The real-world applications and analysis have a significant role in the scientific literature. For instance, mathematical modeling, computer graphics, camera, operating system, Java, disk encryption, web, streaming, and many more are the applications of real-world problems. In this case, we consider disease modeling and its computational treatment. Computational approximations have a significant role in different sciences such as behavioral, social, physical, and biological sciences. But the well-known techniques that are widely used in the literature have many problems. These methods are not consistent with the physical nature and even violate the actual behavior of the continuous model. The structural properties needed for such disciplines, as dynamical consistency, positivity, and boundedness, are the critical requirements of the models in these fields. We studied the transmission of Lassa fever dynamically and numerically. The model’s positivity, boundedness, reproduction number, equilibria, and local stability are investigated in dynamical analysis. In numerical analysis, we developed some explicit and implicit methods. Unfortunately, explicit methods like Euler and Runge Kutta are time-dependent and violate the physical properties of the disease. Then, the proposed implicit method for the said model, the non-standard finite difference, is independent of the time step, dynamically consistent, positive, and bounded. In the end, a comparison of methods is presented.
Lassa fever is an intense hemorrhagic illness caused by the Lassa virus (member of the arenavirus). Lassa virus carries a rat, which is very common in West Africa. It is also known as a zoonosis, which means that disease spreads from animal to human. People usually become infected with Lassa fever because of food and household items’ exposure to urine of infected Mastomys rats. Its symptoms are varied and include cardiac, neurological, and pulmonary problems. In some West Africa, this disease is endemic to the rodent population. It is most common in Liberia, Sierra Leone, Guinea, and Nigeria. Lassa fever is transmitted by playing, touching, and cutting up a rat’s dead body. In 2018, Usman et al. analyzed the Lassa fever virus infection for the transmission dynamics of qualitative and analytic activities of a mathematical model [1]. In 2020, Peter et al. studied Lassa fever’s dynamics, and the solution model stayed and verified boundedness and positivity of basic properties [2]. In 2017, Olayiwola et al. analyzed in the lowliest endemic countries people with an ultimate risk of infection and need for constant investigation develops commanding in the endemic region like Africa with Nigeria at the essential attention will help in no small processes in scheming the scourge [3]. Woyessa et al. investigated the private and public health facilities, control interventions, and prevent infection when feverish patients avoid nosocomial infections [4]. In 2013, Ajayi et al. have studied that Epidemic contained rejoinder schemes and testing to control exertions comprised fright between health staffs, insufficient/poor quality of defensive things, insufficient extra preparation, and poor local laboratory capability [5]. In 2019, Ilori et al. analyzed the activity supporting improving planned for patient care and LLC, emerging infectious diseases, and Medscape through applied epidemiological characteristics and clarifying factors associated with mortality [6]. Adewuyi et al. studied the disease have the probable of actuality organize an infectious menace that essential be controlled by way of biological weapon and currently, vaccine of Lassa fever no available and some natural problems occurs for development of vaccines so prevention the way by control the rodent [7]. Iroezindu et al. analyzed Lassa fever spread; challenges, letdown to use proper defensive tackle, stigmatization of associates, and absence of a purpose-built isolation facility [8]. In 2019, Amodu et al. have studied Lassa fever as an acute disease of scarceness, high endemicity by way of cooperated environmentally-friendly sanitation, and relics susceptible populations to community health problems in Nigeria. Public meeting defense for attractive prevention strategy remains particular sanitation [9]. In 2019, Kangbai et al. highlighted that seasonal epidemics use an effective treatment to make a stratagem, which can control procedures of Lassa fever prevention and control the connection between humans to rodents [10]. Makinde et al. investigated Lassa fever to identify when a nonconformity toward the prestige quo has happened [11]. In 2020, Zhao et al. analyzed quantify of this impact in Nigeria’s presence of Lassa fever significance measure the connotation among local precipitation and infection reproduction number which facts has probable elect applied as a bad sign for Lassa fever epidemics [12]. In 2017, Obabiyi et al. developed a mathematical model for transmission of Lassa fever dynamics with the behavior of susceptible humans, recovered humans divided the population into two parts such as human populations, and rodent population by using the positivity, bounded theorem, and suggested the stability hygiene of environment [13]. In 2018, Akpede et al. studied the necessities for achievement and enduring capacity of the control exertions in Nigeria and the sub-region. In wholly these, the Nigerian administration with NCDC necessity carries a huge responsibility for the organization, supply deployment, and support. If necessary, even persuade sub-regional administrations addicted to action. In addition, there should be expected through determined action [14]. In 2019, Mazzola et al. explored the Diagnostics necessary for acknowledging and controlling epidemics of LASV, unique prevailing and genetically various mediators of VHF, that use scenarios with different performance requirements for text complexity, sensitivity, specificity, and development time [15]. In 2019, Nwafor et al. examined the Lassa fever outbreak in Nigeria; the Health maintenance workers necessity, take a high index of doubt of the infection and follow IPC measures even though provided that maintenance for all patients. Explaining health maintenance workers by the new strategies mentioned above is also significant to reduce the menace of nosocomial transmission of Lassa fever [16]. In 2018, Shehu et al. studied that the Occurrence of rural to urban change of clinical and epidemiological reduced the Lassa fever cases during 2016 for morbidity and mortality [17]. In 2007, Ogbu et al. discussed the situation of Lassa fever in the sub-region of West Africa and suggested strategies for socioeconomic behavior that control the shortage of health care system [18]. In 2020, Tewogbola et al. analyzed the overview and discussed the main reasons it damaged the human population and recommended the control measure of Lassa fever [19]. In 2014, Ajayi et al. reported a case of 59 years that recovers without taking a vaccine such as ribavirin. The symptoms of this disease increase day by day because few people do not use the main precautions to control Lassa fever [20]. Some well-known numerical models related to diseases are studied [21–26].
