Ratio Mathematica                                                                            Volume 43, 2022 

 

              
 

 

 

Controlling Measles Transmission Dynamics 

with Optimal Control Analysis 

 

 

Chinwendu E. Madubueze* 

Isaac O. Onwubuya† 

Iorwuese Mzungwega‡ 
 

 

 

Abstract  

In this paper, a deterministic model for the transmission dynamics of measles infection 

with two doses of vaccination and isolation is studied. The disease-free equilibrium state 

and basic reproduction number, 𝑅0, of the model are computed.  The sensitivity analysis 
of the model parameters is carried out using the Latin Hypercube Sampling (LHS) scheme 

in other to ascertain the parameters that contribute to the spread of measles in the 

population. The result of the sensitivity analysis shows that transmission rates, 

vaccination rates and isolation of the infected persons in the prodromal stage are 

significant parameters to be targeted for the eradication of measles infection. Based on 

the result of sensitivity analysis, an optimal control model with nutritional support as a 

control is developed. The analysis of optimal control model is carried out using 

Pontryagin’s maximum principle to identify the optimal control strategies to be adopted 

by public health practitioners and health policy makers in curtailing the spread of measles 

infection. The result of the optimal control analysis via numerical simulations revealed 

that combined timely implementation of correct administration of the two doses of 

vaccination, isolation of infected persons in the prodromal stage and mass distribution of 

nutritional support would curtail the measles disease outbreak in the population. 

However, in a situation where there is a limited facility to isolate the infected persons in 

 
* Chinwendu E. Madubueze (Joseph Sarwuan Tarka Univerisity, Makurdi, Nigeria); 

ce.madubueze@gmail.com.  
† Isaac O. Onwubuya (Joseph Sarwuan Tarka Univerisity, Makurdi, Nigeria); isaacobiajulu@gmail.com. 
‡ Iorwuese Mzungwega (Joseph Sarwuan Tarka Univerisity, Makurdi, Nigeria). 
‡ Received on April 17th, 2022. Accepted on August 12th, 2022. Published on September 25th, 2022. 

doi:10.23755/rm.v41i0.742. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published 

under the CC-BY licence agreement. 

 

mailto:ce.madubueze@gmail.com
mailto:isaacobiajulu@gmail.com


C. E. Madubueze, I. O. Onwubuya, and I. Mzungwega 

 

 

the prodromal stage, the combined implementation of mass distribution of nutritional 

support and administration of the two doses of vaccination will still eradicate measles 

infection in the population.  

Keywords: Measles; Nutritional Support; Vaccination; Isolation; Sensitivity Analysis; 

Pairwise Comparison; Optimal Control Analysis. 

2010 AMS subject classification:  49K15, 49K40, 90C31, 34D20, 34C60. 

 

1. Introduction  

Measles is a viral infectious disease caused by a single-stranded RNA virus that 

belongs to the group of Morbilliviruses of the Paramyxoviridae family. It is a seasonal 

disease that occurs mostly during the dry season in tropical zones where it is endemic and 

it peaks during late winter and early spring in temperate zones [24]. Non-immune people 

are infected via direct contact with the nasal and oral secretions or inhaling the aerosol 

droplets of an infected person. Ninety percentage of non-immune people are exposed to 

an infective and have the chance of being infected with measles disease [24, 27]. Measles 

has an average of 10 - 12 days incubation period. The incubation period is the interval 

from exposure to the prodromal stage [28], which spans for seven days of the infection 

period, after which the infective recovers with lifelong immunity against the disease. The 

symptoms of measles are based on different stages of the disease. Prodromal stage 

symptoms include high fever, runny nose (coryza), cough, and red eyes (conjunctivitis) 

that lasts 2 to 4 days with a range of 1- 7 days, while the rash stage symptoms occur a few 

days after the initial symptoms. The rash stage can lead to fatal complications or death if 

not treated early [10,12].  

Annually, measles affects up to 20 million people worldwide and most cases are from 

Africa and Asia [24]. According to a report from World Health Organization and United 

State Center for Diseases Control and Prevention (CDC) [44, 45], 869,770 infection cases 

with 207,500 deaths of Measles were recorded globally in 2019, making it the highest 

number since the 1996 outbreak and has 50% increment as of 2016. For sub-Saharan 

Africa, about 134,200 measles deaths were recorded in 2015, while Nigeria recorded a 

significant increase of 28,400 cases in 2019 compared with 5,067 cases in 2018. Despite 

the cases dropping to 9,316 in 2020, the confirmed cases of measles remain high, and the 

case fatality is yet to be eased anytime there is an outbreak. This implies that 

comprehensive efforts and intervention strategies to reduce the menace of measles is 

crucial. Therefore, it is imperative to examine the optimal strategy that can be 

implemented to control measles disease in high-burden countries. 

According to WHO, the two major strategies to eradicate measles are vaccine and 

treatment [21, 23]. Isolation of infected people is also important in preventing further 

spread of the disease. However, increasing population immunity through vaccination 

remains the most effective way to prevent outbreaks of measles in a community [22]. The 

vaccination is mainly based on MMR (measles, mumps, and rubella) and MMRV 

(measles, mumps, rubella, and varicella) vaccines. These vaccines are about 95% 

effective as they globally prevent 4.5 million deaths yearly [16]. There are two doses of 



Controlling measles transmission dynamics with optimal control analysis 

 

 

MMR vaccine. The first dose produces 90% to 95% immunity to measles while the 

second dose produces a stronger immunity for those that do not respond to the first dose 

[17]. Among the childhood vaccine-preventable diseases, measles causes the most deaths 

in children. Measles outbreaks is prevented in a community if 90% to 95% of children 

are vaccinated.  

Mathematical models of infectious diseases are useful in studying transmission 

dynamics of diseases, testing theories, planning, implementing, evaluating and comparing 

various control programs that will prevent the further spread of diseases and their 

epidemics. A notable number of mathematical models have been elaborated and applied 

to infectious diseases like measles [4, 7, 9, 18, 25]. Some authors, like [8, 10, 19], 

developed an SEIR model of measles where testing and diagnosis therapy was 

incorporated as in [10] at the latent period. Authors [15, 20] considered the effect of 

supplemental immunization activities as an optimal policy for measles using an age-

stratified compartmental model. Stephen et al. [17] revealed that the spread of measles 

disease largely depends on the contact rates with infected people within a population and 

the disease dies out in the population if the proportion of the population that is immune 

exceeds the herd immunity level. Vaccination is considered in the autonomous models 

[3, 4, 11, 13, 19, 29] as constant parameter or compartment for vaccinated people, while 

in [27-31], it is examined as a time-dependent control function to determine the optimal 

vaccination strategy that can be implemented to control measles in high-burden countries. 

Although [30 – 35] considered optimal control of vaccination for measles, the effect of 

two doses of vaccination and nutritional support are not studied.  However, the authors 

[3, 37, 40] examined the effect of two doses of vaccination and isolation on measles 

disease as constant parameters without nutritional support impact and optimal control 

analysis. As advised by WHO [46], it is important to consider the effect of two doses of 

vaccination and nutritional support on measles transmission dynamics, which forms the 

study’s motivation. This involves modification of the model by [3] to investigate the 

impact of the two doses of vaccination, nutritional support and isolation on measles 

dynamics using sensitivity analysis and optimal control analysis approaches. This will 

help to provide the mathematical analysis of the possible control strategy (vaccine and 

nutritional support) that will help the public health practitioners to achieve the best 

strategy for the prevention and control of the spread of measles in community. 

The rest of the paper is organized as follows: Section 2 is the model formulation for 

measles with constant control measures. The model analysis is discussed in Section 3 

which includes sensitivity analysis. We obtained the optimal control of the formulated 

model in Section 4. In Section 5, we carried out numerical simulation to verify some 

analytic results and their discussion, while Section 6 is the conclusion. 

 

2. Model formulation 

A deterministic model for measles disease is presented by modifying the model by 

Aldila and Asrianti [3]. The total population at any time (𝑡) denoted by 𝑁(𝑡), is sub-
divided into Susceptible persons, 𝑆(𝑡), Exposed persons, 𝐸(𝑡), Infected persons in 
prodromal stage, 𝑃(𝑡), Infected persons in rash stage, 𝐼(𝑡),  Isolated persons, 𝐽(𝑡), 1st 



C. E. Madubueze, I. O. Onwubuya, and I. Mzungwega 

 

 

dose of vaccinated persons, 𝑉1(𝑡), 2
nd dose of vaccinated persons, 𝑉2(𝑡),  and Recovered 

persons, 𝑅(𝑡) such that      
                                                                                                                         

𝑁 = 𝑆 + 𝐸 + 𝑃 + 𝐼 + 𝐽 + 𝑉1 + 𝑉2 + 𝑅.                 (1) 
 

The susceptible persons,𝑆(𝑡),  decreases when they come in contact with the infected 
persons at a force of infection, 𝜆1 and become exposed person or by vaccination with 1

st 

dose vaccine at a rate,  1.  The persons vaccinated with 1
st dose vaccine may be infected 

at a force of infection, 𝜆2 since vaccine is not 100% efficacy.  The force of infections, 𝜆1 
and 𝜆2, are given by  

𝜆1 =
𝛽1(𝑃+𝑛1𝐼+𝑛2𝐽)

𝑁

𝜆2 =
𝛽2(𝑃+𝑛1𝐼+𝑛2𝐽)

𝑁

}          (2) 

 

where  𝛽1 and 𝛽2 are the transmission rate for the susceptible and 1
st dose vaccinated 

persons respectively, 𝑛1  and 𝑛2 are the parameters that reduce the infectivity of the 
infected persons in the rash stage and isolated persons respectively. 

The 1st dose of vaccinated persons, 𝑉1(𝑡) receive 2
nd dose vaccine at a rate, 2 and 

achieve immunity at the rate, 𝜎. The exposed persons, 𝐸(𝑡), becomes infected persons in 
the prodromal stage, 𝑃(𝑡) at a rate, 𝑘 after incubation period of measles disease. The 
infected persons in the prodromal stage, 𝑃(𝑡),  then progress to rash stage at a rate, 𝛼 
while some are isolated for further treatment at a rate,  𝜑1 or they recovered at a rate, 𝛿1. 
In a similar way, the infected persons at the rash stage, 𝐼(𝑡), are isolated at a rate, 𝜑2 or 
recovered from measles at a rate, 𝛿2. Meanwhile, the isolated persons recovered at a rate, 
 𝛿3. It is assumed that all the subpopulations experience natural death at a rate, 𝜇 and the 
subpopulations, 𝐼(𝑡) and 𝐽(𝑡) may die of measles disease at  𝑑1 and  𝑑2 respectively.      

The model description and details of the model parameters are presented in Figure 1 

and Table 1. respectively. 

 

With Figure 1 and Table 1, the transition within subpopulations are expressed by the 

following system of first order differential equations; 

 

 

 

 



Controlling measles transmission dynamics with optimal control analysis 

 

 

𝑑𝑆

𝑑𝑡
= Λ − 𝜆1𝑆 − ( 1 + 𝜇)𝑆,              

𝑑𝑉1

𝑑𝑡
= 1𝑆 − (𝜆2 + 2 + 𝜇)𝑉1,         

𝑑𝑉2

𝑑𝑡
= 2𝑉1 − (𝜇 + 𝜎)𝑉2,                    

𝑑𝐸

𝑑𝑡
= 𝜆1𝑆 + 𝜆2𝑉1 − (𝜇 + 𝑘)𝐸,          

𝑑𝑃

𝑑𝑡
= 𝑘𝐸 − (𝛼 + 𝜇 + 𝜑1 + 𝛿1)𝑃,      

𝑑𝐼

𝑑𝑡
= 𝛼𝑃 − (𝜑2 + 𝛿2 + 𝜇 + 𝑑1)𝐼,     

𝑑𝐽

𝑑𝑡
= 𝜑1𝑃 + 𝜑2𝐼 − (𝛿3 + 𝜇 + 𝑑2)𝐽,

𝑑𝑅

𝑑𝑡
= 𝜎𝑉2 +  𝛿1𝑃 + 𝛿2𝐼 + 𝛿3𝐽 − 𝜇𝑅}

 
 
 
 
 
 

 
 
 
 
 
 

   (3) 

 

where   𝜆1 =
𝛽1(𝑃+𝑛1𝐼+𝑛2𝐽)

𝑁
,    𝜆2 =

𝛽2(𝑃+𝑛1𝐼+𝑛2𝐽)

𝑁
   and the initial conditions, 𝑆(0) >

0,𝑉1(0) ≥ 0,𝑉2(0) ≥ 0,𝐸(0) ≥ 0,𝑃(0) ≥ 0,𝐼(0) ≥ 0,𝐽(0) ≥ 0, 𝑅(0) ≥ 0. The 
model parameters are assumed to be nonnegative except recruitment rate, Λ, that is strictly 
positive. 

 

 
 

Figure 1.  Model flow diagram for transmission dynamics of measles disease. 

