A mathematical analysis of the impact of maternally derived immunity and double-dose vaccination on the spread and control of measles

Samuel Opoku; Baba Seidu; Philip N. A. Akuka

doi:10.1515/cmb-2023-0106

Article Open Access

A mathematical analysis of the impact of maternally derived immunity and double-dose vaccination on the spread and control of measles

Samuel Opoku , Baba Seidu and Philip N. A. Akuka

Published/Copyright: November 16, 2023

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From the journal Computational and Mathematical Biophysics Volume 11 Issue 1

Abstract

Measles is a highly communicable viral infection that mostly affects children aged 5 years and below. Maternal antibodies in neonates help protect them from infectious diseases, including measles. However, maternal antibodies disappear a few months after birth, necessitating vaccination against measles. A mathematical model of measles, incorporating maternal antibodies and a double-dose vaccination, was proposed. Whenever ℛ 0 < 1 , the model is shown to be locally asymptotically stable. This means that the measles disease can be eliminated under such conditions in a finite time. It was established that ℛ 0 is highly sensitive to β (the transmission rate). A numerical simulation of the model using the Runge-Kutta fourth-order scheme was carried out, showing that varying the parameters to reduce ℛ 0 will help control the measles disease and ultimately lead to eradication. The measles-mumps-rubella (MMR) vaccine dosage should be adjusted for babies from recovered mothers, as maternal antibodies are usually high in such babies and can interfere with the effectiveness of the MMR vaccine.

Keywords: basic reproduction number; Runge-Kutta fourth-order method; stability; bifurcation; sensitivity

MSC 2010: 92D30; 37N25; 34D20; 92B05; 92D25

1 Introduction

Measles is a respiratory infection caused by a paramyxovirus belonging to the genus Morbillivirus [1]. It is a highly communicable viral infection marked by a maculopapular rash erupting all over the body and often accompanied by high fever [1]. According to the study by Okyere-Siabouh and Adetunde [2], paramyxovirus typically develops in the cells lining the back of the esophagus and lungs. Measles, which resides in the mucus of an infected individual’s nose and esophagus, is transmitted through coughing, sneezing, or intimate/direct contact with the secretions of an infected person. It can remain viable for up to 2 h on a contaminated surface or in the air [3]. Infected individuals can transmit measles to others from 4 days before to 4 days after the onset of the rash [4]. According to the study by Leuridan et al. [5], 90 % of vulnerable individuals who come into contact with an infected person are likely to contract the disease if they are not already immune to it. The measles-mumps-rubella (MMR) vaccine consists of two doses and can be used to prevent the infection from spreading within a population [6]. Despite the availability of the MMR vaccine, which has significantly reduced global incidence, measles remains a public health hazard. According to the World Health Organization, in 2018, over 140,000 people worldwide died from measles, with the majority being children aged 5 years and below. Measles does not have a specific therapy; patients require fluids, bed rest, and fever management. Those with additional health challenges may need healthcare tailored to their specific needs [7]. Measles is an immunizing disease; thus, once a person recovers, they will not contract measles again due to the permanent immunity gained from the infection. However, the measles virus weakens the immune system of an infected person, leading to subsequent health problems such as pneumonia, blindness, diarrhea, and encephalitis, among others [8]. Adewale et al. [9] formulated a mathematical model to investigate how the transmission of the measles virus is affected by the distance between infected and non-infected individuals. Their findings revealed that an increase in the distance between infected and susceptible persons leads to a reduction in the number of infected individuals. Furthermore, two studies emphasized the effectiveness of vaccination in both controlling and preventing the transmission of measles [10,11].

Garba et al. [12] conducted an analysis using a compartmental mathematical model to explore how vaccination and treatment affect measles dynamics. In addition, Liu et al. [13] proposed an susceptible-infected-quarantine-recovered epidemic model and performed a numerical investigation to examine the impact of treatment and quarantine on measles dynamics. The research showed that combining both quarantine and treatment is more effective in managing and preventing measles. The study also observed a decrease in measles transmission due to the treatment and quarantine of infected individuals. Kuddus et al.’s [14] findings suggest that increasing the vaccination dose rate leads to a reduction in measles transmission. Therefore, to establish robust herd immunity against the disease, it is advisable to promote large-scale vaccination initiatives that cover a significant portion of the population. This proactive approach will help prevent potential measles outbreaks in Bangladesh. Both vaccine efficacy and coverage play crucial roles in effectively controlling and eliminating the burden of measles in the affected community [15]. Arsal [16] modified the susceptible-exposed-infected-recovered model by introducing vaccinated compartments, two-dose vaccination, and vaccinated infants and immigrants. Individuals who have received two doses of vaccination acquire lifelong immunity, while those who have received only one dose may still be susceptible to measles. This study aims to develop a mathematical model for measles that includes temporal maternally derived immunity, double-dose vaccination, and an exposure component, which were missing in previous mathematical models of measles. The rest of the article is organized as follows: the proposed model is formulated in Section 2; analytical analysis are presented in Section 3; numerical simulations performed to support the analytical results are presented in Section 4; and the conclusion is presented in Section 5.