Formulation of Lassa Fever Model
The variables and parameters are described of the lassa fever model as follows: SH(t): denoted as the susceptible class at any time t, IH(t): characterized as the infectious class at any time t, RH(t): characterized as the recovered class at any time t, SR(t): characterized as the susceptible rodent vectors at any time t, IR(t): characterized as the infectious rodent vectors at any time t, NH(t): characterized as whole humans’ population at any time t, m = NRNH: characterized as the number of infectious rodent vectors by the human host, α1: described as the rate at which contagious rodent vectors and a susceptible class of humans interact with each other, α2: defined as the force of infection, α3: defined as the rate at which sensitive rodent vectors and an infectious class of humans interact with each other, τc: denoted the speed at which infectious human hosts comply with the drug, τnc: indicated the rate at which infectious human hosts do not comply with the drug, rc: denoted the rate at which infectious human hosts are educated to adhere to the medication, δ: indicated the rate of mortality of an infectious class, γ: indicated the rate at which humans may lose their immunity. The leading equations of the model are as follows:
There are two steady states of Eqs. (1) to (5), as follows: disease-free equilibrium (DFE)=(SH,IH,RH,SR.IR)=(ΛHμH,0,0,ΛRμR,0) and endemic equilibrium (EE)=(SH∗,IH∗,RH∗,SR∗,IR∗),
In this section, we shall find the two types of matrices like transmission and transition by assuming the disease-free equilibria in the system (1)–(5) by using the next-generation matrix method, furthermore, considering the infected classes as follows:
Thus, the dominant eigenvalue of the matrix is called reproduction number and denoted as follows:
R0=α1α2ΛHμHNH(τc+rc+τnc+δ+μH).
Local Stability
In this section, we present two well-known theorems for stability in the sense of local. Again, consider the system (1)–(5) as function of A,B,C,DandE as follows:
A=ΛH−α1α2SHIRNH+γRH+τncIH−μHSH.
B=α1α2SHIRNH−τcIH−rcIH−τncIH−δIH−μHIH.
C=τcIH+rcIH−γRH−μHRH.
D=ΛR−α1α3SRIHNH−μRSR.
E=α1α3SRIHNH−μRIR.
Theorem 1:The disease-free equilibrium is locally asymptotically stable for the system (6)–(10). If R0<1 and otherwise unstable if R0>1.
Proof:First, we take the partial derivates of the system (6)–(10) concerning state variables as follows:
Since all the coefficients of the polynomial are positive, therefore, by using Routh Hurwitz Criteria for 2nd order, the disease-free equilibria are locally asymptotically stable.
Theorem 2:The endemic equilibrium is locally asymptotically stable for the system (6)–(10) if R0>1.
Proof: The Jacobian matrix at the endemic equilibria of the system (6)–(10) is as follows:
The Routh Hurwitz criteria of the 5^{th} order are satisfied. Hence, the endemic equilibria are locally asymptomatically stable.
Computational Approximations
In this section, we present the well-known approximations like Euler, Runge Kutta, and non-standard finite difference for the system (1)–(5) as follows:
Euler Approximation
The discretization of the system (1)–(5) under the rules of the Euler approximation is as follows:
Hence, the largest eigenvalue of the Jacobian is less than one, ultimately remaining will also lie in the unit circle when R0>1. Thus, endemic equilibrium is stable.
