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On the Mathematical Modeling of Measles Disease Dynamics with Encephalitis and Relapse Under the Atangana–Baleanu–Caputo Fractional Operator and Real Measles Data of Nigeria

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Abstract

In this paper, a population based mathematical model describing the transmission of measles disease with double vaccination dose, treatment and two groups of measles infected and measles induced encephalitis infected humans with relapse under the fractional Atangana–Baleanu–Caputo (ABC) operator is studied. The existence, uniqueness and positivity analysis of the fractional order model is established, while the model is validated by fitting data on measles prevalence in Nigeria made available by the Nigerian Center for Disease Control relative to the year 2020, using the nonlinear least square algorithm. Using the estimated and fitted parameters, the basic reproduction number \(R_{ms}\) is obtained and found to be \(R_{ms}\approx 1.34\), which reveal that despite the vaccination and treatment as controls, at least an individual is still being infected on the average, which describes the failure of vaccination effort and coverage in the Nigerian nation. Also, the model system equilibria, called the measles-free and measles-endemic equilibrium were obtained and the measles-free equilibrium is shown to be locally and globally asymptotically stable if \(R_{ms}\) is less than unity. A numerical scheme under the ABC operator, which is a mixture of the two-step Lagrange polynomial and the fundamental theorem of fractional calculus, is used to obtain the approximate solutions of the fractional order measles model, which proved to be convergent and efficient.

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Abbreviations

\(R_{ms}\) :

Basic reproduction number of measles disease.

ABC :

Atangana–Baleanu–Caputo

\(\tau \) :

Fractional order

\(^{ABC} D_0^\tau \) :

Atangana–Baleanu–Caputo fractional differential operator

\(S_m\) :

Human individuals susceptible to measles disease

\(V_e\) :

Humans who received vaccinations

\(I_m\) :

Humans infected with measles

\(I_e\) :

Humans infected with measles induced encephalitis

\(T_m\) :

Treated humans

\(R_m\) :

Humans who recovered from measles infection

\(\zeta _t(\cdot )\) :

One parameter Mittag-Leffler function

\(\mathfrak {I}^1\) :

Space function

g, f(t):

Function of t

\(\varsigma \) :

Vector of unknown function

X :

Space function

\(P_i(i=1-6)\) :

Functions satisfying the Lipschitz condition

\(\omega _i(i=1-6)\) :

Lipschitz constants

r :

Constant for triangular inequality

\(\wedge \) :

Set of feasible solutions of the model

\(E_m\) :

Measles-free equilibrium solution

\(E_m^*\) :

Measles-endemic equilibrium solution

F :

Non-negative matrix of appearance of new infections

V :

Non-negative matrix of movement of individuals within compartments

h :

Step length

\(\Gamma \) :

Gamma function

\(\mathfrak {R}(q_i)\) :

Real part of a matrix

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Authors and Affiliations

  1. Department of Mathematics, Ekiti State University, Ado-Ekiti, Ekiti State, Nigeria

    Oluwatayo Michael Ogunmiloro

  2. Department of Mathematics, Faculty of Physical Science, University of Ilorin, Ilorin, Nigeria

    Amos Sesan Idowu & Temitope Olu Ogunlade

  3. Department of Statistics, Faculty of Science, Ekiti State University, Ado-Ekiti, Ekiti State, Nigeria

    Roselyn Opeyemi Akindutire

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  1. Oluwatayo Michael Ogunmiloro
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Ogunmiloro, O.M., Idowu, A.S., Ogunlade, T.O. et al. On the Mathematical Modeling of Measles Disease Dynamics with Encephalitis and Relapse Under the Atangana–Baleanu–Caputo Fractional Operator and Real Measles Data of Nigeria. Int. J. Appl. Comput. Math 7, 185 (2021). https://doi.org/10.1007/s40819-021-01122-2

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