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.
1. Introduction
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 [1]. 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 [6], nanotechnology [7], optics [8], human diseases [9], and chaos soliton theory [10]. 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 [21] 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).
(1)
where .
Table 1.
The meaning of the parameters in the model (1).
Parameters
Meaning
S(t)
Susceptible pregnant women population
I(t)
Infected pregnant women population
R(t)
Recovery of pregnant women population
D(t)
The population of pregnant women dying in that country
u1
The rate at which the women are infected
N
The total number of adult women in the given country
μ
The rate at which the women are pregnant
ω
The rate of death of pregnant women with LHF
l
The rate of death of pregnant women with another disease or with natural death
u2
The rate of recovery of pregnant women from LHF
Also, the model has been studied by Goyal et al. [22]. The q-homotopy analysis transform method (q-HATM) has been applied for solving the LHF model [30]. The LHF model via Atangana-Baleanu fractional derivative has been studied in [31]. The SIR model has been studied through Caputo-Fabrizio fractional operator in [32].
In this paper, we reconsider the system (1) by the following fractional sense of the system (1)
(2)
(3)
(4)
(5)
where D(r) is the Caputo fractional derivative of order r.
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.
2. Preliminaries
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
(6)
where Γ is the known gamma function and α ∈ (0, 1) is the order of this fractional integral.
Definition 2. The Caputo-type fractional derivative of order α is defined as
(7)
Definition 3. The Laplace transform of Caputo fractional derivative of order α [33] is given by
(8)
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 t, we can express the rate of change in susceptible population as follows
(9)
With time t, we can express the rate of change in the population of the infected pregnant women as follows
(10)
Here, u1SI is the number of pregnant women excluded from the group of susceptible.
With time t, the rate of change in recovery population is written as
(11)
Finally, the rate of change of dying population with time t is given as
(12)
Hence, the mathematical model for the LHF is given by model (2), with initial conditions
(13)
Also 0 < αi < 1, i ∈ {1, 2, 3, 4}. Here, S(t), I(t), R(t), and D(t) 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.
Basic reproduction number R0 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 R0. 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 R0 < 1, 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 R0 > 1, the DFE is locally unstable; a sufficiently small number of infected women will produce an outbreak. Here, R0 is estimated from the paradigm of nonlinear FDE’s (2) via the next-generation matrix method [34].
Since the Jacobian matrix is
(14)
At the disease-free equilibrium point (μ − l/lN, 0, 0, 0) and u2 = 0 then by putting the model in the F − V form [34] where
(15)
then
(16)
Then, the basic reproduction number of the given paradigm is R0 = u1(μ − l)/ωlN. If R0 < 1, then, infection cannot develop because an infected woman infects less than one new woman during their infectious period on average. Conversely, if R0 > 1, the disease can enter the population because on average each infected woman will infect more than one new woman.
Hereafter, we will discuss an approximate solution for the proposed model. Taking the inverse of Laplace transformation for (24), (25), (26), and (27) gives
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
(44)
(45)
(46)
(47)
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
(48)
(49)
(50)
(51)
After the compensation by the values of a, b, c, d1, d, e, f, g, h, i, j, p, q, r, r1, s, u, v, w, x, y, z, a1, b1, c1, d1, e1, f1, g1, h1, i1, we obtain S(t) = S0 + S1 + S2 + S3 =
(52)
(53)
(54)
(55)
Now, by taking the first three terms of S(t), I(t), R(t), and D(t) and putting α1 = α2 = α3 = α4 = α, u1 = 0.4, μ = 0.3, N = 1000, l = 0.2, ω = 0.8, u2 = 0.2, S(0) = 900, I(0) = 10, R(0)= D(0) = 0, in the proposed model yield (see [30]).
(56)
(57)
(58)
(59)
Figure 2 exhibits the dynamical behavior of S(t), I(t), R(t), and D(t) against various fractional orders (α = 0, 7; 0.8; 0.9; 1).
The behavior of S(t), I(t), R(t), and D(t) populations versus time with different fractional derivative 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 σ : ℘⟶℘ is a contractive nonlinear operator with for all x, x′ ∈ ℘, ‖σ(x) − σ(x′)‖ ≤ λ‖x − x′‖, 0 < λ < 1. Then, based on Banach contraction principle, there is a unique point x such that σx = x, where x = (s, i, r, d). By applying the ADM, the series in (21) can be written as follows:
(60)
and suppose that x0 = x0 ∈ Aj(x) where Aj(x) = {x′∈℘:‖x − x′‖ < j}, then, we obtain
(61)
(62)
Proof. For the first part (a)xn ∈ Aj(x); by the mathematical induction if n = 1, we obtain
(63)
Suppose the result is true at n − 1, hence,
(64)
We have
(65)
(66)
This shows that xn ∈ Aj(x)..☐
For the second part (b) .
Since ‖xn − x‖ ≤ λn‖x0 − x‖ 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 [35]. 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 S). Assuming that [0, T] is the domain of the solution and n = 0, 1, 2, ⋯, N, h = T/N, tn = nh:
(67)
where
(68)
(69)
A complete study of the stability of the method has been produced in [36].
In this part, we have found the numerical simulation of the proposed paradigm via the predictor-corrector PECE method of ABM [35] coded in MATLAB. The behavior of the obtained solution is shown by taking suitable values of the parameters as in [30]: u1 = 0.4, μ = 0.3, N = 1000, l = 0.2, ω = 0.8, u2 = 0.2, S(0) = 900, I(0) = 10, R(0)= D(0) = 0, and α1 = α2 = α3 = α4 = α. In Figure 3, the behavior of S(t) and I(t) populations versus time with different fractional derivative orders is shown. In Figure 4, the behavior of R(t) and D(t) populations versus time with different fractional derivative orders is shown. In Figure 5, the dynamics of the populations in I(t), R(t) and D(t) 3D space with different fractional derivative orders are shown. In Figure 6, the dynamics of the populations in S(t), I(t) and D(t) 3D-space with different fractional derivative orders are shown. In Figure 7, the dynamics of the populations in S(t), I(t), and R(t) 3D space with different fractional derivative orders are shown. In Figure 8, the dynamics of the populations in S(t), R(t), and D(t) 3D space with different fractional derivative orders are shown. In Figure 9, the Lyapunov exponents concerning the fraction derivative order α 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.
3D dynamics of the susceptible population versus both time and infection rate at fractional derivative order 0.95.
6. Fractional Optimal Control (FOCP) for SIRD Paradigm
In this position, we discuss the optimal control of the fractional-order SIRD paradigm:
(70)
According to references [37–40], consider the model (70) inside R4. The proposed control functions can be formulated as follows:
(71)
and Tf refers to the end of the time period, u1(.) provides the rate of susceptibility per year, and u2(.) provides the rate of curing from infection population to recovered population per year. u1(.) with u2(.) is the control functions. Here, the function space L∞ norm is defined as . This concept of the L∞ norm considered to be equal to taking the limit as p⟶∞ of the Lp norm.
The following is the definition of the objective function (the control variables are squared)
(72)
Such that a, b, and c clarify, respectively, the quantity of infectious, a per year rate of susceptible, and a per year rate of progress from infective to recovered.
Finding the optimal controls u1(.) and u2(.) is the basic goal in FOCPs, in order to minimize the predefined objective function:
(73)
where subjected to the constraint
(74)
where ηi = ηi(S, I, R, D, u1, u2, t), i = 1, 2, 3, 4.
The initial conditions are as follows:
(75)
To construct the FOCP, consider the selected objective function defined in [41–44]:
(76)
Such that i = 1, 2, 3, 4.
In equation (77), (78), we write the Hamiltonian of the cost (objective) function (76) and the control of the fractional-order SIRD paradigm (70):
(77)
then
(78)
From (76) and (78), following the method in [37, 39, 43–45], the FOPC’s necessary and sufficient conditions are formulated as
(79)
(80)
(81)
additionally,
(82)
where λi, i = 1, 2, 3, 4 represent the Lagrange multipliers. In equations (79) and (80), the necessary conditions using FOPC Hamiltonian are represented.
Theorem 5. If u1 and u2 are optimal controls related to the controlled states S∗, I∗, R∗ and D∗, then, the adjoint variables exist that satisfy the following conditions:
(i)
Adjoint conditions (costate conditions)
Considering the related conditions represented in the recorded theorem and using equations (82), see [37, 39, 43, 44], we obtain the following four conditions that can be formulated as
(83)
(84)
(85)
(86)
(ii)
Transversality conditions
(87)
(iii)
Conditions of optimality
(88)
moreover, by applying equations (80), the control functions are given as follows:
(89)
(90)
(91)
(92)
Proof. The system of costate conditions (83)–(86) is produced from equation (81) such that the Hamiltonian H∗ is presented as
(93)
Furthermore, the conditions in equation (82) are guaranteed, and the optimal control given by equations (91)–(92) can be produced via equation (80).☐
Substituting in (70), the state system is formulated as follows:
The existence of the control system (83)–(86) is to be proved here, the same can be found in [37–43] as follows:
Let
(95)
(96)
(97)
Let .
We have, at , , , ,
(98)
(99)
(100)
(101)
(102)
(103)
where k1, k2, k3, k4, k5k22, k23, k24, k2, k31, k33, k34, k41, k42, k43, and k44 are constants greater than zero. Hence, each of the four functions f1, f2, and f4 agrees with the Lipschitz z conditions. Then, each of the four kernels f1, f2 is continuous respecting the four features, then. See for instance [37–43] for more information about the existence and uniqueness.
7. Conclusion
In this work, we have provided the approximate solution constructions for the fractional-order LHF model. In addition, the numerical simulations of the fractional-order LHF model have been experimented with many arbitrary orders. Also, an optimal control case has been investigated. The outcomes acquired by utilizing Laplace transform combined with the ADM are very compatible with results existing in the previous research. Further, the applied method for finding the numerical solution of the proposed model is introduced without considering any perturbation, discretization, or transformations. The current examination enlightens the considered nonlinear manner that rely upon the time history and the time moment by applying the idea of fractional order differentiation. Moreover, the obtained results show that Laplace transform combined with the ADM is amazingly deliberate, more powerful, and exceptionally precise, and which can be applied to break down the different classes of nonlinear marvels that emerge in science and innovation. For future work, we suggest generalizing the studied LHF model to show the effect of the infection from other sources as animals, men, and children.
Conflicts of Interest
The authors declare no conflict of interest.
Authors’ Contributions
Conceptualization was done by M.H.; methodology was done by M.H, A.E., and A.M.; software was done by M.H, A.E., and A.M.; validation was done by M.H, A.E., S.U., A.G., and A.M.; formal analysis was done by M.H, A.E., S.U., A.G., and A.M.; investigation was done by M.H, A.E., S.U., A.G., and A.M.; resources was done by M.H, A.E., S.U., A.G., and A.M.; data curation was done by M.H, A.E., S.U., A.G., and A.M.; writing—original draft preparation was done by M.H. and A.E.; writing—review and editing was done by M.H. and A.E.; visualization was done by M.H. and A.E.; supervision was done by M.H.; project administration was done by M.H.; funding acquisition was done by A.G. and S.U. All authors have read and agreed to the published version of the manuscript.
Acknowledgments
The authors acknowledge the support and funding of Research Center for Advanced Material Science (RCAMS) at King Khalid University through Grant No. RCAMS/KKU/008-21.
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