Computational Approximations
By using the values of the parameters as presented in Tab. 1. The diagrams for the system (1)–(5) for disease-free equilibrium (DFE) and endemic equilibrium (EE) plotted with MATLAB software as follows:
Value of parameters
Parameters
Values
ΛH
0.8
ΛR
0.8
μH
0.8
μR
0.8
α1
1.00166 (DFE)3.00166 (EE)
α2
1.0004 (DFE3.0004 (EE)
α3
0.1
τc
0.7
τnc
0.9
rc
0.2
δ
0.133
γ
0.220
Comparison SectionCombine graphical behaviors of NSFD with Euler and Runge Kutta methods at different time-step sizes (a) Comparison of Euler and NSFD at h = 0.1 (b) Comparison of Euler and NSFD at h = 1 (c) Comparison of Runge Kutta and NSFD at h = 0.1 (d) Comparison of Runge Kutta and NSFD at h = 1Results and Discussion
We investigated the transmission dynamics of Lassa fever disease in humans and rats through the study. The critical point is modeling, terminology related to epidemiology, and Lassa fever disease. Dynamical analysis of the model is investigated. Computational analysis, including well-known methods, is presented. Mostly, methods are valid for only tiny time step sizes. But inappropriately flop for huge time step sizes like Euler and Runge Kutta. Our proposed scheme (NSFD) remains convergent for step sizes like h = 100. Furthermore, Tab. 2 shows the efficiency of the numerical methods.
Comparison analysis of methods at different values of h
h
Euler
RK-4
NSFD scheme
0.01
EE = ConvergenceDFE = Convergence
EE = ConvergenceDFE = Convergence
Convergence
0.1
EE = ConvergenceDFE = Convergence
EE = ConvergenceDFE = Convergence
Convergence
1
EE = DivergenceDFE = Divergence
EE = DivergenceDFE = Divergence
Convergence
100
Divergence(method failed)
Divergence
Convergence
Conclusion
The non-standard finite difference scheme was designed for the communication dynamics of Lassa fever disease. Unfortunately, the earlier methods, like Euler and Runge Kutta of order 4^{th}, are unsuitable because they depend on time step size. So, Euler and Runge Kutta are tentatively convergent. When we increase the time step size, the graph of Euler and Runge Kutta gives variation in result from time to time they display divergent. The new well-known numerical scheme, like the non-standard finite difference scheme independent of time step size. The NSFD scheme is a comfortable tool on behalf of dynamical properties like stability, positivity, boundedness and shows the exact behavior of the continuous model. The graphical behavior of ODE-45, Euler, Runge Kutta, NSFD schemes and comparison of schemes are given in Figs. 1a, 1b, Figs. 2a, 2b, Figs. 3a, 3b, Figs. 4a, 4b and Figs. 5a–5d respectively. In the end, we could extend this idea to all types of nonlinear and complex models. In the future, we could develop our analysis into fuzzy epidemic models and many other types of modeling as given in [27–31].
Combine graphical behaviors of the Lassa fever disease (a) Sub-populations at disease-free equilibrium (DFE) (b) Subpoulations at endemic equilibrium (EE)Euler method for the behavior of infected rats at different time-step sizes (a) Infected rats at h = 0.01 (b) Infected rats at h = 1Runge Kutta method for the behavior of infected rats at different time-step sizes (a) The behavior of infected rats at time step size h = 0.1 (b) The behavior of infected rats at time step size h = 1NSFD method for the behavior of infected rats at different time-step sizes (a) The behavior of Infected rats for EE at h = 0.1 (b) The behavior of infected rats for EE at h = 1000
Thanks, our families and colleagues who supported us morally.
