill \\ \hfill D^{\alpha }I& =& \rho E-(\varphi +\eta +\mu )I,\hfill \\ \hfill D^{\alpha }R& =& \omega I+bV-\mu R\hfill \end{array}$$

(1)

with initial conditions

$$\begin{array}{ccc}\hfill S\left(t_0\right)& =& S_0,V\left(t_0\right)=V_0,E\left(t_0\right)=E_0,I\left(t_0\right)=I_0,R\left(t_0\right)=R_0.\hfill \end{array}$$

(2)

All the parameters are taken from Table 1.

Figure 1. Diagram of fractional Model (1).

Figure 1. Diagram of fractional Model (1).

2.1. Mathematical Analysis

To determine the existence and uniqueness of the model, we rewrite Model (1) as

$$\begin{array}{ccc}\hfill D_t^{\alpha }g\left(t\right)& =& f(t,g(t\left)\right),0<\alpha \le 1,\hfill \end{array}$$

(3)

where

$$\begin{array}{ccc}\hfill f(t,g)& =& {[f_1,f_2,f_3,f_4,f_5]}^T; \mathrm{here}, T \mathrm{denotes} \mathrm{the} \mathrm{transpose}\hfill \end{array}$$

with initial values

$$\begin{array}{ccc}\hfill g\left(0\right)& =& \left[S\right(0),V(0),E(0),I(0),R(0\left)\right]\hfill \end{array}$$

as the initial condition, where the derivative is in the left Caputo sense. Here, $f_1, f_2, f_3, f_4, f_5$ are right-hand-side of System (1). For Model (1), we have

$$\begin{array}{ccc}\hfill f_1& =& p\lambda -\beta SI+\theta V-(\nu +\mu )S,\hfill \\ \hfill f_2& =& (1-p)\lambda +\nu S-(\theta +b+\mu )V,\hfill \\ \hfill f_3& =& \beta SI-(\rho +\mu )E,\hfill \\ \hfill f_4& =& \rho E-(\varphi +\eta +\mu )I,\hfill \\ \hfill f_5& =& \omega I+bV-\mu R.\hfill \end{array}$$

(4)

Function $f(t,.g):R\times R^d\to R^d$ with $d\ge 1$.

Table 1. Parameters and their values for (1).

Parameter	Definition	Value	Reference
p	Proportion of the successively
	vaccinated at birth	0.1	Assumed
$\lambda$	Birth rate	200	[25]
$\beta$	Contact rate	0.09091	[26]
$\theta$	Efficacy rate of 1st dose
	of vaccine to susceptible rate	0.167	Assumed
$\nu$	Vaccination rate from S	0.008754	Assumed
$\mu$	Natural death rate	0.00875	[25]
b	Received 2nd dose of vaccine
	to recovery rate	0.5	Assumed
$\rho$	Recovered form the
	infected class at rate	0.003	[25]
$\varphi$	Receive 1st dose of vaccine
	to susceptibility rate	0.2	[25]
$\eta$	Disease death rate for infected class	0.09	Assumed
$\omega$	Natural rate of progression	0.0013	[25]

Table 1. Parameters and their values for (1).

Parameter	Definition	Value	Reference
p	Proportion of the successively
	vaccinated at birth	0.1	Assumed
$\lambda$	Birth rate	200	[25]
$\beta$	Contact rate	0.09091	[26]
$\theta$	Efficacy rate of 1st dose
	of vaccine to susceptible rate	0.167	Assumed
$\nu$	Vaccination rate from S	0.008754	Assumed
$\mu$	Natural death rate	0.00875	[25]
b	Received 2nd dose of vaccine
	to recovery rate	0.5	Assumed
$\rho$	Recovered form the
	infected class at rate	0.003	[25]
$\varphi$	Receive 1st dose of vaccine
	to susceptibility rate	0.2	[25]
$\eta$	Disease death rate for infected class	0.09	Assumed
$\omega$	Natural rate of progression	0.0013	[25]

2.2. Local Existence and Uniqueness of Solutions

Let us consider $R_{+}^5=[g\in R^5|g\ge 0]$ and $g\left(t\right)={(S,V,E,I,R)}^T$ and the rational field $Q[p,q]$. We have the following theorem.