2 Model formulation

This section presents the derivation of the model of concern in this article. The total population at time t , denoted by N ( t ) , is subdivided into seven different compartments: susceptible individuals (individuals who are uninfected and are at risk of contracting the measles disease), S ( t ) ; maternally derived immune babies (new born babies from day 1 to 6 months), M ( t ) ; exposed individual (individuals who are exposed to the disease), E ( t ) ; individuals who have received the first dose of the MMR vaccine, V 1 ( t ) ; individuals who have received the second dose of the MMR vaccine, V 2 ( t ) ; individuals who are infected and infectious, I ( t ) ; and individuals who have developed permanent immunity to the measles disease, R ( t ) . The total population size N ( t ) is thus given by

N ( t ) = M ( t ) + S ( t ) + E ( t ) + V 1 ( t ) + V 2 ( t ) + I ( t ) + R ( t ) .

The maternally derived immune babies compartment is increased by birth at rate Λ , and neonates who lose maternal antibodies (immunity) few months after birth join the susceptible compartment at rate α . The natural death and measles-induced death rates are given as μ and δ , respectively. The susceptible class also decrease through infection at the rate β I S , where β is the transmission probability per contact between an infected individual and a susceptible person. The susceptible population is also reduced through first dose vaccination at the rate κ 1 S . The parameters ε 1 and ε 2 are assumed to be the efficacy of the first and second dose vaccination, respectively. Those who are exposed become infectious at a rate σ , and those with clinical symptoms recover at rate ω . Those who had the second dose vaccination at rate κ 2 and developed permanent immunity join the recovered compartment at rate ρ . The transition among the various classes considered in the model is illustrated in Figure 1 and mathematically presented in equation (1). The model parameters are defined in Table 1.

(1) d M d t = Λ − ( α + μ ) M , d S d t = α M − β S I N − ( κ 1 + μ ) S , d V 1 d t = κ 1 S − β ( 1 − ε 1 ) V 1 I N − ( κ 2 + μ ) V 1 , d V 2 d t = κ 2 V 1 − β ( 1 − ε 2 ) V 2 I N − ( ρ + μ ) V 2 , d E d t = β I [ S + ( 1 − ε 1 ) V 1 + ( 1 − ε 2 ) V 2 ] N − ( σ + μ ) E , d I d t = σ E − ( ω + μ + δ ) I , d R d t = ρ V 2 + ω I − μ R .

In the rest of the article, the following substitutions are used for convenience:

K 1 = ( α + μ ) , K 2 = ( κ 1 + μ ) , K 3 = ( σ + μ ) , K 4 = ( ω + μ + δ ) K 5 = ( κ 2 + μ ) , K 6 = ( ρ + μ ) , λ * = β I N .

Figure 1

Compartmental diagram of the model.

Table 1

Parameter description

Parameter	Description
Λ	Birth rate
β	Transmission rate
α	Rate of loss of maternal antibodies
σ	Rate at which an exposed become infective
ω	Recovery rate
ρ	Rate at which second vaccination receivers gets permanent immunity
κ 1	First vaccination rate
κ 2	Second vaccination rate
ε 1	Efficacy rate of first dose of MMR vaccine
ε 2	Efficacy rate of second dose of MMR vaccine
μ	Natural death rate
δ	Death rate cause by measles infection

3 Qualitative properties

3.1 Positivity of solutions and invariant region

Model (1) is an epidemiological model, and hence, it is necessary that the associated population sizes be positive. Model (1) should be considered in a feasible region where such property (nonnegative) is preserved. This is provided in Theorem 3.1.

Theorem 3.1

If positive conditions and initial conditions are provided for equation (1), then all its solutions remain positive for t > 0 .

Proof

From the first equation, if at some point t > 0 , we have M ( t ) = 0 , then we have d M d t = Λ > 0 . This shows that M ( t ) > 0 .

Similar arguments can be used to show that the other state variables have nonnegative solutions for all t > 0 , hence completing the proof.□

The next results concern the region within which analysis of the model are considered.

Theorem 3.2

The region Ω ⊂ R + 7 defined by Ω = ( M , S , V 1 , V 2 , E , I , R ) ∈ R + 7 ∣ 0 ≤ N ( t ) ≤ Λ μ is the positively invariant region for the model (1).

Proof

Adding all equations of model (1) yields d N d t = Λ − μ N − δ I ≤ Λ − μ N .

Using standard comparison theorems, we have

N ( t ) ≤ Λ μ + N 0 − Λ μ e − μ t .

It follows that Λ μ ≤ lim t → ∞ inf N ( t ) ≤ lim t → ∞ sup N ( t ) ≤ Λ μ so that lim t → ∞ sup N ( t ) ≤ Λ μ .

This concludes the proof of the theorem. As t → ∞ , we see that 0 ≤ N ( t ) ≤ Λ μ , and hence all solutions of the model remain in the region defined by□

(2) Ω = ( M , S , V 1 , V 2 , E , I , R ) ∈ R + 7 ∣ 0 ≤ N ( t ) ≤ Λ μ .

Thus, the region Ω is positively invariant, and the model can be sufficiently analyzed in Ω inside which the model is mathematically well-posed [17].

3.2 Equilibrium points of the model

The model has two equilibria: the disease-free equilibrium ℰ 0 and an endemic equilibrium ℰ * . The disease-free equilibrium is given by ℰ 0 = Λ K 1 , α Λ K 1 K 2 , κ 1 α Λ K 1 K 2 K 5 , κ 1 κ 2 α Λ K 1 K 2 K 5 K 6 , 0 , 0 , κ 1 κ 2 α Λ ρ μ K 1 K 2 K 5 K 6 .

Using the technique of Seidu et al. [18], the basic reproduction number of model (1) is obtained as follows:

(3) ℛ 0 = β α μ σ [ K 5 K 6 + κ 1 ( 1 − ε 1 ) K 6 + ( 1 − ε 2 ) κ 1 κ 2 ] K 1 K 2 K 3 K 4 K 5 K 6 .

The model can be shown to have an endemic equilibrium ℰ * = ( M * , S * , V 1 * , V 2 * , E * , I * , R * ) , where:

(4) M * = Λ K 1 , S * = α Λ K 1 ( λ * + K 2 ) ; V 1 * = κ 1 α Λ K 1 ( λ * + K 2 ) [ λ * ( 1 − ε 1 ) + K 5 ] , V 2 * = κ 2 κ 1 α Λ K 1 ( λ * + K 2 ) [ λ * ( 1 − ε 1 ) + K 5 ] [ λ * ( 1 − ε 2 ) + K 6 ] , E * = α Λ λ * K 1 K 3 ( λ * + K 2 ) 1 + κ 1 ( 1 − ε 1 ) [ λ * ( 1 − ε 2 ) + K 6 ] + κ 2 ( 1 − ε 2 ) [ λ * ( 1 − ε 1 ) + K 5 ] [ λ * ( 1 − ε 2 ) + K 6 ] , I * = α σ Λ λ * K 1 K 3 K 4 ( λ * + K 2 ) 1 + κ 1 ( 1 − ε 1 ) [ λ * ( 1 − ε 2 ) + K 6 ] + κ 2 ( 1 − ε 2 ) [ λ * ( 1 − ε 1 ) + K 5 ] [ λ * ( 1 − ε 2 ) + K 6 ] , R * = ρ κ 2 κ 1 α Λ K 1 μ ( λ * + K 2 ) [ λ * ( 1 − ε 1 ) + K 5 ] [ λ * ( 1 − ε 2 ) + K 6 ] + ω σ K 4 μ E *

and λ * satisfies the following polynomial equation:

(5) ( A 3 ( λ * ) 3 + A 2 ( λ * ) 2 + A 1 λ * + A 0 ) λ * = 0 ,

where

A 3 = ( α σ δ − K 1 K 3 K 4 ) ( 1 − ε 1 ) ( 1 − ε 2 ) , A 2 = ( β α 2 σ 2 μ δ κ 1 − K 1 K 2 K 3 K 4 ) ( 1 − ε 1 ) ( 1 − ε 2 ) + ( α σ δ − K 1 K 3 K 4 ) ( K 5 ( 1 − ε 2 ) + K 6 ( 1 − ε 1 ) ) , A 1 = ( β α σ μ − K 1 K 2 K 3 K 4 ) ( K 5 ( 1 − ε 2 ) + K 6 ( 1 − ε 1 ) ) + β α σ μ ( κ 1 ( 1 − ε 1 ) ( 1 − ε 2 ) ) + α σ δ κ 1 ( K 6 ( 1 − ε 1 ) + κ 2 ( 1 − ε 2 ) ) + α σ δ K 5 K 6 − K 1 K 3 K 4 K 5 K 6 , A 0 = β α σ μ κ 1 ( K 6 ( 1 − ε 1 ) + κ 2 ( 1 − ε 2 ) ) + β α σ μ K 5 K 6 − K 1 K 2 K 5 K 6 K 3 K 4 .

Equation (5) has four roots, namely λ * = 0, which corresponds to the disease-free equilibrium, and the positive real zeros of the following equation:

(6) A 3 ( λ * ) 3 + A 2 ( λ * ) 2 + A 1 λ * + A 0 = 0 .

By Descartes’ rule of signs, if A i > 0 ∀ i = 0 , 1 , 2 , 3 , then equation (6) has no realistic endemic equilibrium. On the other hand, if A i < 0 ∀ i = 0 , 1 , 2 , 3 , then equation (6) has three realistic endemic equilibria.

3.3 Local stability of equilibria

In this section, we investigate the local asymptotic stability of the equilibria.

Theorem 3.3

The measles-free fixed point ℰ 0 of the model (1) is locally asymptotically stable whenever ℛ 0 ≤ 1 and unstable otherwise.

Proof

The Jacobian matrix of equation (1) at the measles-free equilibrium point is given by ( ℰ 0 ) as follows:

J 1 ( ℰ 0 ) = − K 1 0 0 0 0 0 0 α − K 2 0 0 0 − β S 0 N 0 0 0 κ 1 − K 5 0 0 − β ( 1 − ε 1 ) V 1 0 N 0 0 0 0 κ 2 − K 6 0 − β ( 1 − ε 2 ) V 2 0 N 0 0 0 0 0 0 − K 3 β S 0 N 0 + β ( 1 − ε 1 ) V 1 0 N 0 + β ( 1 − ε 2 ) V 2 0 N 0 0 0 0 0 0 σ − K 4 0 0 0 0 ρ 0 ω − μ

Clearly, λ 1 = − K 1 , λ 2 = − K 2 , λ 3 = − K 5 , λ a 4 = − K 6 , and λ 5 = − μ are eigenvalues of J( ℰ 0 ), which are all negative when ℛ 0 < 1 , and the remaining eigenvalues are those of the following submatrix:

J 2 ( ℰ 0 ) − I λ = − K 3 − λ β S N + β ( 1 − ε 1 ) V 1 N + β ( 1 − ε 2 ) V 2 N σ − K 4 − λ ,

whose characteristic polynomial is given by

(7) λ 2 + λ ( ( σ + μ ) + ( ω + μ + δ ) ) + ( 1 − ℛ 0 ) ( ( σ + μ ) ( ω + μ + δ ) ) = 0 .

Since all the coefficients of equation (7) are positive, the Routh-Hurwitz criterion shows that all zeros of equation (7) have negative real parts.

Therefore, the disease-free equilibrium point ( ℰ 0 ) is locally asymptotically stable if ℛ 0 ≤ 1 and unstable if ℛ 0 > 1 .□

At the endemic equilibrium point, the characteristic polynomial of the Jacobian matrix of the model is given as follows:

(8) λ 6 ζ 6 + λ 5 ζ 5 + λ 4 ζ 4 + λ 3 ζ 3 + λ 2 ζ 2 + λ ζ 1 + ζ 0 = 0 ,

where

ζ 0 = ( ( a 5 a 6 + a 4 a 8 ) a 9 + a 4 a 7 a 10 ) a 1 a 12 a 13 + ( a 7 a 10 + a 8 a 9 ) a 1 a 3 a 12 a 13 ( a 10 a 12 a 14 + a 9 a 11 a 16 − a 9 a 12 a 15 ) a 1 a 3 a 6 ζ 1 = ( ( a 11 + a 16 ) a 9 + a 11 a 16 ) a 1 a 3 a 6 − ( ( ( a 6 + a 9 ) a 3 + a 6 a 9 ) a 1 + a 3 a 6 a 9 ) a 12 a 15 + ( a 8 a 13 + a 10 a 14 ) a 1 a 3 a 12 + ( a 5 a 6 + a 4 a 8 ) a 1 a 12 a 13 + ( a 1 + a 3 ) a 7 a 10 a 12 a 13 + ( a 1 a 5 + a 5 a 6 + a 4 a 8 ) a 9 a 12 a 13 + ( a 1 + a 6 ) a 3 a 9 a 11 a 16 + ( a 6 a 11 a 16 + a 8 a 12 a 13 ) a 1 a 9 + ( a 1 + a 3 ) a 6 a 10 a 12 a 14 + ( a 3 a 8 a 9 a 13 + a 4 a 7 a 10 a 13 ) . ζ 2 = ( a 9 + a 11 + a 16 ) a 1 a 3 a 6 + ( a 3 + a 6 + a 16 ) a 1 a 9 a 11 + ( a 9 + a 11 ) a 1 a 6 a 16 + ( a 1 + a 6 + a 11 ) a 3 a 9 a 16 + ( a 9 + a 16 ) a 3 a 6 a 11 − ( a 6 + a 3 + a 9 ) a 1 a 12 a 15 + ( a 1 + a 6 + a 9 ) a 5 a 12 a 13 + ( a 1 + a 3 + a 9 ) a 8 a 12 a 13 + ( a 1 + a 3 + a 6 ) a 10 a 12 a 14 − ( a 3 a 6 + a 3 a 9 + a 6 a 9 ) a 12 a 15 + ( a 1 a 3 a 11 + a 6 a 9 a 11 ) a 16 + ( a 4 a 8 a 13 + a 7 a 10 a 13 ) a 12 . ζ 3 = ( a 9 + a 11 + a 1 + a 3 ) a 6 a 16 + ( a 6 + a 9 + a 11 + a 16 ) a 1 a 3 + ( a 6 + a 11 + a 16 ) a 3 a 9 + ( a 16 + a 1 + a 6 ) a 9 a 11 + ( a 9 + a 11 ) a 1 a 6 + ( a 6 + a 16 ) a 3 a 11 + ( a 9 + a 11 ) a 1 a 16 − ( a 6 + a 9 + a 1 + a 3 ) a 12 a 15 + ( a 8 a 13 + a 10 a 14 + a 5 a 13 ) a 12 ζ 4 = ( a 3 + a 6 + a 9 + a 11 + a 16 ) a 1 + ( a 6 + a 9 + a 11 + a 16 ) a 3 + ( a 9 + a 11 + a 16 ) a 6 + ( a 9 + a 11 ) a 16 + a 9 a 11 − a 12 a 15 ζ 5 = a 1 + a 3 + a 6 + a 9 + a 11 + a 16 ζ 6 = 1 .

Now, since ζ 6 and ζ 5 are positive, then applying Descartes’ rule of sign, equation (8) has all roots with real parts situated in the left half of the real line provided that ζ i > 0 ∀ i = 0 , 1 , … , 5 . Hence, endemic equilibrium will be stable; otherwise, it would be unstable.

3.4 Sensitivity analysis of ℛ 0

Sensitivity analysis is used to assess the sensitivity of ℛ 0 to changes in the model’s parameter values. The level of infection is determined by the magnitude of ℛ 0 . Therefore, if ℛ 0 is highly sensitive to a specific parameter, then that parameter may be helpful in establishing policies or intervention strategies that lower the occurrence of epidemics. The sensitivity analysis is given by ϒ p ℛ 0 = ∂ ℛ 0 ∂ p × p ℛ 0 , where p is the parameter of interest.

For β (Transmission rate), we have

ϒ β ℛ 0 = ∂ ℛ 0 ∂ β × β ℛ 0 = 1 > 0 .

For α (Rate of loss of maternal antibodies), we have

ϒ α ℛ 0 = ∂ ℛ 0 ∂ α × α ℛ 0 = μ α + μ > 0 .

For σ (Rate at which and exposed become infected), we have

ϒ σ ℛ 0 = ∂ ℛ 0 ∂ σ × σ ℛ 0 = μ σ + μ > 0 .

For ω (Recovery rate), we have

ϒ ω ℛ 0 = ∂ ℛ 0 ∂ ω × ω ℛ 0 = − ω ω + μ + δ < 0 .

For ρ (Rate at which second vaccination receivers get permanent immunity), we have

ϒ ρ ℛ 0 = ∂ ℛ 0 ∂ ρ × ρ ℛ 0 = − ρ ( 1 − ε 2 ) κ 1 κ 2 ( ρ + μ ) { ( κ 2 + μ ) ( ρ + μ ) + κ 1 ( 1 − ε 1 ) ( ρ + μ ) + ( 1 − ε 2 ) κ 1 κ 2 } < 0 .

For ε 1 (Efficacy rate of first dose of MMR vaccine), we have

ϒ ε 1 ℛ 0 = ∂ ℛ 0 ∂ ε 1 × ε 1 ℛ 0 = − ε 1 κ 1 ( ρ + μ ) ( κ 2 + μ ) ( ρ + μ ) + κ 1 ( 1 − ε 1 ) ( ρ + μ ) + ( 1 − ε 2 ) κ 1 κ 2 < 0 .

For ε 2 (Efficacy rate of second dose of MMR vaccine), we have

ϒ ε 2 ℛ 0 = ∂ ℛ 0 ∂ ε 2 × ε 2 ℛ 0 = − ε 2 κ 1 κ 2 ( κ 2 + μ ) ( ρ + μ ) + κ 1 ( 1 − ε 1 ) ( ρ + μ ) + ( 1 − ε 2 ) κ 1 κ 2 < 0 .

From Table 2, results show that parameters β , σ , and α have positive correlation with ℛ 0 , which indicates that increasing (decreasing) these parameters will increase the prevalence of measles disease. On the other hand, parameters ω , ε 1 , ε 2 , ρ , κ 1 , κ 2 , and δ have negative correlation with ℛ 0 , which indicates that increasing (decreasing) these parameters will decrease the prevalence of measles disease. Therefore, to completely eradicate measles infection from the population, more effort should be given on treatment of latent infected individuals and infectious individuals. Second, much priority should be given on first dose vaccination, both the efficacy rate and the proportion of unvaccinated individuals.

Table 2

Sensitivity signs of ℛ 0 to the parameters in equation (3)

Parameter	Numerical sensitivity index
β	+1.0000
σ	+0.0584
ω	‒ 0.9749
α	+0.0382
ε 1	‒ 0.8965
ε 2	‒ 0.7126
ρ	‒ 0.0217
κ 1	‒ 0.8921
κ 2	‒ 0.0657
δ	‒ 0.0098

3.5 Bifurcation analysis

Bifurcation points are expected at ℛ 0 = 1 at which we can choose a bifurcation parameter β * as follows:

β * = ( σ + μ ) ( ω + μ + δ ) ( α + μ ) ( κ 1 + μ ) ( κ 2 + μ ) ( ρ + μ ) α μ σ [ ( κ 2 + μ ) ( ρ + μ ) + κ 1 ( 1 − ε 1 ) ( ρ + μ ) + ( 1 − ε 2 ) κ 1 κ 2 ] .

The center manifold theorem of Castillo-Chavez and Song [27] is often used to study the bifurcation in the model if it can be shown that the Jacobian at the disease-free equilibrium has a simple eigenvalue.

To conduct bifurcation, the model variables are written as x 1 = M , x 2 = S , x 3 = V 1 , x 4 = V 2 , x 5 = E , x 6 = I , x 7 = R so that the model equations are written as follows:

(9) d x 1 d t = Λ − ( α + μ ) x 1 , d x 2 d t = α x 1 − β x 2 x 6 x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 − ( κ 1 + μ ) x 2 , d x 3 d t = κ 1 x 2 − β ( 1 − ε 1 ) x 3 x 6 x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 − ( κ 2 + μ ) x 3 , d x 4 d t = κ 2 x 3 − β ( 1 − ε 2 ) x 4 x 6 x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 − ( ρ + μ ) x 4 , d x 5 d t = β x 6 { x 2 + ( 1 − ε 1 ) x 3 + ( 1 − ε 2 ) x 4 } x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 − ( σ + μ ) x 5 , d x 6 d t = σ x 5 − ( ω + μ + δ ) x 6 , d x 7 d t = ρ x 4 + ω x 6 − μ x 7 .

In order to apply the theorem, the left and right eigenvectors ( v and w , respectively) associated with the simple eigenvalue are computed. If w = ( w 1 , w 2 , w 3 , w 4 , w 5 , w 6 , w 7 ) T is the right eigenvalue of J ( ℰ M F E ) associated with the zero eigenvalue, then J ( ℰ M F E ) w = 0 , which gives