 

 

 

 



C. E. Madubueze, I. O. Onwubuya, and I. Mzungwega 

 

 

Table 1. Parameters and their descriptions 

Parameters  Parameters Description  (Ranges)Nominal 

values 

Sources 

𝚲 

𝜷𝟏 

𝜷𝟐 

𝜺𝟏 

𝜺𝟐 

𝒏𝟏 

𝒏𝟐 

𝝁 

𝒌 

𝜶 

𝜹𝟏 

𝜹𝟐 

𝜹𝟑 

𝒅𝟏 

𝒅𝟐 

𝝎𝟏 

𝝎𝟐 

𝝈 

Recruitment rate 

Transmission rate for 𝑆(𝑡) class 

Transmission rate for 𝑉(𝑡) class 

Vaccination rate of first dose vaccine 

Vaccination rate of second dose vaccine 

Infectivity reduction rate for 𝐼(𝑡) class 

Infectivity reduction rate for 𝐽(𝑡) class 

Natural death rate 

Progression rate from 𝐸(𝑡) to 𝑃(𝑡) 

Progression rate from 𝑃(𝑡) to 𝐼(𝑡) 

Recovery rate for 𝑃(𝑡) class  

Recovery rate for 𝐼(𝑡)class 

Recovery rate for 𝐽(𝑡) class  

Disease-related death rate for 𝐼(𝑡)class 

Disease-related death rate for 𝐽(𝑡) class 

Isolation rate for 𝑃(𝑡) class 

Isolation rate for 𝐼(𝑡) class 

Immunity rate due to 2nd dose of 

vaccine 

(−)2000 

(0.0004 − 0.5)0.6 

(0.0003 − 0.4)0.5 

(0.01 − 0.95)0.6 

(0.01 − 0.95)0.01 

(−)0.1 

(−)0.01 

(−)1 65.365⁄  

(−)0.09 

(−)0.003 

(−)0.2 

(−)0.06 

(−)0.3121 

(−)0.125 

(−)0.1 

(0.0001 − 0.05)0.01 

(0.001 − 0.5)0.001 

(−)0.01 

[1] 

Assumed 

Assumed 

[38] 

Assumed 

[3] 

[3] 

[3] 

[1] 

Assumed 

[3] 

Assumed 

[36] 

[39] 

[1] 

[1] 

Assumed 

[38] 

  

 

3. Model analysis 

Here, the well-poseness of system (3) is established which implies that the model 

makes biological sense. This is done by proving the existence of nonnegative solutions 

and boundedness of the model (3) when given initial solutions of the model. 

 

Theorem 1. With the initial solutions, 𝑆(0) > 0,  𝑉1(0) ≥ 0,  𝑉2(0) ≥ 0, 𝐸(0) ≥ 0,
𝑃(0) ≥ 0, 𝐼(0) ≥ 0, 𝐽(0) ≥ 0,𝑅(0) ≥ 0, the model equation (3) has non-negative 
solutions for all time, 𝑡 > 0.  

Proof. Let 𝑡1 = 𝑠𝑢𝑝{𝑡 > 0: 𝑆(0) > 0,  𝑉1(0) ≥ 0,  𝑉2(0) ≥ 0, 𝐸(0) ≥ 0, 𝑃(0) ≥ 0,
𝐼(0) ≥ 0, 𝐽(0) ≥ 0,𝑅(0) ≥ 0} ∈ [0,𝑡].  



Controlling measles transmission dynamics with optimal control analysis 

 

 

From the first equation of system (3), we have  

𝑑𝑆

𝑑𝑡
= Λ − 𝜆1𝑆 − 1𝑆 −  𝜇𝑆 ≥ −( 1 +  𝜇 + 𝜆1)𝑆. 

Applying the method of integrating factor with initial condition, 𝑆(0), we have 
 

𝑆(𝑡) ≥  𝑆(0)exp {−∫ ( 1 + 𝜇 + 𝜆1)
𝑡

0

𝑡1} > 0 

which is always positive for 𝑡 > 0. 
In similar way, 𝑉1(𝑡) > 0,𝑉2(𝑡) > 0,𝐸(𝑡) > 0, 𝑃(𝑡) > 0, 𝐼(𝑡) > 0, 𝐽(𝑡) > 0,

𝑅(𝑡) > 0 for 𝑡 > 0. This means that the solution set 
𝑆(𝑡),𝑉1(𝑡),𝑉2(𝑡),𝐸(𝑡),𝑃(𝑡),𝐼(𝑡),𝐽(𝑡),𝑅(𝑡) of the system (3) is non-negative for all 𝑡 >
0.  

To show the boundedness of the solutions of the system (3), we state and prove 

feasible region of the system (3).  

  

Theorem 2.  The solutions of system (3) are contained in the feasible region, Ω =

{(𝑆, 𝑉1,𝑉2,𝐸,𝑃,𝐼, 𝐽,𝑅) ∈ ℜ+
8 : 𝑁 ≤

Λ

𝜇
} with the non-negative initial conditions.  

 

Proof.  To obtain the total population, 𝑁(𝑡), we sum up the equations of system (3) 
to yields 

          
𝑑𝑁

𝑑𝑡
= Λ − 𝜇𝑁 − 𝑑1𝐼 − 𝑑2𝐽 ≤  Λ − 𝜇𝑁.           (4) 

  

Applying Gronwall’s inequality with the initial condition, 𝑁(0) = 𝑁0 in equation (4) 
gives  

𝑁(𝑡) ≤
Λ

𝜇
 + [𝑁0 −

Λ

𝜇
 ]𝑒−𝜇𝑡.           (5) 

If 𝑁0 > (<)
Λ

𝜇
 , the total population, 𝑁, tends to 

Λ   

𝜇
  as 𝑡 → ∞.  Thus, in either case, the 

total population,  𝑁(𝑡) →
Λ 

𝜇
   as 𝑡 → ∞ in (5). Hence, the solution set of system (3) will 

enter the feasible region, Ω that is positively invariant.  

 

3.1 Existence of disease-free equilibrium state and basic 

reproduction number  

Disease-free equilibrium state occurs when there is no infection in the population, 

that is when the infected state variables are zero.  

Solving simultaneously at equilibrium state, 
𝑑𝑆

𝑑𝑡
= 0,

𝑑𝑉1

𝑑𝑡
=  0,

𝑑𝑉2

𝑑𝑡
=  0,

𝑑𝐸

𝑑𝑡
= 0,

𝑑𝑃

𝑑𝑡
= 0,

𝑑𝐼

𝑑𝑡
= 0,

𝑑𝐽

𝑑𝑡
= 0,

𝑑𝑅

𝑑𝑡
= 0 of the system (3) gives the disease-free equilibrium state,   

 



C. E. Madubueze, I. O. Onwubuya, and I. Mzungwega 

 

 

𝐸0 = (𝑆
0,𝑉1

0,𝑉2
0,𝐸0,𝑃0, 𝐼0, 𝐽0,𝑅0) 

= (
Λ

(𝜀1+𝜇)
,

𝜀1Λ

𝑓(𝜀1+𝜇)
,

𝜀1𝜀2Λ

𝑓(𝜎+𝜇)(𝜀1+𝜇)
,0,0,0,0,

𝜎𝜀1𝜀2Λ

𝜇𝑓(𝜎+𝜇)(𝜀1+𝜇)
)   (6) 

 

with 𝑓 = 2 + 𝜇. 
 