Theorem 1.

For $h\left(x\right)\in Q[p,q]$ and $D_t^{\alpha }h\left(x\right)\in Q[p,q]$; then, $0<\alpha \le 1,$

$$\begin{array}{ccc}\hfill h\left(g\right)& =& h\left(p\right)+\frac{1}{\Gamma \left(\alpha \right)}\left(D_p^{\alpha }f\right)\left(\varphi \right){(g-p)}^{\alpha },\hfill \end{array}$$

Here, $p\le \varphi \le l,\forall l\in (p,q]$ [27]. Let $g\left(x\right)\in Q[p,q]$ and $D_p^{\alpha }h\left(g\right)\in Q[p,q],$ for $0<\eta \le 1.$ If $D_p^{\alpha }h\left(g\right)\ge 0,$ for all $g\in (p,q),$$h\left(g\right)$ is non-decreasing for each $g\in (p,q)$ if $D_p^{\alpha }h\left(g\right)\le 0,$∀$g\in (p,q);$ then, $h\left(g\right)$ is non-increasing for each $g\in (p,q)$.

We consider $M\left(t\right)$ such that $M\left(t\right):R_{0,+}\to R_{0,+}$ as follows:.

$$\begin{array}{ccc}\hfill M\left(t\right)& =& f_1+f_2+f_3+f_4+f_5,\forall t\ge t_0.\hfill \end{array}$$

(5)

where $f_i,i=1,2,3,4,5$. Function $M\left(t\right)$ is defined and differentiable for $[0,t_f]$, and we have

$$\begin{array}{ccc}\hfill M\left(t\right)& \le & p\lambda -\delta (S+V+E+I+R).\hfill \end{array}$$

(6)

From (6), we have

$$\begin{array}{ccc}\hfill f\left(t\right)& \le & p\lambda -\delta \left[S\left(t\right)+V\left(t\right)+E\left(t\right)+I\left(t\right)+R\left(t\right)\right],\forall t\in (0,t_f).\hfill \end{array}$$

(7)

The local existence of the solution of fractional ordered System (1) can be proved using the following theorem [28].

Theorem 2.

For $Q=[t_0-b,t_0+b],W=g\in R^d\left|\right|g-g_0\left|\right|\le b,U=\{(t,g)\in R\times R^d|t\in Q,Q\in W\}.$

And $f:U\to R^d$ satisfies the following three properties:

(a)

$f(t,g)$ is Lebesgue measurable with $t\in Q$ with t,

(b)

$f(t,g)$ is continuous on W,

(c)

$m\left(t\right)\in L^5\left(Q\right)$ is a real function satisfying $f(t,g)\le m\left(t\right)$ for $t\in Q$ with $g\in W.$

Now, $\alpha >\frac{1}{2}$; we have only one solution of $D_t^{\alpha }g\left(t\right)=f(t,g\left(t\right)),0<\alpha <1$ within $[t_0-s,t_0+s],s>0.$ Now, the $(f_1,f_2,f_3,f_4,f_5)$ of Model (1) are measurable $\forall t\in [t_0-s,t_0+s].$ Thus, $f(g,t)$ satisfies Theorem 3 with $m\left(t\right)=\lambda -\delta g\left(t\right)$, with $g=S+V+E+I+R$. Hence, there exists a solution for Model (1) in $(0,t_f).$

Theorem 3.

Suppose the conditions of Theorem 2 hold. Now, consider a real-valued function $\kappa \left(t\right)\in L^5\left(Q\right)$ for which

$$\begin{array}{c}\hfill \left|\right|f(t,g)-f(t,\kappa )\left|\right|\le \kappa \left(t\right)\left|\right|g-\kappa \left|\right|,\end{array}$$

(8)

with $t\in Q$ and $g,k\in W$; then, system $D_t^{\alpha }g\left(t\right)=f(t,g\left(t\right)),0<\alpha \le 1$ possesses a unique solution on the interval $[t_0-s,t_0+s]$ for some $s>0$. From Equations (1) and (6), we have

$$\begin{array}{ccc}\hfill \left|\right|f(t,g)-f(t,\kappa )\left|\right|& \le & \kappa \left(t\right)\left|\right|g-\kappa \left|\right|,\hfill \end{array}$$

(9)

with $\kappa \left(t\right)\le \frac{\lambda }{\delta }$, thus satisfying fractional ordered System (1) with the uniqueness of the solution. With the help of Theorem 3.1 [28], we verify the global existence of the solution.