w 1 = 0 w 2 = − β * S 0 N 0 ( κ 1 + μ ) w 6 w 3 = − β * κ 1 S 0 N 0 ( κ 2 + μ ) ( κ 1 + μ ) + β * ( 1 − ε 1 ) V 1 0 N 0 ( κ 2 + μ ) w 6 w 4 = − 1 ρ + μ β * ( 1 − ε 2 ) V 2 0 N 0 + β * κ 1 κ 2 S 0 N 0 ( κ 2 + μ ) ( κ 1 + μ ) + β * κ 2 ( 1 − ε 1 ) V 1 0 N 0 ( κ 2 + μ ) w 6 w 5 = ( ω + μ + δ ) σ w 6 w 7 = − ρ μ ( ρ + μ ) β * ( 1 − ε 2 ) V 2 0 N 0 + β * κ 1 κ 2 S 0 N 0 ( κ 2 + μ ) ( κ 1 + μ ) + β * κ 2 ( 1 − ε 1 ) V 1 0 N 0 ( κ 2 + μ ) w 6 + ω μ w 6 .

Similarly, if v = ( v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , v 7 ) is the left eigenvalue of J ( ℰ M F E ) associated with the zero eigenvalue, then v J ( ℰ M F E ) = 0 , which gives

v 1 = v 2 = v 3 = v 4 = v 7 = 0

v 5 = σ σ + μ v 6 .

The left and right eigenvectors have been expressed in terms of v 6 and w 6 .

Since w ⋅ v = 1 , this leads to the condition that w 6 v 6 = 1 .

Bifurcation coefficients are obtained as follows:

a = 2 β ( M 0 + ε 1 V 1 0 + ε 2 V 2 0 + R 0 ) ( M 0 + S 0 + V 1 0 + V 2 0 + R 0 ) 2 w 6 w 2 v 5 − 2 β ( ε 1 M 0 − M 0 + ε 1 S 0 + ε 1 V 2 0 − ε 2 V 2 0 + ε 1 R 0 − R 0 ) ( M 0 + S 0 + V 1 0 + V 2 0 + R 0 ) 2 w 6 w 3 v 5 − 2 β ( ε 2 M 0 − M 0 + ε 2 S 0 − ε 1 V 1 0 + ε 2 V 1 0 + ε 2 R 0 − R 0 ) ( M 0 + S 0 + V 1 0 + V 2 0 + R 0 ) 2 w 6 w 4 v 5 − 2 β ( S 0 + ( 1 − ε 1 ) V 1 0 + ( 1 − ε 2 ) V 2 0 ) ( M 0 + S 0 + V 1 0 + V 2 0 + R 0 ) 2 w 6 w 5 v 5 − 2 β ( S 0 + ( 1 − ε 1 ) V 1 0 + ( 1 − ε 2 ) V 2 0 ) ( M 0 + S 0 + V 1 0 + V 2 0 + R 0 ) 2 w 6 w 6 v 5 − 2 β ( S 0 + ( 1 − ε 1 ) V 1 0 + ( 1 − ε 2 ) V 2 0 ) ( M 0 + S 0 + V 1 0 + V 2 0 + R 0 ) 2 w 6 w 7 v 5 .

It is obvious that our expressions for w 2 w 3 , and w 4 are negative and w 5 and v 5 are positive. In addition, 0 < ε 1 , ε 2 < 1 , and ε 1 < ε 2 , and therefore, ( ε 1 M 0 − M 0 ) , ( ε 1 V 2 0 − ε 2 V 2 0 ) , and ( ε 1 R 0 − R 0 ) are negative. Hence, a < 0 (because in the second and third terms of a , ( ε 1 R 0 − R 0 ) exceed ε 1 S 0 in terms of magnitude).

b = σ w 6 v 6 σ + μ S 0 + V 1 0 ( 1 − ε 1 ) + V 2 0 ( 1 − ε 2 ) M 0 + S 0 + V 1 0 + V 2 0 + R 0 .

It is clear that b > 0 .

Since a < 0 and b > 0 is obtained with ℛ 0 − 1 < 0 , then the origin (0) is stable, but it becomes unstable when ℛ 0 − 1 > 0 .

Also, since a < 0 and b > 0 , then according to Castillo-Chavez and Song [27], the model undergoes forward bifurcation at ℛ 0 = 1 .

4 Numerical simulation and discussion

In this section, numerical simulations are carried out to support the analytical results and to assess the impact of some model parameters. Numerical simulation for the model of equation (1) is done using Runge-Kutta fourth-order method. The parameter values are given in Table 3, and the initial conditions are M ( 0 ) = 0.1 , S ( 0 ) = 0.98 , V 1 ( 0 ) = 0.2 , V 2 ( 0 ) = 0.1 , E ( 0 ) = 0 , I ( 0 ) = 0.2 , and R ( 0 ) = 0.1 . We study the impact of β , κ 1 ε 1 , α , and κ 2 on some compartments of the model by varying the value of one control ( β , κ 1 , ε 1 , α κ 2 ) at a time while all other parameters remain constant.

