A nonlinear fractional epidemic model for the Marburg virus transmission with public health education

In this study, a deterministic model for the dynamics of Marburg virus transmission that incorporates the impact of public health education is being formulated and analyzed. The Caputo fractional-order derivative is used to extend the traditional integer model to a fractional-based model. The model’s positivity and boundedness are also under investigation. We obtain the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {R_0}$$\end{document}R0 and establish the conditions for the local and global asymptotic stability for the disease-free equilibrium of the model. Under the Caputo fractional-order derivative, we establish the existence-uniqueness theory using the Banach contraction mapping principle for the solution of the proposed model. We use functional techniques to demonstrate the proposed model’s stability under the Ulam-Hyers condition. The numerical solutions are being determined through the Predictor-Corrector scheme. Awareness, as a form of education that lowers the risk of danger, is reducing susceptibility and the risk of infection. We employ numerical simulations to showcase the variety of realistic parameter values that support the argument that human awareness, as a form of education, considerably lowers susceptibility and the risk of infection.

i.A novel fractional model defined in the Caputo sense is used to examine the dynamics of Marburg virus transmission incorporating the impact of public health education.ii.The reproduction number R 0 for the proposed model is being derived along with the disease free equi- librium points for the system.iii.We demonstrate the existence and uniqueness solutions of the dynamics of Marburg virus transmission incorporating the impact of public health education by employing the Banach contraction mapping principle.iv.The fractional controlling system of equations underwent a stability study using the Hyers-Ulam-type stability criteria.v.To validate the theoretical components and conclusions of the suggested model, an effective numerical technique is adopted.vi.The obtained results demonstrate the effectiveness of the suggested model in providing some new insights into the dynamics of the Marburg virus infection as well as some preventative actions.
The paper is structured in the manner described below.In Section "Preliminaries", the basic definitions and lemmas are presented.The formulation of the Marburg virus infection model based on the system of a deterministic mathematical model and the Caputo fractional derivative in Section "Model formulation".Section "Basic qualitative properties of the model" deals with equilibrium points and the basic reproduction number.The mathematical analysis of the existence-uniqueness of our suggested Marburg virus transmission model is covered in Section "Existence and uniqueness".In Section "HU stability", the stability results of the Marburg virus transmission model is shown and discussed.Sections "Numerical simulation" and "Numerical simulation and discussion" deal with the numerical framework and simulations, respectively.Section "Conclusion" of the paper concludes the work.

Preliminaries
n this section, we recall some critical concepts, lemmas, and definitions to study our proposed model.
Definition 2.1 33 The Caputo fractional derivative of order γ (γ > 0 ) of u is given by The Riemann-Liouville fractional integral of γ order of u is given by where n = [γ ] + 1 , [γ ] denotes the integer part of number γ , provided that the right side is pointwise defined on (0, 1).

Model formulation
In this section, we present the formulation of the model, which we will be studying in this paper.Here, the β is the transmission rate.γ 1 and γ 2 are the modification parameters of I u and H, respectively.Consid- ering the interrelationship, the infection model used in analyzing the dynamics of Marburg virus transmission incorporating the impact of public health education is given by the following deterministic system of nonlinear differential equations The flow diagram of the model is presented in Fig. 1 while the description of the rest of parameters are presented in Table 1.According to the explanation of time-dependent kernel defined by the power law correlation function, presented in Ref. 35 , our considered Caputo fractional order derivative model for the dynamics of Marburg virus transmission is defined as follows; . (1) Where c D γ is Caputo fractional derivative, 0 < γ ≤ 1 .Memory and heredity traits, which are complex behavioural patterns of biological systems, are goal of dealing with fractional order systems in our newly designed the Marburg virus transmission with public health education impact, these together allows us more realistic approach to biological systems.The memory function allows fractional order models to incorporate more knowledge from the past, allowing for more accurate prediction and translation.In addition, the hereditary property specifies the genetic profile, as well as the age and status of the immune system.

Basic qualitative properties of the model
In this section, we give positivity, boundedness and existence of unique solution to projected model.