Theorem 4.

Vector filed $V(t,g)$ exists along with the conditions for Theorems (2) and (3) in the global space and $\left|\right|f(t,g)\left|\right|\le \omega +l\left|\right|x\left|\right|$ for almost every $t\in R$ and $g\in R^d$ with $\omega ,l>0$. Then, function $g\left(t\right)$ can be defined on $(-\infty ,+\infty )$ by solving (1) with an initial condition. Using (6) along with the norm property, we have $\left|\right|f\left(g\right(t\left)\right)\left|\right|\le \lambda +\delta \left|\right|g\left|\right|$. Thus, the global existence condition is satisfied with $\omega =\lambda$ and $l=\delta$.

2.3. Basic Reproduction Ratio

To calculate the basic reproduction number for System (1), we consider F and G to be two three-dimensional matrices corresponding to vectors $\mathcal{F}$ and $\mathcal{G}$ defined. To formulate the matrix, we only consider the infected compartments of the system, which are $E,I,R$. At the infection-free state, we have

$$\begin{array}{ccc}\hfill \mathcal{F}& =& \left(\begin{array}{c}\beta SI\\ 0\\ 0\end{array}\right),\mathcal{G}=\left(\begin{array}{c}(\rho +\mu )E\\ (\varphi +\eta +\mu )I-\rho E\\ -\omega I-bV+\mu R\end{array}\right).\hfill \end{array}$$

(10)

Then, we have

$$\begin{array}{ccc}\hfill F& =& \left(\begin{array}{ccc}0& \beta S& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right),G=\left(\begin{array}{ccc}(\rho +\mu )& 0& 0\\ -\rho & (\varphi +\eta +\mu )& 0\\ 0& -\omega & \mu \end{array}\right),\hfill \end{array}$$

Now,

$$\begin{array}{ccc}\hfill G^{-1}& =& \left(\begin{array}{ccc}\frac{1}{(\rho +\mu )}& 0& 0\\ \frac{\rho }{(\rho +\mu )(\varphi +\eta +\mu )}& \frac{1}{(\varphi +\eta +\mu )}& 0\\ \frac{\rho \omega }{\mu (\rho +\mu )(\varphi +\eta +\mu )}& \frac{\omega }{\mu (\varphi +\eta +\mu )}& \frac{1}{\mu }\end{array}\right),\hfill \end{array}$$

which is a non-singular matrix.

Then, $F{G}^{-1}$ at $E^0$ is also a non-negative next-generation matrix, which provides the estimated number of the new infection,

$$\begin{array}{ccc}\hfill F{G}^{-1}& =& \left[\begin{array}{ccc}0& \beta S^0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\times \left[\begin{array}{ccc}\frac{1}{(\rho +\mu )}& 0& 0\\ \frac{\rho }{(\rho +\mu )(\varphi +\eta +\mu )}& \frac{1}{(\varphi +\eta +\mu )}& 0\\ \frac{\rho \omega }{\mu (\rho +\mu )(\varphi +\eta +\mu )}& \frac{\omega }{\mu (\varphi +\eta +\mu )}& \frac{1}{\mu }\end{array}\right],\hfill \end{array}$$

$$\begin{array}{ccc}& =& \left[\begin{array}{ccc}\frac{\beta S^0\rho }{(\rho +\mu )(\varphi +\eta +\mu )}& \frac{\beta S^0}{(\varphi +\eta +\mu )}& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right].\hfill \end{array}$$

(11)

According to [29,30], we obtain the basic reproduction number $R_0$ as

$$\begin{array}{ccc}\hfill {\mathcal{R}}_0& =& \frac{\beta S^0\rho }{(\rho +\mu )(\varphi +\eta +\mu )}.\hfill \end{array}$$

(12)

Theorem 5.

System (1) of measles infection has a threshold parameter basic reproduction number $R_0=\frac{\beta S^0\rho }{(\rho +\mu )(\varphi +\eta +\mu )}$ at $E^0$. For $R_0>1$, Model (1) switches to the endemic state.

3. Model Analysis

In this model, (1), we have two equilibrium points.

• $E^0(S^0,V^0,0,0,0)$ represents no infection.

  $$\begin{array}{ccc}\hfill S^0& =& \frac{\lambda (\nu +\mu +p\mu )}{[{\mu }^2+\mu (\theta +b)+\nu (\mu +b)]}>0,\hfill \\ \hfill V^0& =& \frac{\lambda (\mu +\nu -\mu p)}{[{\mu }^2+\mu (\theta +b)+\nu (\mu +b)]}>0,\hfill \end{array}$$

  (13)

  and $V^0$ exist when $p<\frac{\mu +\nu }{\mu }$.
• With the endemic equilibrium point $E^{*}(S^{*},V^{*},E^{*},I^{*},R^{*})$, we obtain

  $$\begin{array}{ccc}\hfill S^{*}& =& \frac{(\rho +\mu )(\varphi +\eta +\mu )}{\beta \rho },\hfill \\ \hfill V^{*}& =& \frac{\beta (1-\rho )\lambda +\nu (\mu +\rho )(\varphi +\eta +\mu )}{\beta \rho (\theta +b+\mu )},\hfill \\ \hfill E^{*}& =& \frac{(\varphi +\eta +\mu )I^{*}}{\rho },\hfill \\ \hfill I^{*}& =& \frac{p\lambda +(\nu +\mu )S^{*}-\theta V^{*}}{\beta S^{*}},\hfill \\ \hfill R^{*}& =& \frac{\omega I^{*}+b{V}^{*}}{\mu }.\hfill \end{array}$$

  (14)
  $E^{*}$ exits when $\theta <\frac{p\lambda +(\nu +\mu )S^{*}}{\beta S^{*}{V}^{*}}$.

Stability Analysis

At $E^0$, we have the Theorem stated below.

Theorem 6.

When $R_0<1$, disease-free equilibrium $E^0$ for (1) is locally asymptotically stable if $R_0<1$ and unstable if $R_0>1$

To verify the local stability of System (1) at $E^0$, we have

$$\begin{array}{ccc}\hfill J& =& \left(\begin{array}{ccccc}-(\beta I+\nu +\mu )& \theta & 0& -\beta S& 0\\ \nu & -(\theta +b+\mu )& 0& 0& 0\\ 0& 0& -(\rho +\mu )& \beta S& 0\\ 0& 0& \rho & -(\varphi +\eta +\mu )& 0\\ 0& b& 0& \omega & -\mu \end{array}\right),\hfill \end{array}$$

(15)

Now,

$$\begin{array}{ccc}\hfill |J-\xi I_{4\times 4}{|}_{E^0}& =& \left(\begin{array}{ccccc}-(\nu +\mu )-\xi & \theta & 0& -\beta S& 0\\ \nu & -(\theta +b+\mu )-\xi & 0& 0& 0\\ 0& 0& -(\rho +\mu )-\xi & \beta S& 0\\ 0& 0& \rho & -(\varphi +\eta +\mu )-\xi & 0\\ 0& b& 0& \omega & -\mu -\xi \end{array}\right),\hfill \end{array}$$

(16)

and the characteristic equation for (16) is

$$\begin{array}{c}\hfill {\xi }^4+A_{11}{\xi }^3+A_{22}{\xi }^2+A_{33}\xi +A_{44}=0\end{array}$$

(17)

where

$$\begin{array}{ccc}\hfill A_{11}& =& (a_1+a_2+a_3+a_4),\hfill \\ \hfill A_{22}& =& (a_3{a}_4-\beta \nu S)+(a_1+a_2)(a_3+a_4)+(a_1{a}_2-\theta \nu ),\hfill \\ \hfill A_{33}& =& (a_1+a_2)(a_3{a}_4-\beta \nu S)(a_2{a}_1-\theta \nu )(a_3+a_4),\hfill \\ \hfill A_{44}& =& (a_1{a}_2-\theta \nu )(a_3{a}_4-\beta \nu S),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill a_1& =& (\theta +b+\mu ),a_2=(\mu +\nu ),a_3=(\mu +\rho ),a_4=(\varphi +\eta +\mu ),\hfill \end{array}$$