 

Basic Reproduction Number, 𝑅0   
 

The basic reproduction number is a threshold quantity that determines the persistence 

and eradication of the infectious disease in the population, making it the most important 

quantity in infectious disease epidemiology. It is defined as the mean number of persons 

infected when a single infective is introduced into a wholly susceptible population [6].  

𝑅0 is computed using the next-generation matrix approach [6].  
Following the approach in [6], the rate of new infection, ℱ𝑖,  and the rate of 

transitional terms, 𝒱𝑖, in compartment 𝑖, of the system (3) are given as  
  

ℱ𝑖 =

(

 

𝛽1(𝑃+𝑛1𝐼+𝑛2𝐽)𝑆

𝑁
+
𝛽2(𝑃+𝑛1𝐼+𝑛2𝐽)𝑉1

𝑁

0
0
0 )

 , 𝒱𝑖 = (

𝑔𝐸
−𝑘£ + ℎ𝑃
−𝛼𝑃 + 𝑝𝐼

−𝜑1𝐼 − 𝜑2𝑃 + 𝑞𝐽

), 

 

where 𝑖 = 1,…,4 is the number of infected compartments and 
 

𝑔 = (𝜇 + 𝑘),ℎ = (𝛼 + 𝜑1 + 𝛿1 + 𝜇),𝑝 = (𝜑2 + 𝛿2 + 𝜇 + 𝑑1),𝑞 = (𝛿3 + 𝜇 + 𝑑2).   (7) 
 

Taking the partial derivative of ℱ𝑖 and 𝒱𝑖 with respect to 𝐸,𝑃,𝐼 𝑎𝑛𝑑 𝐽 at DFE, 𝐸0, we 
have respective Jacobian matrices  

 

𝐹 =

(

 
 
0

(𝛽1𝑆
0+𝛽2𝑉1

0)

𝑁0

𝑛1(𝛽1𝑆
0+𝛽2𝑉1

0)

𝑁0

𝑛2(𝛽1𝑆
0+𝛽2𝑉1

0)

𝑁0

0 0 0 0
0 0 0 0
0 0 0 0 )

 
 

, 

 

𝑉 = (

𝑔 0 0 0
−𝑘 ℎ 0 0
0 −𝛼 𝑝 0
0 −𝜑1 −𝜑2 𝑞

), 

 

where 

𝑁0 = 𝑆0 + 𝑉1
0 + 𝑉2

0 + 𝐸0 + 𝑃0 + 𝐼0 + 𝐽0 + 𝑅0 =
Λ

𝜇
 . 



Controlling measles transmission dynamics with optimal control analysis 

 

 

 

The inverse of 𝑉 is given as   
  

𝑉−1 =

(

 
 
 
 

1

𝑔
0 0 0

𝑘

𝑔ℎ

1

ℎ
0 0

𝛼𝑘

𝑔ℎ𝑝

𝛼

ℎ𝑝

1

𝑝
0

𝑘(𝛼𝜑2+𝑝𝜑1)

𝑔ℎ𝑝𝑞

(𝛼𝜑2+𝑝𝜑1)

ℎ𝑝𝑞

𝜑2

𝑝𝑞

1

𝑞)

 
 
 
 

. 

 

With definition of basic reproduction number, 𝑅0, as the spectral radius of matrix, 
𝐹𝑉−1, we have   

 

𝑅0 =
(𝛽1𝑆

0+𝛽2𝑉1
0)

𝑁0
[
𝑘

𝑔ℎ
+
𝛼𝑘𝑛1

𝑔ℎ𝑝
+
𝑘𝑛2(𝛼𝜑2+𝑝𝜑1)

𝑔ℎ𝑝𝑞
]. 

 

Upon substitution of  𝑆0 =
Λ

𝜀1+𝜇
 ,  𝑉1

0 =
𝜀1Λ

𝑓(𝜀1+𝜇)
  and  𝑁0 =

Λ

𝜇
 ,  we have  

 

𝑅0 =
𝜇(𝛽1𝑓+𝛽2𝜀1)

𝑓(𝜀1+𝜇)
[
𝑘

𝑔ℎ
+
𝛼𝑘𝑛1

𝑔ℎ𝑝
+
𝑘𝑛2(𝛼𝜑2+𝑝𝜑1)

𝑔ℎ𝑝𝑞
]. 

 

By the virtue of next-generation matrix approach [6], the disease-free equilibrium, 

𝐸0, of system (3) is locally asymptotically stable if 𝑅0 < 1 and unstable if 𝑅0 > 1. This 
means that the measles infection will die out in the population if 𝑅0 < 1 while it will 
persist in the population when 𝑅0 > 1.  

 

3.2 Sensitivity analysis  

Sensitivity analysis plays an important role in examining the effect, influence and 

contribution of the parameters of a mathematical model to the model output. To know the 

type of intervention strategies to adopt in reducing the transmission and prevalence of any 

infectious disease, sensitivity analysis is carried-out to determine the biological 

significance of the model parameters in relation to the reproduction number, 𝑅0. We adopt 
the Latin Hypercube Sampling (LHS) scheme used by [41,42,43] with the Partial Rank 

Correlation Coefficients (PRCCs) procedure to assess the biological implications of each 

input parameter to the output parameter, the disease threshold, 𝑅0. This type of sensitivity 
analysis approach provides numerical results that enable us to explore the entire 

parameter space simultaneously, thereby producing an unbiased selection of the 

parameter values. The signs (positive or negative) of the PRCCs indicate the precise 

strength of the relationship between the input variables (parameters of the model) and the 

output variable, 𝑅0 in this case. It also provides an insight to the degree of monotonicity 
between the parameters of the model and 𝑅0. Thus, comparing the values of PRCCs 
enabled us to directly evaluate the impact of the model parameters on 𝑅0.  



C. E. Madubueze, I. O. Onwubuya, and I. Mzungwega 

 

 

Figure 2 shows the PRCCs for some important parameters of the model. The 

parameters 𝛽1 and 𝛽2 have positive PRCCs meaning increasing their values increase 𝑅0, 
which in return increase the spread of measles infection in the population.   Whereas, the 

parameters 1 , 2,𝜔1 and 𝜔2 with negative PRCCs reduce the value of 𝑅0 when they are 
increased. They have the capacity of ameliorate the spread of measles infection in the 

population, which leads to the eradication of the disease in the population. However, the 

parameter 𝜔2 has a small magnitude of PRCC that is non-monotonically related to 𝑅0 but 
it can still produce a change in the transmission dynamics of measles infection. In other 

to identify the model parameters that are significant in curtailing or enhancing the spread 

of measles disease, the Fisher Transformation is applied to the PRCCs to compute the p-

values of each of the model parameters as used in [42]. This is shown in Table 2. It is 

observed in Table 2 that the parameters (𝛽1,𝛽2, 1 , 2,𝜔1) have p-values that are 
significant while  the parameter, 𝜔2, has an insignificance p-value. This is further shown 
in Figure 3 as scatterplots for 𝑅0 against some model parameters. From Figure 3, it is 
observed that the parameters (𝛽1,𝛽2, 1 , 2,𝜔1) have a significant impact on 𝑅0 than 𝜔2.    

 

 
Figure 2. Tornados plot for some significant model parameters.  



Controlling measles transmission dynamics with optimal control analysis 

 

 

 
Figure 3. Monte Carlo simulations for some important parameters of the model 

generated using the parameter values in Table 1. In each simulation run, 1000 

randomly selected parameters are used. 

 

 

Table 2. Parameter PRCC significance (unadjusted p-value) 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 0.5 1
-6

-4

-2

0

2


1

lo
g
(R

0
)

0 0.5 1
-6

-4

-2

0

2


2
 

lo
g
(R

0
)0

0 0.5 1
-6

-4

-2

0

2


1

lo
g
(R

0
)

0 0.5 1
-6

-4

-2

0

2


2

lo
g
(R

0
)

0 0.05 0.1
-6

-4

-2

0

2


1

lo
g
(R

0
)

0 0.005 0.01
-6

-4

-2

0

2


2

lo
g
(R

0
)

Parameter PRCC p-value Keep 

𝜷𝟏 0.65564141 0.0000 TRUE 

𝜷𝟐 0.57787305 0.0000 TRUE 

𝜺𝟏 −0.67496222 0.0000 TRUE 

𝜺𝟐 −0.65937154  0.0000 TRUE 

𝝎𝟏 −0.54171573  0.0000 TRUE 

𝝎𝟐 −0.01142558  0.7049 FALSE 



C. E. Madubueze, I. O. Onwubuya, and I. Mzungwega 

 

 

Table 3. Pairwise PRCC Comparisons (unadjusted p-values) 

 𝜷𝟏 𝜷𝟐 𝜺𝟏 𝜺𝟐 𝝎𝟏 

𝜷𝟏  0.00506 0 0 0 

𝜷𝟐   0 0 0 

𝜺𝟏    0.5314 2.048 × 10
−6 

𝜺𝟐     3.743 × 10
−5 

𝝎𝟏      

 

 

Table 4. Pairwise PRCC Comparisons (FDR Adjusted p-values) 

 𝜷𝟏 𝜷𝟐 𝜺𝟏 𝜺𝟐 𝝎𝟏 

𝜷𝟏  0.005622 0 0 0 

𝜷𝟐   0 0 0 

𝜺𝟏    0.5314 2.926 × 10
−6 

𝜺𝟐     4.679 × 10
−5 

𝝎𝟏      

 

 

Table 5. Parameters different after FDR adjustment? 

 𝜷𝟏 𝜷𝟐 𝜺𝟏 𝜺𝟐 𝝎𝟏 

𝜷𝟏  TRUE TRUE TRUE TRUE 

𝜷𝟐   TRUE TRUE TRUE 

𝜺𝟏    FASLE TRUE 

𝜺𝟐     TRUE 

𝝎𝟏      

 

 

Tables 3 and 4 show the pairwise comparison of the important parameters of the 

model, whose p-values are less than 0.05.𝑇ℎ𝑖𝑠 𝑖𝑠 to establish if there exist any difference 
between the processes describing the compared parameters. The results of the pairwise 

PRCC comparison for the unadjusted p-values and the false discovery rate (FDR) 

adjusted p-values are presented in Table 3 and Table 4, respectively. With the FDR 

adjusted p-values in Table 4, we present the parameters different in Table 5. If the p-

values of the compared pair of significant parameters are less than 0.05, we say that they 



Controlling measles transmission dynamics with optimal control analysis 

 

 

are different (TRUE); otherwise not different (FALSE). We also noted from Table 5, that 

apart from  1 − 2  pair, all other pairs of parameters are significantly different. Thus, 
the parameters 𝛽1,𝛽2, 1, 2, 𝜔1 play a vital role in the eradication of the measles disease.  

Hence, the spread of measles infection will reduce drastically if  the value of 𝑅0 is 
less than a unity (𝑅0 < 1), which implies reducing the values of 𝛽1 and 𝛽2 as well as 
increasing the values of 1, 2, 𝜔1 . This establishes that isolating the infected 
individuals in the prodromal stage and minimizing contact with infected persons (both at 

the prodromal and rash stage) will eradicate the spread of measles in the population. Also, 

increasing the rate of correct administration of vaccines (first and second dose) will go a 

long way in reducing the number of infected individuals as many susceptible people will 

be protected by vaccination thereby minimize the spread of measles before and during the 

epidemic.  

 

4. Optimal control analysis 

Optimal control has been extensively applied as a strategy in controlling many 

epidemic outbreaks. The main idea of applying the optimal control to disease epidemics 

is to choose among the available strategies, the most suitable and effective strategies that 

will reduce disease infection rate to a minimum level while optimizing the cost of 

deploying these strategies [26]. In terms of measles epidemics, such strategy can include 

therapies, vaccines, isolation and educational campaigns [5]. 

Based on the result of the sensitivity analysis, the functions, 

𝑢1(𝑡),𝑢2(𝑡),𝑢3(𝑡),𝑢4(𝑡), are considered as time-dependent control functions where 
𝑢1(𝑡) is mass distribution of nutrition (supplement) support that reduces the transmission 
rates, 𝑢2(𝑡) is the first dose vaccination control, 𝑢3(𝑡) is the second dose vaccination 
control and 𝑢4(𝑡) is the isolation of infected people in the prodromal stage. The nutritional 
support is to boost the immune system of the body.  Thus, the optimal control model of 

the system (3) is given by 

 
𝑑𝑆

𝑑𝑡
 = Λ − 

(1−𝑢1(𝑡))𝛽1(𝑃+𝑛1𝐼+𝑛2𝐽)𝑆

𝑁
− 1𝑢2(𝑡)𝑆 − 𝜇𝑆,                                            

𝑑𝑉1

𝑑𝑡
= 1𝑢2(𝑡)𝑆 − 

(1−𝑢1(𝑡))(1−𝑢2(𝑡))𝛽2(𝑃+𝑛1𝐼+𝑛2𝐽)𝑉1

𝑁
− 2𝑢3(𝑡)𝑉1 − 𝜇𝑉1,        

𝑑𝑉2

𝑑𝑡
= 2𝑢3(𝑡)𝑉1 − (𝜇 + 𝜎)𝑉2,

𝑑𝐸

𝑑𝑡
= 

(1−𝑢1(𝑡))𝛽1(𝑃+𝑛1𝐼+𝑛2𝐽)𝑆

𝑁
+
(1−𝑢1(𝑡))(1−𝑢2(𝑡))𝛽2(𝑃+𝑛1𝐼+𝑛2𝐽)𝑉1

𝑁
− (𝜇 + 𝑘)𝐸,

𝑑𝑃

𝑑𝑡
= 𝑘𝐸 − (𝛼 + 𝜇 + 𝜑1 + 𝑢4(𝑡) + 𝛿1)𝑃,                                                              

𝑑𝐼

𝑑𝑡
= 𝛼𝑃 − (𝜑2 + 𝛿2 +  𝜇 + 𝑑1)𝐼,                                                                            

𝑑𝐽

𝑑𝑡
= 𝜑1𝑃 + 𝑢4(𝑡)𝑃 + 𝜑2𝐼 − (𝛿3 +  𝜇 + 𝑑2)𝐽,                                                      

𝑑𝑅

𝑑𝑡
=  𝜎𝑉2 + 𝛿2𝐼 + 𝛿1𝑃 + 𝛿3𝐽 − 𝜇𝑅.                                                                       }