Theorem 4.1 Assume
) , the solution of proposed system (2) is nonnegative and bounded.
Proof Since the coefficients of (2) are positive constants, we have Hence, the solution of ( 2) is non-negative.For the boundedness of solution, one get where Then, we can easily get Apply the Laplace transform method to the inequality (3) with N (t 0 ) ≥ 0 , one can see that (2) www.nature.com/scientificreports/Taking the Laplace inverse, we infer that from E γ ,1 (−µt γ ), E γ ,γ +1 (−µt γ ) are the series of the Mittag-Leffler function which converges for any argument.Hence, the solution to the model is bounded.

Existence and uniqueness
Let us write the system (2) in the compact form for easy description as follows: Let C[0, T] be the set of all real continuous function on the interval [0, T] with the norm ||z|| = sup{|z|, z ∈ C[0, T]}.
We show the analysis for S and for others it is similar.Hence we consider the initial value system according to the definition and lemma of Caputo fractional calculus, we can obtain the equivalent integral solution of the above system (6) as there exists a constant L > 0 , such that Then the system (6) has a unique solution. .
, Step 1: Prove that the operator T is completely continuous by Arzela-Ascoli theorem.
(1) Prove that T is bounded, where is a bounded subset of B .For any S, P, E, I u , I d , H, R ∈ , by the condition (H 1 ) , ∃m > 0, such that Then one can see that (2) Show that T is equicontinuous.

D uo to
is unifor m ly cont inuous on any b ounde d subs et , for any Then it holds that (3) State T : B → B is continuous.
From (H 1 ) and the function > 0 , we get T : B → B .Set S n , S ∈ B and S n → S as n → +∞ .From the Lebesgue dominated convergence theorem and the continuity of the function G 1 , we can get Then we get lim n→+∞ TS n (y) → TS(y).
Step 2: Show that the system (6) has a unique solution.
For any S 1 , S 2 ∈ C, by (H 2 ) , we observe that Then by From contraction mapping principle, T has a unique fixed point.That is, (6) has a unique solution.
Proof For solution S of system ( 6), according to (H 4 ) , it gives Hence, one gets Let S be unique solution to system (6), we have Hence, we deduce where K = T γ Ŵ(γ +1)−T γ L .By (H 3 ) of Definition 6.1, the system ( 6) is HU stable.

Numerical simulation
In this section, the numerical solution of system ( 2) is given by corrector-predictor iterative scheme in Caputos sense.Let the proposed model can be written into the following one as Choose step length h = T M and using the integral equation equivalent to system (2), U a (t j+1 ) (j = 0, 1, − G 1 (y, S(y), P(y), E(y), I u (y), I d (y), H(y), R(y)) dy where The Predictor formula is derived as follows: Thus the corrector formula for system (2) is