Now, the characteristic polynomial $D\left(ϵ\right)$ with

$$\begin{array}{ccc}\hfill ϵ\left(\xi \right)& =& {\xi }^4+A_{11}{\xi }^3+A_{22}{\xi }^2+A_{33}{\xi }^2+A_{44}=0\hfill \end{array}$$

(18)

is

$$\begin{array}{ccc}\hfill D\left(ϵ\right)& =& \left(\begin{array}{ccccccc}1& A_{11}& A_{22}& A_{33}& A_{44}& 0& 0\\ 0& 1& A_{11}& A_{22}& A_{33}& A_{44}& 0\\ 0& 0& 1& A_{11}& A_{22}& A_{33}& A_{44}\\ 4& 3A_{11}& 2A_{22}& A_{33}& 0& 0& 0\\ 0& 4& 3A_{11}& 2A_{22}& A_{33}& 0& 0\\ 0& 0& 4& 3A_{11}& 2A_{22}& A_{33}& 0\\ 0& 0& 0& 4& 3A_{11}& 2A_{22}& A_{33}\end{array}\right).\hfill \end{array}$$

(19)

We have the following properties.

Proposition 1.

At equilibrium point $E^{*}$ for the system in $R_{+}^4$, we have the following.

(i)

There exist ${\kappa }_1,{\kappa }_2$, and ${\kappa }_3$ where ${\kappa }_1=A_{11};{\kappa }_2=A_{11}{A}_{22}-A_{33}$ and

$$\begin{array}{ccc}\hfill {\kappa }_3& =& \left(\begin{array}{ccc}{A}_{11}& 1& 0\\ A_{33}& A_{22}& A_{11}\\ 0& A_{44}& A_{33}\end{array}\right).\hfill \end{array}$$

When $\alpha =1$, then it satisfies the following condition:

$$\begin{array}{ccc}\hfill {\kappa }_1& >& 0,{\kappa }_2>0,{\kappa }_3=0and{\kappa }_4>0,\hfill \end{array}$$

which shows the locally asymptotically stable condition for $E^{*}$.

(ii)

$E^{*}$ is unstable when $D\left(ϵ\right)>0$, $A_{11},A_{22}<0$, and $\alpha >\frac{2}{3}$.

(iii)

$E^{*}$ is locally asymptotically stable when $D\left(ϵ\right)<0,A_{11}>0,A_{22}>0,A_{33},A_{44}$, and $\alpha <\frac{1}{3}$ exist. Also at $E^{*}$, the system is unstable. When $D\left(ϵ\right)<0,A_{11}<0,A_{22}>0,A_{33}<0$, and $A_{44}>0$.

(iv)

$E^{*}$ is locally asymptotically stable for $\alpha \in (0,1)$ along with condition $D\left(ϵ\right)<0,A_{11}>0,A_{22}>0,A_{33}>0,A_{44}>0$, and $A_{22}=\frac{A_{11}{A}_{44}}{A_{33}}+\frac{A_{33}}{A_{11}}$.

(v)

The necessary condition for locally asymptotically stable $E^{*}$ is $A_{44}>0$.

Remark 1.

Endemic equilibrium $E^{*}$ is locally asymptotically stable for $\alpha =1$ when it satisfies the R-H conditions. But this condition is not sufficient when $\alpha \in [0,1)$.

4. The Fractional Optimal Control Problem

This section deals with control strategies. Here, our main aim is to reduce the side effects of the vaccine and the cost of treatment. The control input is defined as $u\left(t\right)=\{u_1\left(t\right),u_2\left(t\right)\}$. Here, we assume $u_1\left(t\right)$ as the first dose and $u_2\left(t\right)$ as the second dose of the vaccine.

We take the objective function as

$$\begin{array}{ccc}\hfill J\left[u\right(t\left)\right]& =& {\int }_0^T[A{u}_1^2+B{u}_2^2+C{E}^2+D{I}^2]dt,\hfill \end{array}$$

(20)

where C and D represent the penalty multipliers on the benefit of the cost, and A and B stand for weighting constants on the profit of the price of making. We take a quadratic cost functional on the control factors as an approximation to the real nonlinear functional that depends on the fact that the cost takes a nonlinear form to overcome the bang-bang or singular optimal control cases.

Here, the aim is to minimize the above functional equations. Thus, the problem can be represented as follows:

$$\begin{array}{ccc}\hfill minJ\left[u\right(t\left)\right]& =& {\int }_0^T[A{u}_1^2+B{u}_2^2+C{E}^2+D{I}^2]dt,\hfill \end{array}$$

(21)

subject to state System (22),

$$\begin{array}{ccc}\hfill D^{\alpha }S& =& p\lambda -(1-u_1)\beta SI+\theta V-(\nu +\mu )S,\hfill \\ \hfill D^{\alpha }V& =& (1-p)\lambda +\nu S-(\theta +b+\mu )V,\hfill \\ \hfill D^{\alpha }E& =& (1-u_1)\beta SI-(1-u_2)(\rho +\mu )E,\hfill \\ \hfill D^{\alpha }I& =& (1-u_2)\rho E-(\varphi +\eta +\mu )I,\hfill \\ \hfill D^{\alpha }R& =& \omega I+bV-\mu R\hfill \end{array}$$

(22)

with the following initial conditions:

$$\begin{array}{ccc}\hfill S\left(0\right)& =& S_0,V\left(0\right)=V_0,E\left(0\right)=E_0,I\left(0\right)=I_0,R\left(0\right)=R_0.\hfill \end{array}$$

(23)

The objective functional is described by $D_t^{\alpha }=(S,V,E,I,R)$. The weight constants C and D are associated to the infected class, while A and B are related to control variables $u_1\left(t\right)$ and $u_2\left(t\right)$. Eradication of the disease by minimizing the infected population is our main focusing area. Control input $(u_1^{*},u_2^{*})$ is derived from

$$\begin{array}{ccc}\hfill \mathcal{J}(u_1^{*},u_2^{*})& =& min\{\mathcal{J}(u_1,u_2),u_i\in \mathcal{U},fori=1,2\},\hfill \end{array}$$

(24)

where control function $\mathcal{U}$ is defined as

$$\begin{array}{c}\hfill \mathcal{U}=\left\{(u_1,u_2)\right|u_i \mathrm{is} \mathrm{the} \mathrm{Lebesgue} \mathrm{measurable} \mathrm{on}[0,T],0\le u_i\left(t\right)\le 1,i=1,2\}.\end{array}$$

(25)

Conditions for Optimalilty

The fractional order control problem is defined as

$$\begin{array}{ccc}\hfill D_t^{\alpha }x& =& f(x,u,t),withx\left(0\right)=x_0.\hfill \end{array}$$

(26)

Here, state vector $x\left(t\right)={[S\left(t\right),V\left(t\right),E\left(t\right),I\left(t\right),R\left(t\right)]}^T$ and control vector $u\left(t\right)={[u_1\left(t\right),u_2\left(t\right)]}^T$. J is given as

$$\begin{array}{ccc}\hfill J\left(u\right)& =& {\int }_0^{t}g(x,u,t)dt.\hfill \end{array}$$

(27)

Thus, we have

$$\begin{array}{ccc}\hfill infJ\left(u\right(.\left)\right)& =& J\left(u^{*}(.)\right),\hfill \end{array}$$

(28)

with state system

$$\begin{array}{ccc}\hfill D_t^{\alpha }& =& f(x,u,v),x\left(0\right)=x_0.\hfill \end{array}$$

(29)

Here, $V\left(t\right)$ is the co-vector satisfy

$$\begin{array}{ccc}\hfill D_{tf}^{\alpha }V& =& \frac{\partial g}{\partial x}+V^T\frac{\partial f}{\partial x},V\left(t_f\right)=0.\hfill \end{array}$$

(30)

Thus, $u^{*}$ is calculated with the help of

$$\begin{array}{ccc}\hfill \frac{\partial g}{\partial x}+V^T\frac{\partial f}{\partial u^{*}}& =& 0.\hfill \end{array}$$

(31)

Equations (29)–(31) represent the Euler–Lagrange optimally condition.

To determine the solution, we construct the Hamiltonian function as

$$\begin{array}{ccc}\hfill H& =& g+V^{T}f\hfill \end{array}$$

(32)

where

$$\begin{array}{ccc}\hfill g& =& (A{u}_1^2+B{u}_2^2+C{E}^2+D{I}^2),\hfill \\ \hfill V& =& {(v_1,v_2,v_3,v_4,v_5)}^T,\hfill \\ \hfill f& =& {(f_1,f_2,f_3,f_4,f_5)}^T.\hfill \end{array}$$

By the help of the Euler–Lagrange property, we can minimize (14).

The adjoint system for the control input $(u_1\left(t\right),u_2\left(t\right))$ for Model (22) is given by

$$\begin{array}{ccc}\hfill D_{tf}^{\alpha }{\xi }_1& =& -[(1-u_1)\beta I({\xi }_3-{\xi }_1)+{\xi }_2\nu -{\xi }_1(\nu +\mu )],\hfill \\ \hfill D_{tf}^{\alpha }{\xi }_2& =& -[\theta ({\xi }_1-{\xi }_2)+b({\xi }_5-{\xi }_2)-{\xi }_2\mu ],\hfill \\ \hfill D_{tf}^{\alpha }{\xi }_3& =& -(1-u_2)[{\xi }_4\rho -{\xi }_3(\rho +\mu )],\hfill \\ \hfill D_{tf}^{\alpha }{\xi }_4& =& -[\beta S(1-u_1)({\xi }_3-{\xi }_1)-{\xi }_4(\varphi +\eta +\mu )+{\xi }_5\omega ],\hfill \\ \hfill D_{tf}^{\alpha }{\xi }_5& =& {\xi }_5\mu .\hfill \end{array}$$

(33)

Thus, we have

$$\begin{array}{ccc}\hfill u_1^{*}\left(t\right)& =& \frac{({\xi }_3-{\xi }_1)\beta SI}{2A},\hfill \\ \hfill u_2^{*}\left(t\right)& =& -\frac{{\xi }_3\left(\mu E\right)}{2B}.\hfill \end{array}$$

(34)

The standard form of control is given below:

$$\begin{array}{ccc}\hfill u_1^{*}\left(t\right)& =& \left\{\begin{array}{cc}0,\hfill & \frac{({\xi }_3-{\xi }_1)\beta SI}{2A}<0,\hfill \\ \frac{({\xi }_3-{\xi }_1)\beta SI}{2A},\hfill & 0<\frac{({\xi }_3-{\xi }_1)\beta SI}{2A}<1,\hfill \\ 1,\hfill & \frac{({\xi }_3-{\xi }_1)\beta SI}{2A}>1.\hfill \end{array}\right.\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill u_2^{*}\left(t\right)& =& \left\{\begin{array}{cc}0,\hfill & -\frac{{\xi }_3\left(\mu E\right)}{2B}<0,\hfill \\ -\frac{{\xi }_3\left(\mu E\right)}{2B},\hfill & 0<-\frac{{\xi }_3\left(\mu E\right)}{2B}<1,\hfill \\ 1,\hfill & -\frac{{\xi }_3\left(\mu E\right)}{2B}>1.\hfill \end{array}\right.\hfill \end{array}$$

(36)

The compact form of $u_1^{*}\left(t\right)$ is

$$\begin{array}{ccc}\hfill u_1^{*}\left(t\right)& =& max\left(min\left(1,\frac{({\xi }_3-{\xi }_1)\beta SI}{2A}\right),0\right).\hfill \end{array}$$

(37)

Similarly, for $u_2^{*}\left(t\right)$, we have

$$\begin{array}{ccc}\hfill u_2^{*}\left(t\right)& =& max\left(min\left(1,-\frac{{\xi }_3\left(\mu E\right)}{2B}\right),0\right).\hfill \end{array}$$

(38)

5. Numerical Findings

This section deals with numerical findings based on our analytical results. We follow the iterative schemes as established by Cao et al. [31] for numerically solving the proposed Model (1) and the optimal system (i.e., state System (33) together with adjoint System (22)).