 
 
 
 
 
 

 
 
 
 
 
 

             (8) 

 



C. E. Madubueze, I. O. Onwubuya, and I. Mzungwega 

 

 

These control functions are bounded, Lebesgue integrable functions that satisfy 0 ≤ 𝑢1 ≤
1, 0 ≤ 𝑢2 ≤ 0.95, 0 ≤ 𝑢3 ≤ 0.95 and 0 ≤ 𝑢4 ≤ 1 with assumption that the highest 
vaccination coverage will be 95%.

 The goal is to reduce the number of infected people (𝑃(𝑡),𝐼(𝑡),𝐽(𝑡)) and increase 

the number of susceptible people 𝑆(𝑡) while minimizing the cost of implementing 
controls. Therefore, the objective function is given as   

 

Γ(𝑢1 ,𝑢2 ,𝑢3 ,𝑢4 ) = ∫ (𝑏𝑃 + 𝑐𝐼 + 𝑑𝐽 + 
1

2
 ∑ 𝑚𝑖𝑢𝑖

2(𝑡)4𝑖=1  )
𝑡𝑓
0

𝑑𝑡          (9) 

and is subject to equation (8) with the initial conditions of the system (3).  

In equation (9), the constants, 𝑏,𝑐,𝑑,𝑚1,𝑚2,𝑚3,𝑚4, are positive weights to balance 
the size of the terms attached with them and 𝑡𝑓 is the final time to implement the controls, 

𝑢1(𝑡),𝑢2(𝑡),𝑢3(𝑡),𝑢4(𝑡). The terms, 𝑏𝑃,𝑐𝐼,𝑑𝐽 are the cost related to reducing the 
number of infected people (𝑃,𝐼,𝐽) such as cost of the mass distribution of nutrition 
(supplement) support, isolation and first dose and second dose vaccination at the due time. 

 

We seek optimal controls 𝑢1
∗,𝑢2

∗,𝑢3
∗,𝑢4

∗  such that   
 

Γ(𝑢1
∗,𝑢2

∗,𝑢3
∗,𝑢4

∗) = 𝑚𝑖𝑛{Γ(𝑢1 ,𝑢2 ,𝑢3 ,𝑢4 )|𝑢1 ,𝑢2 ,𝑢3 ,𝑢4 ∈ 𝑈}.       (10) 

With the application of Pontryagin’s maximum principle [14], the equations (8) and 

(9) are converted into a problem of minimizing pointwise a Hamiltonian, 𝐻 with respect 
to 𝑢1 ,𝑢2 ,𝑢3 ,𝑢4 . This is given by  

 

𝐻 = 𝑏𝑃 + 𝑐𝐼 + 𝑒𝐽 +
𝑚1𝑢1

2(𝑡)

2
+ 
𝑚2𝑢2

2(𝑡)

2
+ 
𝑚3𝑢3

2(𝑡)

2
+
𝑚4𝑢4

2(𝑡)

2

+ 1 (Λ − 
(1 − 𝑢1(𝑡))𝛽1(𝑃 + 𝑛1𝐼 + 𝑛2𝐽)𝑆

𝑁
− 1𝑢2(𝑡)𝑆 − 𝜇𝑆)

+ 2 ( 1𝑢2(𝑡)𝑆 − 
(1 − 𝑢1(𝑡))(1 − 𝑢2(𝑡))𝛽2(𝑃 + 𝑛1𝐼 + 𝑛2𝐽)𝑉1

𝑁
− 2𝑢3(𝑡)𝑉1

− 𝜇𝑉1) + 3( 2𝑢3(𝑡)𝑉1 − (𝜇 + 𝜎)𝑉2)

+ 4 (
(1 − 𝑢1(𝑡))𝛽1(𝑃 + 𝑛1𝐼 + 𝑛2𝐽)𝑆

𝑁

+
(1 − 𝑢1(𝑡))(1 − 𝑢2(𝑡))𝛽2(𝑃 + 𝑛1𝐼 + 𝑛2𝐽)𝑉1

𝑁
− (𝜇 + 𝑘)𝐸)

+ 5 (𝑘𝐸 − (𝛼 + 𝜇 + 𝜑1 + 𝑢4(𝑡) + 𝛿1)𝑃) + 6 (𝛼𝑃 − (𝜑2 + 𝛿2 +  𝜇 + 𝑑1)𝐼)

+ 7 (𝜑1𝑃 + 𝑢4(𝑡)𝑃 + 𝜑2𝐼 − (𝛿3 +  𝜇 + 𝑑2)𝐽)

+ 8( 𝜎𝑉2 + 𝛿2𝐼 + 𝛿1𝑃 + 𝛿3𝐽 − 𝜇𝑅) 



Controlling measles transmission dynamics with optimal control analysis 

 

 

 

with 1, 2, 3, 4, 5, 6, 7, 8  as respective adjoint variables for the state variables, 
𝑆,𝑉1,𝑉2,𝐸,𝑃,𝐼,𝐽,𝑅.  
 

The system of adjoint variables are derived by taking the partial derivative of  𝐻 with 
respect to each of their corresponding state variables.  This is given by   

 
𝑑𝜁1

𝑑𝑡
= −

𝜕𝐻

𝜕𝑆
= 𝑎 + 𝐴(1 −

𝑆

𝑁
)( 1 − 4) + 𝐵𝑉1( 4 − 2)                  

 + 1𝜇 + 1𝑢2(𝑡)( 1 − 2),                                                      
𝑑𝜁2

𝑑𝑡
= −

𝜕𝐻

𝜕𝑉1
= 𝐵𝑁(1 −

𝑆

𝑁
)( 2 − 4) +

𝐴𝑆

𝑁
( 4 − 1)+ 2𝜇           

+ 2𝑢3(𝑡)( 2 − 3),                                                                   
𝑑𝜁3