Funding Statement: The authors received no specific funding for this study.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
ReferencesS.Usman and I. I.Adamu, “Modelling the transmission dynamics of the Lassa fever infection,” O. J.Peter, A. I.Abioye, F. A.Oguntolu, T. A.Owolabi, M. O.Ajisopeet al., “Modelling and optimal control analysis of Lassa fever disease,” J. O.Olayiwola and A. S.Bakarey, “Epidemiological trends of Lassa fever outbreaks and insights for future control in Nigeria,” A. B.Woyessa, L.Maximore, D.Keller, J.Dogba, M.Pajiboet al., “Lesson learned from the investigation and response of Lassa fever outbreak,” N. A.Ajayi, C. G.Nwigwe, B. N.Azuogu, B. N.Onyire, E. U.Nwonwuet al., “Containing a Lassa fever epidemic in a resource-limited setting: Outbreak description and lessons learned from Abakaliki,” E. A.Ilori, Y.Furuse, O. B.Ipadeola, C. C.Dan-Nwafor, A.Abubakaretet al., “Epidemiologic and clinical features of Lassa fever outbreak in Nigeria,” G. M.Adewuyi, A.Fowotade and B. T.Adewuyi, “Lassa fever: Another infectious menace,” M. O.Iroezindu, U. S.Unigwe, C. C.Okwara, G. A.Ozoh, A. C.Nduet al., “Lessons learnt from the management of a case of Lassa fever and follow-up of nosocomial primary contacts in Nigeria during ebola virus disease outbreak in West Africa,” S. E.Amodu and S. O.Fapohunda, “Lassa fever and the Nigerian experience: A review,” J. B.Kangbai, F. K.Kamara, R. C.Lahai and F.Gebeh, “Lassa fever in post-ebola sierra leone socio-demographics and case fatality rates of in-hospital patients admitted at the Kenema Government hospital Lassa fever ward between 2016-2018,” O. A.Makinde, “As ebola winds down, Lassa fever reemerges yet again in West Africa,” S.Zhao, S. S.Musa, H.Fu, D.He and J.Qin, “Large-scale Lassa fever outbreaks in Nigeria: Quantifying the association between disease reproduction number and local rainfall,” O. S.Obabiyi and A. A.Onifade, “Mathematical model for Lassa fever transmission dynamics with variable human and reservoir population,” G. O.Akpede, D. A.Asogun, S. A.Okogbenin and P. O.Okokhere, “Lassa fever outbreaks in Nigeria,” L. T.Mazzola and C.Kelly-Cirino, “Diagnostics for Lassa fever virus: A genetically diverse pathogen found in low-resource settings,” C. C. D.Nwafor, Y.Furuse, E. A.Ilori, O.Ipadeola, K. O.Akabikeet al., “Measures to control protracted large Lassa fever outbreak in Nigeria,” N. Y.Shehu, S. S.Gomerep, S. E.Isa, K. O.Iraoyah, J.Mafukaet al., “Nigeria-the changing epidemiology and clinical presentation,” O.Ogbu, E.Ajuluchukwu and C. J.Uneke, “Lassa fever in West African sub-region: An overview,” P.Tewogbola and N.Aung, “Lassa fever: History, causes, effects, and reduction strategies,” N. A.Ajayi, K. N.Ukwaja, N. A.Ifebunandu, R.Nnabu, F. I.Onweet al., “Lassa fever-full recovery without ribavirin treatment: A case report,” W.Shatanawi, A.Raza, M. S.Arif, M.Rafiq, M.Bibiet al., “Essential features preserving dynamics of stochastic dengue model,” A.Raza, M. S.Arif, M.Rafiq, M.Bibi, M.Naveedet al., “Numerical treatment for stochastic computer virus model,” M. S.Arif, A.Raza, K.Abodayeh, M.Rafiq and A.Nazeer, “A numerical efficient technique for the solution of susceptible infected recovered epidemic model,” W.Shatanawi, A.Raza, M. S.Arif, M.Rafiq, M.Bibiet al., “Essential features preserving dynamics of stochastic dengue model,” M. A.Noor, A.Raza, M. S.Arif, M.Rafiq, K. S.Nisaret al., “Non-standard computational analysis of the stochastic COVID-19 pandemic model: An application of computational biology,” K.Abodayeh, A.Raza, M. S.Arif, M.Rafiq, M.Bibiet al., “Numerical analysis of stochastic vector-borne plant disease model,” R. E.Mickens, “Dynamic consistency: A fundamental principle for constructing non-standard finite difference schemes for differential equations,” A.Raza, A.Ahmadian, M.Rafiq, S.Salahshour and M.Ferrara, “An analysis of a nonlinear susceptible-exposed-infected-quarantine-recovered pandemic model of a novel coronavirus with delay effect,” G.Neha, A.Ghosh, S. P.Mondal, M. Y.Bajuri, A.Ahmadianet al., “Identification of dominant risk factor involved in spread of COVID-19 using hesitant fuzzy MCDM methodology,” Z.Muhammad, K.Shah, F.Nadeem, M. Y.Bajuri, AliAhmadianet al., “Threshold conditions for global stability of disease-free state of COVID-19,” A.Ahmadian, N.Senu, F.Larki, S.Salahshour, M.Suleimanet al., “A Legendre approximation for solving a fuzzy fractional drug transduction model into the bloodstream,”