In order to solve Model (1), the iterative scheme is developed:

$$\begin{array}{ccc}\hfill S\left(n\right)& =& [p\lambda -(1-u_1)\beta S(n-1)I(n-1)+\theta V(n-1)-(\nu +\mu )S(n-1)]h^{\eta }-\sum _{r=1}^{n}l\left(r\right)S(n-r),\hfill \\ \hfill V\left(n\right)& =& [(1-p)\lambda +\nu S(n-1)-(\theta +b+\mu )V(n-1)]h^{\eta }-\sum _{r=1}^{m}l\left(r\right)V(n-r),\hfill \\ \hfill E\left(n\right)& =& [(1-u_1)\beta S(n-1)I(n-1)-(1-u_2)(\rho +\mu )E(n-1)]h^{\eta }-\sum _{r=1}^{n}l\left(r\right)E(n-r),\hfill \\ \hfill I\left(n\right)& =& [(1-u_2)\rho E(m-1)-(\varphi +\eta +\mu )I(n-1)]h^{\eta }-\sum _{r=1}^{n}l\left(r\right)I(n-r),\hfill \\ \hfill R\left(n\right)& =& [\omega I(n-1)+bV(n-1)-\mu R(n-1)]h^{\eta }-\sum _{r=1}^{n}l\left(r\right)R(n-r).\hfill \end{array}$$

(39)

The last term of the above set of equations is the memory term. Here, $l\left(0\right)=1$ and $l\left(m\right)=\frac{1-\eta }{r}{l}_{r-1},n\ge 0$, and all the model variables are all positive. The system behaviors are represented by different values of $\alpha$ in Figure 2. Here, we demonstrate an integer order system with $\alpha =1$ and a fractional order system for $\alpha =0.98$. The figure clearly shows that the linear order system converges to its steady state in a shorter time. Figure 3 shows the effect of $\beta$ on model dynamics. Here, we consider $\beta =0.0002,0.00002$. This figure shows that decreasing the value of the infection rate is useful for controlling the disease.

To analyze fractional order Model (22), we follow the algorithm from the article of Cao et al. [32]. To solve optimal System (22), we consider (33). Here, we use the forward iteration method to solve (22), and for (33), we use the backward iteration method.

The numerical solution of Model (22) is represented in Figure 4 and Figure 5 in the presence of control. Figure 4 shows the system behavior when control is incorporated, and the optimal control input is represented by Figure 5. From this simulation, we observe that control reduces the infected population and increases the susceptible population.

6. Discussion

This work deals with the study of measles disease infection in the presence of vaccination. We formulated an SVEIR-type compartmental model. The model is extended to the SEIR model to gat an idea of the effect of vaccination. In the model with vaccination, we studied the effectiveness of the first dose of vaccination to control the infection in a cost-effective way.

Analytical and numerical analyses confirm that the system has a disease-free state when the basic reproduction number is $R_0<1$, but when $R_0>1$, the system switches its state to an endemic state (Figure 3 and Figure 6). An endemic equilibrium point exists for $R_0>1$ and is asymptotically stable. Also, optimal control therapy can improve the susceptible population and reduce the infection level (Figure 4 and Figure 5) in a cost-effective way.

In summary, the stability analysis shows that the disease is endemic when the basic reproduction number is greater than unity. The optimal control problem is solved using an iterative method. Simulation results show that the role of fractional order parameter $\alpha$ is effective on model dynamics. The FOCP shows that the vaccination is administered in an optimal order.

7. Conclusions

In this article, the focus is to formulate a fractional order mathematical model for the dynamics of the measles disease.

The physical–dynamical problem consists of memory or aftereffects. Also, we know that the fractional order differential equation model plays a more significant role and has more advantages compared to the integer-order mathematical model in handling such problems. Thus, we incorporate a FODE model to study the vaccination effect on the measles infection. Here, we extend our model based on control input. Non-medical interventions (like self-isolation and contact tracing) and medical interventions (like vaccination) are considered control inputs, and we try to optimize the control effect on model dynamics.

In a nutshell, it can be concluded that the fractional order mathematical model is more realistic than the integer order model. The control measures through vaccination have a significant impact on the disease when applied in an optimal way.

The present work can be extended using impulse drug dosing [33]. This method will enable us to apply the two consecutive doses one by one with an interval.