Numerical Methods for Differential and Integral EquationsView this Special Issue
Numerical, Approximate Solutions, and Optimal Control on the Deathly Lassa Hemorrhagic Fever Disease in Pregnant Women
This paper is devoted to the model of Lassa hemorrhagic fever (LHF) disease in pregnant women. This disease is a biocidal fever and epidemic. LHF disease in pregnant women has negative impacts that were initially appeared in Africa. In the present study, we find an approximate solution to the fractional-order model that describes the fatal LHF disease. Laplace transforms coupled with the Adomian decomposition method (ADM) are applied. In addition, the fractional-order LHF model is numerically simulated in terms of a varied fractional order. Furthermore, a fractional order optimal control for the LHF model is studied.
Infectious diseases remain a major threat to human health and welfare throughout history. Their spread is affected by several factors such as mode of transmission, infectious agent, infectious periods, incubation period, susceptibility, and resistance . Some devastating infectious diseases like Lassa fever are endemic in many parts of the world and continue to emerge.
Lassa fever, also known as Lassa hemorrhagic fever (LHF), is an animal-borne, or zoonotic, acute viral hemorrhagic illness caused by Lassa virus, a member of the family Arenaviridae. The reservoir is the rats Mastomys natalensis, M. erythroleucus, and Hylomyscus pamfi. Food contaminated with rodent urine, saliva, or feces is the main cause of human cases. Human-to-human transmission can occur via exposure to the blood, feces, urine, saliva, or vomitus of infected patients [2, 3]. The incubation period is ca. 2-21 days. LHF is basically endemic in some countries of West Africa, and outbreaks have occurred in Benin, Ghana, Guinea, Liberia, Nigeria, Sierra Leone, and Togo. Cases have been imported to Germany, United Kingdom, United States, and Sweden. This disease with high mortality rates (e.g., 80% among pregnant women) is likely affecting between 100,000 and 300,000 people every year, and it kills around 5,000 people, almost all in West Africa alone [4, 5].
Fractional Calculus (FC) is a fruitful field of mathematical research with numerous applications in engineering , nanotechnology , optics , human diseases , and chaos soliton theory . Application of FC is also adopted in biology, heat transfer, system identification, genetic algorithms, traffic systems, telecommunications, physics as well as finance, and economics [11–20].
Herein, we are interested in the paradigm showing the nature of LHF, since it is lethal and transmissible [1, 2, 21, 22]. Using mathematical modeling to probe this fatal illness may provide insight, answers, and guidance useful for controlling its spread, and the originating applications. The given model is studied through Caputo fractional operator. In this model, the Caputo fractional operator is used since it permits both initial and boundary conditions to involve in modeling problems. The Caputo fractional operator can be applied on sufficiently differentiable functions only. One of the advantages of the fractional derivative of Caputo compared with the fractional derivative 3 of Riemann-Liouville is that the fractional derivative of Caputo for the constant is equal to zero. We develop an approximate solution to the fractional-order model describing the LHF affecting pregnant women using the Laplace transform coupled with the ADM. Numerical simulations are also presented. A fractional optimal control for the SIRD model is ultimately offered. For a good survey on the approximating of the differential equations, see [23–26]. Some recent research fractional optimal controls can be found in [27–29]. Atangana  studied the LHF using beta differential operator, where the model was proposed as follows in (1). Table 1 shows the meaning of the parameters in the model (1). where .
Also, the model has been studied by Goyal et al. . The -homotopy analysis transform method (q-HATM) has been applied for solving the LHF model . The LHF model via Atangana-Baleanu fractional derivative has been studied in . The SIR model has been studied through Caputo-Fabrizio fractional operator in .
In this paper, we reconsider the system (1) by the following fractional sense of the system (1) where is the Caputo fractional derivative of order
We organize the rest of our paper as follows. In Section 2, we provide some basic mathematical concepts that will be necessary for our work. Also, the description of the proposed model is presented. We then follow by presenting the series solution for the proposed model in Section 3. We provide the approximate solution construction in Section 4. The numerical solution of the proposed model, having arbitrary order, is given in Section 5. The optimal control case is studied in Section 6. In Section 7, a concise conclusion of our paper results is presented.
For studying the proposed model (2), we present in this section the basic definitions concerned with fractional calculus. For a good survey on the basic definitions and several applications on fractional calculus, see [6, 7, 33].
2.1. Some Important Definitions
Definition 1. The Riemann–Liouville integral is represented by where is the known gamma function and is the order of this fractional integral.
Definition 2. The Caputo-type fractional derivative of order is defined as
Definition 3. The Laplace transform of Caputo fractional derivative of order  is given by
2.2. Mathematical Formulation and Description for the Proposed Model
The meaning of the parameters is shown in Table 1. Now, the description and the mathematical formulation of the model can be shown as follows.
With time we can express the rate of change in susceptible population as follows
With time we can express the rate of change in the population of the infected pregnant women as follows
Here, is the number of pregnant women excluded from the group of susceptible.