Numerical simulation and discussion
For the purpose of validating our created iterative scheme, we present our numerical simulation in this part.For this, we start with initial values for each compartment of our proposed model (1); S = 100; P = 70; E = 55; I u = 20; I d = 35; H = 30; R = 15.We have employed Adam-Bashforth-Moulton scheme to obtain numerical solution to the system.We compare the effects of various fractional order values with a step size 0.2 throughout the time range [0,300] against the suitable parameter values listed in Table 1.Figures 2 and 3 represent the numerical simulation results for the individuals.It is clear that the outcomes and the changes of the fractional-order γ fit well, which indicate that the method is effective, thus when the operator γ is varied the dynamism of each state variable has the same trend.However, their values are slightly differ- ent.For example, when the fractional order γ is reduced from 1 in Fig. 2a and b show the differences between susceptible and susceptible individuals received health education.These figures demonstrate that the number of susceptible educated individuals increases over time until it achieves a carrying capacity while the number of susceptible individuals decreases over time as more and more people contract the infection.In Fig. 2c, the exposed population initially increase with fast speed corresponding to small fractional order then became slow.Similar to Fig. 2d, the undetected infected people.This graph shows a rapid decrease in the initial 50 days and later on it goes towards stability.In Fig. 3a there is sharp increase for all values of fractional operator γ due to the high transmissibility of Marburg virus of the disease advocated according to WHO.In Fig. 3b, the hospitalised population initially increase up with fast speed corresponding to small fractional order then became slow and later on it goes towards stability.During this time the recovery population also achieve their maximum peak in the initial 40 days as shown in Fig. 3c.These figures demonstrate that the Caputo derivative generates global dynamics of the suggested model, where lower orders reach stability more quickly than integer orders.
Furthermore, considering the contributions of some of the sensitive parameters in our proposed model, we maintain the fractional operator to be fixed at γ = 0.95 and varied the information dissemination and the rate of detecting unknown Marburg virus infected individuals in Figs. 4 and 5, respectively.Figure 4 depicts the impact of the various values of information dissemination or awareness rate on the dynamics of S, P, E, I u , and I d .In general, this figure reflects that when information dissemination increases, the number of S, E, I u , and I d decreases rapidly.This indicates how educating susceptible people about their health can help to stop the spread of the Marburg virus.As a result, in order to stop and limit the spread of the Marburg virus, public policymakers www.nature.com/scientificreports/must concentrate on enhancing the value of information dissemination.Fig. 5 depicts the impact of the various values of detecting unknown Marburg virus infected individuals on the dynamics of I u , I d and R. We noticed that dynamics has significant impact on our proposed model.These results give public policymakers a note on how and where resources allocate are needed in order to prevent and control Marburg virus spread.

Conclusion
In recent years, numerous deadly diseases have appeared in many countries around the globe.If the limitations of established methodologies, ideas, and procedures are updated, questioned, and amended in response to contemporary scientific findings and the emergence of unforeseen physical phenomena, the dynamics of infectious diseases can be better understood and even predicted.We have comprehensively analyzed a new deterministic mathematical model for Marburg virus in a homogeneously mixing human population under Caputo fractional order derivative in this paper.We have investigated the qualitative aspect of the spread of the Marburg virus by analysing the positiveness, boundedness, equilibrium point, and fundamental reproductive number R 0 .The Banach contraction mapping principle is used to show the system existence and uniqueness analysis.The fractional controlling system of equations underwent a stability study using the Hyers-Ulam-type stability criteria.
Using the Adam-Bashforth-Moulton scheme, numerical trajectories are constructed to test the effectiveness of the suggested fractional-order model.We looked at the impact of some critical parameters.Based on the trajectories, we hypothesized that the memory index or fractional order can be use by the public health policymakers to comprehend and foresee the dynamics of the transmission of the Marburg virus.It is also seen that if information dissemination and availability of resources to detect infected individuals can reduce the spread of Marburg virus infection.This study consider no real data which is our limitation.Despite this limitation, this model provides a good description of ongoing Marburg outbreak.The model looks into the use of public health education, which is a crucial component of disease control in the modern era.

Figure 1 .
Figure 1.Transfer diagram for the Marburg virus transmission.

Figure 2 .
Figure 2. Numerical trajectory of Marburg transmission under Caputo fractional operator.

Figure 3 .
Figure 3. Numerical trajectory of Marburg transmission under Caputo fractional operator.
For this, we have considered only human population and divided it into seven different compartments, which are susceptible humans without public health education about the Marburg virus transmission (at a time) S(t); susceptible humans with public health education P(t); Exposed individuals E ; Undetected infected individuals I u ; Detected Marburg virus infection I d ; Hospitalisation of Marburg virus infected individuals H ; Recovery from Marburg virus infection R .We assume that, at any time, the educated susceptible group may act as ignorantly and enter the class of susceptible at a constant rate ξ .The total population as a whole is provided by Susceptible humans enter the population, either through birth or migration, at a rate and are infected with the Marburg virus at rate , where

Table 1 .
Interpretation of parameters in the model.
dt = φS(t) − (ν + µ + ξ)P(t), S is the unique solution for system (6) if and only if S is the unique solution for TS = S.