𝑑𝑡
= −

𝜕𝐻

𝜕𝑉2
= 𝐵𝑉1( 4 − 2)+

𝐴𝑆

𝑁
( 4 − 1) + 𝜎( 3 − 8) + 3𝜇,

𝑑𝜁4

𝑑𝑡
= −

𝜕𝐻

𝜕𝐸
= 𝐵𝑉1( 4 − 2) +

𝐴𝑆

𝑁
( 4 − 1)+ ( 4 − 5)𝑘 + 4𝜇,

𝑑𝜁5

𝑑𝑡
= −

𝜕𝐻

𝜕𝑃
= −𝑏 + 𝐵𝑉1( 4 − 2)+

𝐴𝑆

𝑁
( 4 − 1) + ( 5 − 8)𝛿1 

+
(1−𝑢1(𝑡))(1−𝑢2(𝑡))𝛽2𝑉1

𝑁
( 2 − 4) + ( 5 − 7)(𝜑1 + 𝑢4(𝑡))

 +
(1−𝑢1(𝑡))𝛽1𝑆

𝑁
( 1 − 4)+ ( 5 − 6)𝛼 + 5𝜇,                          

𝑑𝜁6

𝑑𝑡
= −

𝜕𝐻

𝜕𝐼
= −𝑐 + 𝐵𝑉1( 4 − 2)+

𝐴𝑆

𝑁
( 4 − 1)+ 6(𝜇 + 𝑑1)   

+
(1−𝑢1(𝑡))(1−𝑢2(𝑡))𝑛1𝛽2𝑉1

𝑁
( 2 − 4)+ ( 6 − 7)𝜑2             

+
(1−𝑢1(𝑡))𝑛1𝛽1𝑆

𝑁
( 1 − 4)+ ( 6 − 8)𝛿2,                               

 
𝑑𝜁7

𝑑𝑡
= −

𝜕𝐻

𝜕𝐽
= −𝑑 + 𝐵𝑉1( 4 − 2) +

𝐴𝑆

𝑁
( 4 − 1)+ 7(𝜇 + 𝑑2 ) 

+
(1−𝑢1(𝑡))(1−𝑢2(𝑡)) 𝑛2𝛽2𝑉1

𝑁
( 2 − 4)+ 8𝜇  + ( 7 − 8)𝛿3

+
(1−𝑢1(𝑡))𝑛2𝛽1𝑆

𝑁
( 1 − 4),                                                          

𝑑𝜁8

𝑑𝑡
= −

𝜕𝐻

𝜕𝑅
= 𝐵𝑉1( 4 − 2) +

𝐴𝑆

𝑁
( 4 − 1),                                        }

 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 

                (11) 

 

where 𝐴 =
(1−𝑢1(𝑡))𝛽1(𝑃+𝑛1𝐼+𝑛2𝐽)

𝑁
   and     𝐵 =

(1−𝑢1(𝑡))(1−𝑢2(𝑡))𝛽2(𝑃+𝑛1𝐼+𝑛2𝐽)

𝑁2
  with 

tranversality conditions 

 

1(𝑡𝑓) = 2(𝑡𝑓) = 3(𝑡𝑓) = 4(𝑡𝑓) = 5 (𝑡𝑓) = 6(𝑡𝑓) = 7(𝑡𝑓) = 8(𝑡𝑓) = 0.     (12) 

 

Furthermore, the respective controls, 𝑢1
∗,𝑢2

∗,𝑢3
∗,𝑢4

∗ are obtained by solving  
𝜕𝐻

𝜕𝑢1
= 0,

𝜕𝐻

𝜕𝑢2
= 0,

𝜕𝐻

𝜕𝑢3
= 0,

𝜕𝐻

𝜕𝑢4
= 0 and these are given by   

  



C. E. Madubueze, I. O. Onwubuya, and I. Mzungwega 

 

 

𝑢1
∗ =

(𝑃+𝑛1𝐼+𝑛2𝐽)(𝛽1𝑆(𝜁4− 𝜁1)+(1−𝑢2(𝑡))𝛽2𝑉1(𝜁4− 𝜁2))

𝑚1𝑁
,  

 

𝑢2
∗ =

(𝑃+𝑛1𝐼+𝑛2𝐽)(1−𝑢1(𝑡))𝛽2𝑉1(𝜁4− 𝜁2)

𝑚1𝑁
+
(𝜁1−𝜁2)𝜀1𝑆

𝑚2
 , 

 

𝑢3
∗ =    

(𝜁2−𝜁3)𝜀2𝑉1

𝑚3
,     𝑢4

∗ =
(𝜁5−𝜁7)𝑃

𝑚4
 . 

 

With the controls 𝑢1
∗,𝑢2

∗,𝑢3
∗,𝑢4

∗, the optimality condition is given by 

 

𝑢1
𝑜𝑝𝑡

= 𝑚𝑎𝑥{0 ,𝑚𝑖𝑛(1 ,𝑢1
∗)},      

𝑢2
𝑜𝑝𝑡

= 𝑚𝑎𝑥{0 ,𝑚𝑖𝑛(0.95 ,𝑢2
∗)},

𝑢3
𝑜𝑝𝑡

= 𝑚𝑎𝑥{0 ,𝑚𝑖𝑛(0.95 ,𝑢3
∗)},

𝑢4
𝑜𝑝𝑡

= 𝑚𝑎𝑥{0 ,𝑚𝑖𝑛(1 ,𝑢4
∗)}.     }

 
 

 
 

            (13) 

 

The optimality system consists of the state system (8), the adjoint system (11) with initial 

conditions of (3) and transversality condition (12) together with the characterization of 

the optimality condition (13).  

 

The restrictions of obtaining the uniqueness of the optimal control based on the length 

of time follow the approach in [2, 9, 14]. 

 

5. Numerical simulations 

In this section, the solutions of the optimality system are solved numerically using 

the forward and backward fourth-order Runge-Kutta method that is coded in MatLab 

software. The parameter values in Table 1 with the constants 𝑏 = 𝑐 = 𝑑 = 100,𝑚1 =
10000,𝑚2 = 2000, 𝑚3 = 2000,𝑚4 = 5000  and the initial conditions, 𝑆(0) =
200000, 𝑉1(0) = 2000, 𝑉2(0) = 1800, 𝐸(0) = 80,  𝑃(0) = 60, 𝐼(0) = 100,   𝐽(0) =
20, 𝑅(0) = 10000 are used for the numerical simulations purpose.   

 

5.1 Discussion 

Figure 4 shows the population dynamics of infected persons in the prodromal stage, 
𝑃(𝑡), infected persons in the rash stage, 𝐼(𝑡), isolated persons, 𝐽(𝑡), for with and without 
control and the control profile. According to Figure 4𝑎, with control measures, measles-
free population is achieved for 𝑃(𝑡) population faster, and thus reducing the number of 
persons moving to the infected people in the rash stage, while Figure 4b, the number of 

infected persons in the rash stage, 𝐼(𝑡), decrease to zero within 25 weeks with control 
measures in place. Also, Figure 4𝑐 reveals that with the implementation of the control 
measures, there is a sharp increase in the number of isolated persons before decreasing to 

zero by 25  weeks and achieves measles-free population.  



