Numerical solution of nonlinear fractional SEIR epidemic model by using Haar wavelets

Highlights

• A numerical method is proposed using Haar wavelets on [0,c) for SEIR model.

• The method is simple and easy to implement.

• The used SEIR model is based on system of fractional differential equations.

• Theoretical error bounds are derived.

• The derived error bounds are validated through numerical computations.

Abstract

Throughout the world people have been suffering from the infectious diseases like Rubella, Herpes Simplex, Hepatitis B, Chagas, and HIV(AIDS) which have been causing loss of millions of lives and billions of rupees in cure. These diseases get transmitted via both horizontal and vertical transmission routes. One among the most recent mathematical models for modeling the above infectious diseases is fractional order nonlinear SEIR model [3] with non-constant population. In this paper, we have proposed a computationally faster and simpler numerical method based on Haar wavelets to solve the SEIR model. The error bounds have also been derived and validated.

Introduction

One can broadly classify the infectious diseases into two classes: diseases that confer immunity and diseases without immunity. Both can be modeled through different compartmental mathematical models. The compartmental mathematical models enable us for the study of transport between different compartments of a system. Like in medicine, compartment models have been used to study the transport of chemicals (nutrients, drugs, hormones, etc.) between different parts of the human body. In compartment based epidemic models, each individual person is considered as being in a particular state known as compartments. At no given time, one particular individual can be found in two different compartments. Hamer [1] was the first who postulated that the course of an epidemic depends on the rate of contact between susceptibles and infectious individuals. This became one of the most important concepts in Mathematical epidemiology. The simplest compartment model is based on the assumption that a person can be in only one of two states, either susceptible (S) or infectious (I) but not all diseases are accurately described by a model with only two states. In the diseases Paramyxovirus (measles) and Viral Parotitis (mumps), the infectious person may reach to a third state called recovered (R). An SIR model is appropriate where a lifelong immunity is obtained in the recovered state. One of the basic SIR model was proposed by Kermack–McKendrik [2]. It is important to model one more state called the Exposed (E) state whenever there exists a delay in between the time at which an individual is infected and the time at which that individual becomes infectious.

The main concern in epidemic is whether the infection will spread or not. In case it spreads, then how its transmission get affected with time and when it will start to decline etc. In earlier time, all the epidemic models were based on integer-order derivatives. But over the years, more complex models have been introduced which include fractional order models. Not only that the fractional order mathematical models are the generalization of integer-order mathematical models, but also it can benefit us to reduce the errors which arises on neglecting the parameters in modeling the real life problems [3], [4].

The fractional order nonlinear SEIR model with vertical transmission [3] is defined as

$$\begin{array}{c}{\mathcal{D}}_{*}^{\alpha }S(t)=\mathrm{\Delta }-\frac{p\mathrm{\Delta }E}{N}-\frac{q\mathrm{\Delta }I}{N}-\frac{\mathrm{r\; S}I}{N}-\mathrm{dS},\hfill \\ {\mathcal{D}}_{*}^{\alpha }E(t)=\frac{p\mathrm{\Delta }E}{N}+\frac{q\mathrm{\Delta }I}{N}+\frac{\mathrm{r\; S}I}{N}-\mathrm{dE}-\beta E,\hfill \\ {\mathcal{D}}_{*}^{\alpha }I(t)=\beta E-\mathrm{d\; I}-\theta I-\gamma I,\hfill \\ {\mathcal{D}}_{*}^{\alpha }R(t)=\gamma I-\mathrm{dR},\hfill \end{array}$$

with initial conditions

$$S(0)=S_0,E(0)=E_0,I(0)=I_0,R(0)=R_0,$$

where ${\mathcal{D}}_{*}^{\alpha }$ is Caputo fractional differential operator such that 0 < α ≤ 1, N(t) is size of nonconstant population such that N(t) = S(t) + E(t) + I(t) + R(t), S(t) is the proportion of susceptible, E(t) is the proportion of exposed, I(t) is the proportion of infectious, R(t) is the proportion of recovered, for a specific $t,(S,E,I,R)\in {ℝ}_{+}^4$, Δ is the number of recruits per unit of time, 0 ≤ d is the natural death rate, 0 < r is the transmission rate at which horizontal transmission of the disease is assumed to take place with direct contact between infectious and susceptible hosts, p is the probability of offspring from exposed class born in exposed class, q is the probability of offspring from infectious class born in exposed class, 0 < β is the rate at which the exposed individual become infectious, 0 < γ is the rate at which the exposed individual recover, 0 ≤ θ is infection related death rate.

The well-posedness of above model (1) is proved by Özalp and Demirci [3]. Once the infectious disease is Mathematically modeled by Eq. (1), the next challenge is to solve it efficiently. Özalp and Demirci [3] have solved Eq. (1) by using Adams-type predictor corrector method. In the recent times, wavelets based methods [5], [6], [7], [8] are developing rapidly for solving fractional differential equations.

The highlights of this paper can be summarized as follows:

• The system of nonlinear fractional differential equations based SEIR model has been considered.

• A numerical method is proposed based on Haar wavelets over an arbitrary interval [0, c).

• The theoretical error bounds are derived to show the convergence of the proposed numerical scheme.

• Numerical validation has been performed through test example.