With time the rate of change in recovery population is written as
Finally, the rate of change of dying population with time is given as
Hence, the mathematical model for the LHF is given by model (2), with initial conditions
Also Here, and are sufficiently differentiable functions. Model (2) has not yet been studied by using the Laplace transform coupled with the ADM. Also, we introduce numerical simulations and fractional optimal controls for the SIRD model.
The interaction between different compartments of LHF model (2) can be viewed via the following signal flow graph shown in Figure 1.
2.3. Calculating the Basic Reproduction Number
Basic reproduction number is the expected production of newly infected females, in a totally susceptible women, by meeting a typical infective woman. The local stability and instability of the disease-free equilibrium (DFE) depend on the value of . Also, it defines the local stability threshold for the DFE. Moreover, it is a very important tool for controlling the disease, and it is an essential epidemiological criteria of disease. When the DFE is locally asymptotically stable; a small number of infections into the population may cause it to evolve into an endemic prevalence. From another point of view, when the DFE is locally unstable; a sufficiently small number of infected women will produce an outbreak. Here, is estimated from the paradigm of nonlinear FDE’s (2) via the next-generation matrix method .
Since the Jacobian matrix is
At the disease-free equilibrium point and then by putting the model in the form  where
Then, the basic reproduction number of the given paradigm is . If then, infection cannot develop because an infected woman infects less than one new woman during their infectious period on average. Conversely, if the disease can enter the population because on average each infected woman will infect more than one new woman.
3. Series Solution for the Proposed Model
Applying Laplace transformation of (2) gives which gives where .
By dividing the th equation of (18) by , we get
Taking the inverse of Laplace transformation on (19) gives
Now, we will find the series solution by putting
Decomposing the terms gives
The polynomial is known as “Adomian polynomial” with the form
By substituting from (21) and (22) into (20), we get
Note that in (29).
4. Approximate Solution
4.1. Construction of the Approximate Solution
Hereafter, we will discuss an approximate solution for the proposed model. Taking the inverse of Laplace transformation for (24), (25), (26), and (27) gives
Since then, if we take the first four terms from the above series and take into consideration the equations from (30) to (46), we get
After the compensation by the values of we obtain
Now, by taking the first three terms of , , , and and putting , , , , , , , , = , in the proposed model yield (see ).
Figure 2 exhibits the dynamical behavior of and against various fractional orders ().
4.2. Convergence and Error Analysis
It is clear that the previous solution is a series. This solution converges rapidly and uniformly to the exact solution. We will use classical techniques to prove the convergence of the series (21), and hence, we can get the sufficient conditions for the convergence of the used method.
Theorem 4. Suppose is the Banach space and is a contractive nonlinear operator with for all Then, based on Banach contraction principle, there is a unique point such that where By applying the ADM, the series in (21) can be written as follows: and suppose that where then, we obtain
Proof. For the first part by the mathematical induction if we obtain Suppose the result is true at hence, We have This shows that .☐
For the second part (b) .
Since and So, we obtain
5. Fractional-Order Numerical Simulations of the SIRD System
First, we recall the basics of the applied numerical technique that have been given for the numerical simulation of fractional IVPs with Caputo derivatives (2). The technique is an extension of the well-known Adams-Bashforth-Moulton (ABM) integrator which is common for the numerical simulation of 1st order differential equations . This technique comes from the idea that the IVP is equal to the equation of Volterra integral. The fractional ABM technique is fully presented by the following formulas (all other states can be found same as ). Assuming that is the domain of the solution and , , : where
A complete study of the stability of the method has been produced in .
In this part, we have found the numerical simulation of the proposed paradigm via the predictor-corrector PECE method of ABM  coded in MATLAB. The behavior of the obtained solution is shown by taking suitable values of the parameters as in : , , , , , , , , = , and . In Figure 3, the behavior of and populations versus time with different fractional derivative orders is shown. In Figure 4, the behavior of and populations versus time with different fractional derivative orders is shown. In Figure 5, the dynamics of the populations in and 3D space with different fractional derivative orders are shown. In Figure 6, the dynamics of the populations in and 3D-space with different fractional derivative orders are shown. In Figure 7, the dynamics of the populations in , and 3D space with different fractional derivative orders are shown. In Figure 8, the dynamics of the populations in , and 3D space with different fractional derivative orders are shown. In Figure 9, the Lyapunov exponents concerning the fraction derivative order are displayed. From Figure 9, we can conclude that the system is unstable in the transient period since LE1 is positive. But as shown LE1 loses its positivity after a short period which indicates that the system settles to its fixed point. In Figure 10, we show the 3D dynamics of the infected population concerning both time and infection rate at fractional derivative order 0.95. In Figure 11, we show the 3D dynamics of the susceptible population concerning both time and infection rate at fractional derivative order 0.95. There is a quick convergence between the obtained and the exact solution. In complex models, it is well known that a small variation in physical behavior stimulates several new results to understand and analyze nature in a systematic manner.
6. Fractional Optimal Control (FOCP) for SIRD Paradigm
In this position, we discuss the optimal control of the fractional-order SIRD paradigm: