STABILITY ANALYSIS AND SEMI-ANALYTIC SOLUTION TO A SEIR-SEI MALARIA TRANSMISSION MODEL USING HE’S VARIATIONAL ITERATION METHOD

We have considered a SEIR-SEI Vector-host mathematical model which captures malaria transmission dynamics, described and built on 7-dimensional nonlinear ordinary differential equations. We compute the basic reproduction number of the model; examine the positivity and boundedness of the model compartments in a region using well established methods viz: Cauchy’s differential theorem, Birkhoff & Rota’s theorem which verifies and reveals the wellposedness, and carrying capacity of the model respectively, the existence of the Disease-Free (DFE) and Endemic (EDE) equilibrium points were determined and examined Using the Gaussian elimination method and the Routh-hurwitz criterion, we convey stability analyses at DFE and EDE points which indicates that the DFE (malaria-free) and the EDE (epidemic outbreak) point occurs when the basic reproduction number is less than unity (one) and greater than unity (one) respectively. We obtain a solution to the model using the Variational iteration method (VIM) (an unprecedented method) to each population compartments and verify the efficacy, reliability and validity of the proposed method by comparing the respective solutions via tables and combined plots with the computer in-built Runge-kutta-Felhberg of fourth-fifths order (RKF-45). Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 31 May 2020 doi:10.20944/preprints202005.0484.v1 © 2020 by the author(s). Distributed under a Creative Commons CC BY license. 2 We illustrate the combined plot profiles of each compartment in the model, showing the dynamic behavior of these compartments; then we speculate that VIM is efficient and capable to conduct analysis on Malaria models and other epidemiological models.


Introduction
Malaria is a mosquito-borne infectious disease that is life threatening to humans and other animals (Malaria fact sheet, 2014) [16]. This infectious disease is widely spread throughout the globe and predominantly present in tropical and sub-tropical regions of the earth including some parts of Europe.
The wide spread of this vector-borne disease (malaria) has urged numerous researchers and health organizations to study the epidemiology and transmission dynamics of the disease; so as to be able to implement an appropriate intervention strategy on its ubiquitous nature.
Because of its nature of being a fatal disease, this is why 25 th of April is set aside as the world's annual malaria day for the global alertness against the disease. Malaria causes symptoms that typically include fever, tiredness, vomiting, and headaches (Caraballo, 2014). In severe cases it can cause yellow skin, seizures, coma, or death. (Caraballo, 2014) [15]. These symptoms usually begin ten to fifteen days after being bitten by an infected mosquito and if not properly treated, Of all the semi-analytical methods implemented to solve epidemic models including malaria, none have solved the malaria model using the variational iteration method and as a result, less attention has been paid using this method on malaria models. This method is unprecedented.
The main reason of this paper is to validate the efficiency of variational iteration method and also speculate its capability as alternative approach in solving and analyzing epidemiological models including malaria.
The huge advantage of this method over other methods include: the simplicity and straightforwardness, less computational stress or efforts of the method with no linearization of the nonlinear term, no computation of Adomian or He's polynomials, yet yielding highly accurate and rapidly convergent results devoid of errors when compared numerically and graphically.
In this research, we consider an existing SEIR model of Osman et al (2017), conduct a stability analysis, and then obtain semi-analytic solution via Variational iteration method (VIM).
The model presented here in this research is of two compartmental system of nonlinear ordinary differential equation involving the host which is the human and the Vector which is the mosquito. The human (host) is described by four differential equations and the mosquito by three differential equations. 5 The subsequent organization of this research work is structured as follows: Section 2 elucidates the compartmental model of the malaria transmission dynamics as well as the flow diagram of the model; Section 3 focuses on the mathematical analysis of the model which includes the analysis on the feasible region  of the model, so as to verify the epidemiological validity of the model; the disease-free equilibrium point (DFE), basic reproduction number, the endemic equilibrium point (EDE), stability of the DFE via Gaussian elimination method and the EDE with theorems, lemmas, and proofs were all computed here.
Semi-analytic solution was then proffered to the seven (7) compartments of the vector-host model using He's variational iteration method (VIM) in Section 4.
Lastly, numerical result comparison were made for the solved compartments via tables and combined plots of Runge-Kutta-Felhberg 45 (RKF-45) and VIM, results were then interpreted and discussed before the final conclusion in section 5 and 6 respectively.

The Model
The model consists of two classes of population, the human population and the mosquito population. The human population is subdivided into four compartments, the susceptible, the exposed, the infected, and the recovered. While the mosquito population is subdivided into three compartments, the susceptible, the exposed, the infected as it is assumed that mosquitoes don't recover. We then have that the model for the humans (host) and the model for the mosquito (vector). (Table 1)

Model Assumptions
The Population of the susceptible human ) (t S H is increased by the recruitment of individuals at a rate H  , and by the recovered individuals returning back to the compartment due to loss of immunity at a rate , they acquire infection at a rate , the population is then decreased by natural death of humans at a rate . (Fig 1) The population of the Exposed human ) (t E H is generated by the infection of the susceptible individuals at a rate , decreased by 6 humans whose infection has developed to the infectious compartment at a rate 1 , and further decreased by natural death . (Fig 2) The population of the infected ) (t I H is generated by humans who are infectious at a rate 1 , increased by newborn baby with infection at rate , then decreased by natural death , malaria induced death, and humans who have recovered at rates , , and 2 respectively. (Fig 3) The Recovered population ) ( R H t is generated by those who are infected but are being treated and recovering from malaria at a rate 2 . It is then decreased by those who die naturally and lose their immunity at rates and respectively. (Fig 4) The susceptible mosquito population ) (t S V is generated by the recruitment of mosquitoes into the compartment at a rate ᴧ , decreased by infection and death by natural cause with rates and . (Fig 5) The Exposed mosquito's population is generated by susceptible mosquitoes exposed to the malaria pathogen infection at a rate , decreased by mosquitoes that have developed into the infectious state, and by natural cause at rates 3 , and . (Fig 6) The Infected mosquito's population ) (t I V is generated by exposed mosquito whose state has moved to the infectious state at the rate 3 , and decreased by natural cause . (Fig 7) ( ) (

Positivity and Boundedness of Solution
Here, results are presented and verifications are made as to guarantee that the malaria model governed by the system (1) is epidemiologically and mathematically well-posed in a feasible region  ; given by: The feasible region of the system (1) given by is the human net population.
Now from the derivatives of sums; This implies that, By solving the first order linear differential inequality (4) using integrating factor method we have; () Where p is a constant of integration. Then by applying Birkhoff and Rota's theorem [31] on the differential inequality (5), it follows that lim ( ) This is commonly known as the carrying capacity of the system and hence shows Boundedness.

It then follows that
This proves the boundedness of the solution inside the region H  Now for other classes of the population we have;

Other Compartments
We consider the rate of change of the population in the Susceptible Human compartment

Mosquito Model (Vector)
We consider the governing equation of the vector (SEI) model which is the Mosquito.
The total Population density gives From Cauchy's differential theorem, We have that We then have, By solving the differential inequality by method of integrating factor and apply Birkhoff and (20) It then follows that
This completely proves our theorem 1.

Disease-Free equilibrium points and the Reproduction Number
The points at which the differential equation is equal to zero are referred to as the equilibrium points or steady-state solutions.
The model consists of just two equilibrium points which is the disease-free and the Endemic equilibrium points The point or time at which the disease wiped out and the entire population is susceptible is the Disease-free equilibrium point while the point at which the disease persists in the population (Epidemic outbreak) is the Endemic equilibrium point. 14 At Equilibrium, By substituting (21) into the system of equations (1),  The assumption that the population is susceptible implies that the reproduction number would be  (23) Let the reproduction number of the model be denoted by .

The Basic Reproduction Number
where  is an identity matrix.
Using the next generation matrix method,  (27) By splitting the matrix in the equation (27) This is now in the form For the Reproduction number, we only need terms in the Exposed and the infected compartments Then we have the matrix ; I is an identity matrix.
By computing the spectral radius, the reproduction number is given as; Similarly, by considering the nonlinear system in the Mosquito's model Similarly, using the Next generation matrix approach on the vectors system of equations above we have the Mosquito's reproduction number as From equation (25) (30) and (31) into (25) we have the general reproduction number of the SEIR-SEI system as: This gives the reproduction number of the complete system By alternative notations, if we let Then, 13 2

Existence of the Endemic Equilibrium Points
The SEI-SEI model of Malaria transmission possesses an endemic equilibrium point At this point, there is persistence of the disease in the system and hence an epidemic outbreak.
At equilibrium, Similarly by solving the system (36)

Stability of the Disease-Free Equilibrium
We now check for the stability of the model at DFE by taking the jacobian of the seven dimensional ODES in equation (1) and obtaining its corresponding Eigen values.
The SEIR-SEI is stable if all of the Eigen values obtained from the linearized system are negative real values.
We have the jacobian of the model to be given as: By inserting our alternative notation By applying the same matrix techniques which was applied on (51) accordingly and replacing alternative notations, we have the 7 Eigen values for the model as: , Clearly all our Eigen values are negative real values, and then the Disease-free equilibrium (DFE) is stable.

Stability of the Endemic Equilibrium Point
We evaluate for the Host and Vector respectively.
From the Human 4-dimensional differential equations we have the Jacobian as: By substituting equation (35) and alternative notations in (49)   satisfied. This is trivial as well and has been verified; hence the endemic equilibrium is stable and we can hence have the Lemma.

Lemma 3:
The endemic equilibrium

Lemma 4:
The positive equilibrium * H E of Human is locally asymptotically stable if > 1.

Proof:
Using the Routh-Hurwitz criterion in the Characteristic polynomial in equation (2), we have that Here, 27 The Inequality (68) will hold if and only if 0 > 1. This Proves the Lemma 4∎ Similarly for the Vector Population (Mosquitoes), considering the Vector system of differential By taking the Jacobian at the Endemic point * = ( * , * , * ) we have; Characteristic equation of the above matrix is given as: It has been clearly verified using Routh-Hurwitz criterion that the system has all its eigen value to be negative if and only if 0 holds. This is very trivial and hence the Endemic equilibrium is stable.

Lemma 5:
The positive endemic equilibrium * V E of the vector (mosquito) population is locally

Proof:
By considering the condition for Routh-Hurwitz stability The inequality (5) holds if and only if 0 > 1.
This completes the proof and hence proves the Lemma 5∎

Theorem 3:
The Endemic equilibrium This proves the Theorem 3.

Transmission by Variational Iteration Method (VIM)
The implementation of semi-analytical algorithms or methods has spontaneously developed over the years in the field of numerical analysis and computational mathematics.

Basic Idea of VIM
Consider a non-linear differential equation N is the nonlinear operator and ) (t g is a known analytic function. We construct a correction functional for the equation (1) which is given as: Where  is a lagragian multiplier which can be obtained optimally and expressed as: Where n is the highest order of the differential equation.

Solution of the Model using Variational Iteration Method
We consider the model's system of equations in (1), Subject to the initial condition (State Variables) Adopted from [1]. We have,

VIM Application to Model Equations
We have the correction functional of the system of equation governing this malaria model as: Subject to the initial conditions By putting the model parameters and the value of the lagrangian multiplier in eq. (72); we obtain an iteration formula for the seven (7) compartments as:

Interpretation of Results
The Results obtained from the Numerical simulation and the stability analysis of the proposed SEIR-SEI Vector-host Malaria Transmission model shows that the Disease-free equilibrium is stable when a mosquito doesn't infect more than one individual. That is, when 0 does not exceed unity ( 0 < 1); and moment the converse happens (when 0 > 1) then there is an epidemic outbreak (endemic equilibrium).
Furthermore, the population of the susceptible human undergoes an exponential growth pattern, and the graph is perfectly linear.
The Population of the exposed and the infected human decays (decreases) in a logistic manner and hence a logistic decay which implies that there is a possibility of zero population (At DFE).
While the Recovered Human, Susceptible, Exposed and infected mosquito follows a logistic growth pattern.
All these imply that the populations in the system are prone to the infection since the population in the susceptible compartment is increasing.
As a result, serious attention should be focused on this compartment as regards an appropriate intervention strategy to combat the contact of malaria infection at a time when the infected and the exposed population is stable (when 0 < 1).

Discussion of Results
A SEIR-SEI Vector-Host model of malaria transmission built on 7-dimensional system of ordinary differential was analyzed and solved numerically using the Variational iteration method Having used our proposed method, it is now crystal clear that the variational iteration method is suitable, perfect and efficient in conducting and conveying analysis on Malaria models.

Conclusion
We have studied a nonlinear 7-dimensional ordinary differential equation that describes the transmission dynamics of malaria by carrying asymptotic stability analyses at the malaria-free and endemic equilibriums using the Gaussian elimination method and the routh-hurwitz criterion with three (3) theorems and three (3) lemmas. We also have solved the model using the variational iteration method (VIM). This method is unprecedented.
From all these, we have come to a conclusion that the variational is an efficient alternative for conducting analyses on malaria models and other epidemiological models and as a matter of fact can be implemented in the classroom to provide solutions to a wider class of ordinary differential equations and epidemiological models.

Declaration of Interest
Authors have declared that there is no conflict of interest relevant to this research