Controlling measles transmission dynamics with optimal control analysis 

 

 

 
Figure 4. The population dynamics of (a) infected persons in the prodromal stage, 𝑃(𝑡), 
(b) infected persons in the rash stage, 𝐼(𝑡),  (c) Isolated persons, 𝐽(𝑡), (d) Controls, 𝑢2,
𝑢3, and (e) Controls, 𝑢1, 𝑢4. Here, W/C means “With Control” while W/O/C means 
“Without Control”.  

 

0 50 100
0

500

1000

1500

2000

Time (Weeks)

P
(t

)
(a)

 

 

W/C

W/O/C

0 50 100
0

50

100

Time (Weeks)

I(
t)

(b)

 

 

W/C

W/O/C

0 50 100
0

20

40

60

Time (Weeks)

J
(t

)

(c)

 

 
W/C

W/O/C

0 50 100
0

2

4

6

8
x 10

-4

Time (Weeks)

C
o
n
tr

o
l 
p
ro

fi
le

(d)

 

 

u
2

u
3

0 10 20 30 40 50 60 70 80 90 100
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (Weeks)

C
o
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tr

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p
ro

fi
le

(e)

 

 

u
1

u
4



C. E. Madubueze, I. O. Onwubuya, and I. Mzungwega 

 

 

 
Figure 5. The population dynamics of (a) infected persons in the prodromal stage, 𝑃(𝑡), 
(b) infected persons in the rash stage, 𝐼(𝑡),  (c) Isolated persons, 𝐽(𝑡) when triple controls 
are implemented together. Here, the numbers 1,2,3, 𝑎𝑛𝑑 4, are subscripts of the control 
functions, 𝑢1,𝑢2,𝑢3,𝑢4  while W/O/C means “Without Control”.  
 

 

 
Figure 6. Control profile for implementation of triple optimal controls.  

0 50 100
0

500

1000

1500

2000
(a)

 

 
123

124

134

234

W/O/C

0 50 100
0

50

100

Time (Weeks)

I(
t)

(b)

 

 

123

124

134

234

W/O/C

0 50 100
0

20

40

60

Time (Weeks)

J
(t

)

(c)

 

 
123

124

134

234

W/O/C

0 10 20 30 40 50 60 70 80 90 100
0

0.5

1

1.5
x 10

-4

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C
o
n
tr

o
l 
p
ro

fi
le

u
4
=0

 

 

u
2

u
3

0 10 20 30 40 50 60 70 80 90 100
0

0.5

1

Time (Weeks)

u
1

u
4
=0

 

 

u
1



Controlling measles transmission dynamics with optimal control analysis 

 

 

The control profile for achieving the result in Figures 4a-4c are displayed in Figures 4d 

and 4e. They show that the upper bounds for 𝑢1,𝑢2,𝑢3 are 9.9,    6.1 × 10
−4 and  

2 × 10−5, respectively, where the control, 𝑢3 maintains a bound of 6.9 for 50 weeks 
before it gradually increases to 0.9 as at 90 weeks.  For without control measures, Figures 

4a – 4c show the endemicity of the measles infection in the population.  

In Figure 5, the implementation of triple control measures for the dynamics of the 

infected compartments (𝑃(𝑡),𝐼(𝑡),𝐽(𝑡)) are evaluated.  We observed in Figures 5a -5b 
that simultaneous implementation of any triple control measures reduce the number of 

infected persons (𝑃(𝑡),𝐼(𝑡)) in the population as they achieve a measles-free population 

within a short time while for isolated persons, 𝐽(𝑡), the combine implementation of 
𝑢1,𝑢2,𝑢3(123) yields a faster and better result in achieving a measles-free population 
compared with other combinations (see Figure 5c). With this, it implies that combined 

implementation of control measures, 𝑢1,𝑢2,𝑢3(123), reduces the number of infected 
persons in the prodromal stage, 𝑃(𝑡), rash stage, 𝐼(𝑡) and the isolated persons, (𝐽(𝑡)) 

compare with any other combination of control measures. The control profile for triple 

control measures are display in Figure 6. The control profile when 𝑢4 = 0 is the combined 
implementation of  𝑢1,𝑢2,𝑢3(123) that gives the best result in Figure 5, which indicates 
that mass distribution of nutritional (supplement) support, administration of first and 

second dose vaccine control measures have much effect on controlling measles in the 

population. To achieved this, 𝑢1 maintains an upper bound that declines after 85 weeks, 
whereas 𝑢2 maintains a bound of 1.2 × 10

−4 that decreases gradually till 85 weeks where 
it declines to the final time. For 𝑢3, it starts with a bound of 3.2 × 10

−5 that slightly 

increases to 4.0 × 10−5 at 70 weeks before declining to the final time.  
The numerical simulations imply that combined implementation of mass distribution 

of nutritional support, complete vaccination with the first and second dose of vaccine and 

isolation of infected persons in the prodromal stage will help eradicate the spread of 

measles in the population. This is in agreement with the sensitivity analysis result and the 

results in [3]. This indicates that implementation of control measures will help prevent 

the spread of measles infection in the population. However, if there are limited facilities 

to isolate the infected persons in the prodromal stage, the triple control measures, mass 

distribution of nutritional support and complete vaccination with first and second doses 

of vaccine will reduce the spread of measles infection as fewer people will be infected 

and thus help the health practitioners achieve the best strategy in the control of the spread 

of measles in the community. 

 

6. Conclusion 

In this paper, an autonomous system for the transmission dynamics of measles disease 

involving isolated persons and two doses of vaccination is formulated. The model disease-

free equilibrium and basic reproduction number (𝑅0) are computed. The sensitivity 
analysis of the basic reproduction number, which includes the pairwise comparison and 

scatterplots of important parameters of the basic reproduction number is carried out using 

Latin Hypercube Sampling (LHS) scheme. LHS scheme is also used to compute and 



C. E. Madubueze, I. O. Onwubuya, and I. Mzungwega 

 

 

compare the values of Partial Rank Correlation Coefficients (PRCCs) of the model 

parameters.  The result of the sensitivity analysis indicates that transmission rates, first 

and second-dose vaccination rates and isolation rate of the infected persons in the 

prodromal stage are significant parameters in eradicating the spread of measles infection.  

Furthermore, the optimal control model of the autonomous system is developed and 

analysed with four control measures, mass distribution of nutritional support, 

administration of first and second dose vaccination and isolation of infected persons in 

the prodromal stage. From the numerical simulations, we found out that the combined 

implementation of the four control measures achieves a measles-free population on time 

than without control measures. However, if there are limited facilities to isolate the 

infected persons in the prodromal stage, the triple control measures, mass distribution of 

nutritional support and complete vaccination will reduce the spread of measles infection. 

Thus, this offers the public health practitioners the best strategy that can control the spread 

of measles in the community.  

 

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