Table 3

Parameter values for the model

Parameter	Value per year	Source
Λ	0.02967	[19]
β	7.45 × 1 0 ‒ 7	[14]
α	0.39	[5]
σ	0.25	[20]
ω	0.6	[21]
ρ	0.92	[22]
κ 1	0.94	[23]
κ 2	0.93	[23]
ε 1	0.93	[24,25]
ε 2	0.97	[24,25]
μ	0.01428	[26]
δ	0.125	[21]

From Figure 2, an increase in β leads to an increase in the exposed population, and vice versa, whereas an increase in κ 1 or ε 1 leads to a decrease in the susceptible population and eventually a decrease in the exposed population.

$Figure 2 Time series plot depicting the impact of β κ 1 \beta \hspace{0.33em}{\kappa }_{1} and ε 1 {\varepsilon }_{1} on exposed compartment.$

Figure 2

Time series plot depicting the impact of β κ 1 and ε 1 on exposed compartment.

In Figure 3, it is observed that as the rate of loss of maternal antibodies ( α ) increases, maternally derived immune baby compartments lose population to the susceptible compartment, and this should be accompanied by an increase in the first vaccination compartment; otherwise, it will increase the risk of an outbreak.

$Figure 3 Time series plot showing the impact of α \alpha on babies with maternal antibodies, susceptible and first vaccination compartment.$

Figure 3

Time series plot showing the impact of α on babies with maternal antibodies, susceptible and first vaccination compartment.

Available data show that most people gain permanent immunity through the second dose of the MMR vaccine. Figure 4 confirms this fact, as it is used to demonstrate that the number of individuals with permanent immunity increases with an increase in the rate of the second dose vaccination, κ 2 .

$Figure 4 Effect of κ 2 {\kappa }_{2} on removed with immunity.$

Figure 4

Effect of κ 2 on removed with immunity.

5 Conclusions

The M S V 1 V 2 E I R model formulated revealed that the first dose of MMR vaccine plays a major role in controlling and eliminating measles. This was confirmed with a sensitivity index of − 0.8921 . While about 99% of people infected with measles recover and gain permanent immunity [28], the number of those who achieve permanent immunity through vaccination far exceeds those who gain it through recovery [28,29]. It has also been established that maternal antibodies reduce the susceptible population by protecting neonates while their immune systems are still developing. Sensitivity analysis and numerical simulations confirm that the implementation of a double-dose vaccination will help minimize the spread of measles. Therefore, every possible avenue should be explored to attain herd immunity and ultimately succeed in eliminating measles disease. The disease-free equilibrium and conditions necessary for stability in relation to ℛ 0 have been established. Numerical simulations of the model reveal that an increase in vaccination leads to a decrease in the spread of measles. Based on the results of the study, it is recommended that:

Vaccination centers should not be situated only in urban areas but should also be made available in rural areas.
Because maternal antibodies can interfere with the MMR vaccine and are normally high in babies from recovered mothers, the time frame for administering the MMR vaccine should be adjusted for such babies.
The first dose of the MMR vaccine should be a requirement before a child starts schooling.
Whenever there is an outbreak, protective measures such as wearing a nose mask and regularly washing hands should be practiced to prevent the spread of measles disease throughout the population until the MMR vaccine is administered to the population.

Acknowledgments

This paper was prepared from the MPhil thesis of the first Author at the C. K. Tedam University of Technology and Applied Sciences, Ghana.

Funding information: This research did not receive any funding.
Author contributions: Baba Seidu: conceptualization, methodology, formal analysis, writing-original draft and writing-review & editing. Samuel Opoku: conceptualization, methodology, formal analysis, writing-original draft, and writing-review & editing. Philip N. A. Akuka: methodology, formal analysis, writing-original draft, and writing-review & editing.
Conflict of interest: The authors declare that they have no conflicts of interest to report regarding the present study.
Ethical approval: This research did not require ethical approval.
Data availability statement: The parameter values (data) used to support the findings of this study is presented in Table 3.

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Received: 2022-11-08

Revised: 2023-09-07

Accepted: 2023-10-25

Published Online: 2023-11-16

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/cmb-2023-0106

Keywords for this article

basic reproduction number; Runge-Kutta fourth-order method; stability; bifurcation; sensitivity

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BY 4.0