# Asymptotic Solution for a Class of Epidemic Contagion Ecological Nonlinear Systems

Authors: Yi-Hu Feng; Lei Hou
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

In this paper, a class of systems for epidemic contagion is considered. An epidemic virus ecological model is described. Using the generalized variation iteration method, the corresponding approximate solution to the nonlinear system is obtained and the method for this approximate solution is pointed out. The accuracy of approximate solution is discussed, and it can control the epidemic virus transmission by using the parameters of the system. Thus, it has the value for practical application.


---

## Body

## 1. Introduction

Many results on epidemic virus transmission have been studied in the medical science and ecological environment [1–4]. There is a model of the nonlinear differential system for its basic physical phenomena. Currently, the solving method of the nonlinear epidemic virus population system is an important field [5–8].The nonlinear differential system is an attractive investigated subject. Adams et al. considered the possible “shock” patterns that can exist in the solution to a singular perturbed, third-order nonlinear ordinary differential equation arising as the travelling-wave reduction of the Kuramoto–Sivashinsky equation. In particular, the existence (or otherwise) of oscillatory shocks and multiple shocks made up of combinations of oscillatory and monotonic shocks is examined, using an optimal truncation strategy to track crucial exponential small terms lying beyond all orders of the (divergent) algebraic expansion. The results provide further understanding of numerical solutions previously obtained by others, as well as giving a methodology which is applicable to much broader classes of differential equations exhibiting multiscale phenomena; they also afford the same new insight into the multiscales technique [9]. Bell and Deng studied singular perturbation of N-front traveling waves in the Fitzhugh–Nagumo equations [10]. Many approximate methods have been developed, such as the boundary layer method, the matched asymptotic method, and the multiple scales method. Many scholars have performed a great deal of work [11–21]. The authors also studied a class of ecological environment systems, such as HIV transmission, the solitary wave, the reaction diffusion system, and the model of prey and predator [22–28].He proposed a novel variational approach for limit cycles of a kind of nonlinear oscillators. Some examples are given to illustrate the effectiveness and convenience of the method. The obtained results are valid for the whole solution domain with high accuracy [29]. Tao et al. discussed the variational approach and the Hamiltonian approach to nonlinear oscillators systematically. They gave an extension of the methods to a singular nonlinear oscillator by the iteration perturbation method [30]. In this paper, using the generalized variational iteration method, we study a class of epidemic virus population systems.The rest of this paper is organized as follows. In Section2, we studied a class of epidemic virus transmission population systems and then introduced the names of the variables in the equation and the boundary conditions. In Section 3, we constructed the generalized variation iterations, obtained the first approximations, used the same method, from the generalized variation iterations and the variation principle, and obtained higher-order approximate solution sequences to the system, and these sequences converge uniformly. In Section 4, we gave an example, using the generalized variation iteration method and we obtained the approximate solution to the system. In Section 5, we solved asymptotic expansions to the epidemic virus transmission population dynamic system by using the perturbation method. Comparing the obtained approximate solutions by using two methods, we see that the approximate solution to the epidemic virus transmission population dynamic system by using the method of the generalized functional variational iteration possesses a good accuracy. In Section 6, we considered the virus transmission parameters, obtained the simulated curves for the approximate solution, and took measures to control the virus transmission. In Section 7, we obtained a conclusion, simplified the basic model, and solved it by using the approximate method. The method of the generalized variational iteration is simple and valid.


## 2. Epidemic Virus Population System

The following is a class of epidemic virus transmission population systems [5, 6]:(1)dSdt−α+βKSI+aS=0,(2)dIdt−βKSI+a+b+cI−dR=0,(3)dRdt−γKSI+a+e+dR=0,where S, I, and R are the numbers of the sufferer population, susceptible population, and potential population in the broken area, respectively, t is the time, α is an immigrate rate of the sufferer population, a is a\the natural death rate, c and e are the death rate of susceptible and potential populations, respectively, b is the change rate of the sufferer population changed into the potential population, and d is the lost rate of immunity in potential population. The abovementioned e is a nonnegative constant, other parameters are positive constants, a+b+c>d, and βKS and γKS are nonlinear infectiousness for the sufferer population and potential population, respectively. Also, KS is a sufficiently smooth function in −∞,∞. Specially, nonlinear function KS possesses the following bounds:(4)aSb,b>0;bSna+Sn,a,b>0,n≥1;a1−exp−bS,a,b>0;aS1+bS+1+2bS,a,b>0;1−exp−bS1+aexp−bS,a,b>0,and so on.


## 3. Generalized Variation Iteration System

In order to obtain the approximate solution to systems (1)–(3), we introduce a set of functional Fii=1,2,3 [27, 28]:(5)F1=S−∫0tλ1τdSdτ−α+βKS˜I˜+aSdτ,(6)F2=I−∫0tλ2τdIdτ−βKS˜I˜+a+b+cI−dR˜dτ,(7)F3=R−∫0tλ3τdRdτ−γKS˜I˜+a+d+eRdτ,where S˜,I˜, and R˜ are the restricted variables of S,I, and R, respectively, while λii=1,2,3 are the Lagrange multiplicators.Compute the variationsδFii=1,2,3 of functional and equate to zero as follows:(8)δF1=δS−λ1δS|τ=t+∫0t∂λ1∂τ+aλ1δSdτ=0,(9)δF2=δI−λ2δI|τ=t+∫0t∂λ2∂τ+a+b+cλ2δIdτ=0,(10)δF3=δR−λ3δR|τ=t+∫0t∂λ3∂τ+a+d+eλ3δRdτ=0.From equations (8)–(10), we have(11)∂λ1∂τ+aλ1=0,∂λ2∂τ+a+b+cλ2=0,∂λ3∂τ+a+d+eλ3=0,(12)λiτ|τ=t=1,i=1,2,3.From problems (11) and (12), we obtain(13)λ1t=expat−τ,(14)λ2t=expa+b+ct−τ,(15)λ3t=expa+d+et−τ.From equations (5)–(7) and (13)–(15), we can construct the following generalized variation iterations:(16)Sn+1=Sn−∫0texpat−τdSndτ−α+βKSnIn+aSndτ,n=0,1,…,(17)In+1=In−∫0texpa+b+ct−τdIndτ−βKSnIn+a+b+cIn−dRndτ,n=0,1,…,(18)Rn+1=Rn−∫0texpa+d+et−τdRndτ−γKSnIn+a+d+eRndτ,n=0,1,⋯.First, we select a set of initial approximationsS0t,I0t,R0t. Also, it is a set of solutions to the linear system as follows:(19)dS0dt=α−aS0,(20)dI0dt=−a+b+c−dI0+dR0,(21)dR0dt=−a+d+eR0.It is easy to obtain a set of solutions to systems (19)–(21), so we have(22)S0t=C10exp−at+αt,(23)I0t=C20exp−a+b+c−dt−dC30a+d+eexp−a+d+et,(24)R0t=C30exp−a+d+et,where Ci0i=1,2,3 are the arbitrary constants, which can be decided by the conditions of systems (1)–(3).From equations (16)–(18), we can obtain the first approximations:(25)S1t=C10exp−at+αt−β∫0texpat−τKC10exp−aτ+ατ×C20exp−a+b+c−dτ−dC30a+d+eexp−a+d+eτdτ,(26)I1t=C20exp−a+b+c−dt−dC30a+d+eexp−a+d+et+β∫0texpa+b+ct−τKC10exp−aτ+ατ×C20exp−a+b+c−dτ−dC30a+d+eexp−a+d+eτdτ,(27)R1t=C30exp−a+d+et+γ∫0texpa+d+et−τKC10exp−aτ+ατ×C20exp−a+b+c−dτ−dC30a+d+eexp−a+d+eτdτ.Using the same method, from generalized variation iterations (16)–(18), we can obtain higher-order approximate solution to systems (1)–(3).From the generalized variation iterations (16)–(18) and the variation principle, we can obtain sequences Snt,Int, and Rnt. Also, these sequences converge uniformly [31, 32].Let(28)St=limn⟶∞Snt,It=limn⟶∞Int,Rt=limn⟶∞Rnt.Thus, we have the following theorem.Theorem 1.
IfKS is a sufficiently smooth bounded function in −M,M, where M is a sufficiently large constant, and(29)aSb,b>0,bSna+Sn,a,b>0,n≥1,a1−exp−bS,a,b>0,aS1+bS+1+2bS,a,b>0,1−exp−bS1+aexp−bS,a,b>0,then St,It,Rt is the exact solution to systems (1)–(3) in −M,M.


## 4. Example

For convenience, from (1)–(3), we set that a=b=c=d=α=δ=1,e=0,0<β=γ=ε≪1,andKS=S. Thus, we consider the following example:(30)dSdt−1+S+εIS=0,(31)dIdt+3I−R−εIS=0,(32)dRdt+2R−εIS=0.Using the generalized variation iteration method, we can obtain the approximate solution to systems (30)–(32).From equations (22)–(24), we have the initial approximate solution S0t,I0t,R0t to systems (30)–(32). Also, we have(33)S0t=exp−t+t,(34)I0t=12exp−2t,(35)R0t=exp−2t.where we can take Ci0=1,i=1,2,3.From equations (25)–(27) and (33) and (34), we can obtain the first approximations:(36)S1t=exp−t+t−ε∫0texpt−τexp−τ+τ12exp−2τ+1dτ,(37)I1t=12exp−2t+ε∫0texp3t−τexp−τ+τ12exp−2τ+1dτ,(38)R1t=exp−2t+ε∫0texp2t−τexp−τ+τ12exp−2τ+1dτ.Using the same method, from generalized variation iterations (33)–(35), we can obtain the higher-order approximate solution Snt,Int,Rntn=2,3,⋯ to systems (5)–(7), and their constructions are omitted.


## 5. Accuracy of Approximate Solution

Now, we assume the absolute nondimensional parameters to systems (30)–(32). From Figures 1–3, we show the simulated curves: the real curve is the exact solution St,It,Rt and the approximation curve is the first approximation solution S1t,I1t,R1t.Figure 1
Illustration for the curves ofSt (real curve) and S1t (approximation curve).Figure 2
Illustration for the curves ofIt (real curve) and I1t (approximation curve).Figure 3
Illustration for the curves ofRt (real curve) and R1t (approximation curve).On the other hand, we can also solve asymptotic expansions to epidemic virus transmission population dynamic systems (30)–(32) by using the perturbation method.Let the solutionS¯,I¯,R¯ to systems (30)–(32) have the following form:(39)S¯=∑n=0∞S¯ntεn,I¯=∑n=0∞I¯ntεn,R¯=∑n=0∞R¯ntεn,0<ε≪1.Substituting equation (39) into systems (30)–(32), developing nonlinear terms in ε, equating coefficients of the same powers of ε, and letting initial values be S0,I0,R0=1,1,1 for systems (30)–(32), we can obtain S¯n,I¯n,R¯nn=0,1,⋯ successively. For n=0, we have(40)S¯0t=exp−t+t,(41)I¯0t=12exp−2t,(42)R¯0t=exp−2t.Substituting equations (39)–(42) into expansions (30)–(32), we obtain the first-order perturbed solution S¯1app,I¯1app,R¯1app to systems (30)–(32):(43)S¯1appt=exp−t+t−ε∫0texpt−τexp−τ+τ12exp−2τ+1dτ+Oε2,0<ε≪1,(44)I¯1appt=exp−2t−12exp−2t+ε∫0texp3t−τexp−τ+τ×12exp−2τ+1dτ+Oε2,0<ε≪1,(45)R¯1appt=exp−2t+ε∫0texp2t−τexp−τ+τ×12exp−2τ+1dτ+Oε2,0<ε≪1.Comparing obtained approximate solutions (36)–(38) with (43)–(45) by using the above two methods, respectively, we observe that they are the same for the first-order precision Oε,0<ε≪1. Therefore, we can see that the approximate solution S¯t,I¯t,R¯t to epidemic virus transmission population dynamic systems (30)–(32) by using the method of the generalized functional variational iteration possesses a good accuracy.


## 6. Change Virus Transmission Parameters

We can obtain the simulated curves for the approximate solution and take measures to control the virus transmission.Now, we assume the absolute nondimensional parameters of systems (25)–(27) as follows:(1)
Take the virus transmission parameters asa=b=c=1,α=d=e=0, β=γ=0.5 or β=γ=5, and KS=S, respectively. We can obtain the simulated curves St,It, and Rt (see Figures 4–6). Also, we show the simulated curves of St,It, and Rt, respectively, where the real curve is the case as β=γ=0.5 and the ′+′ curve is the case as β=γ=5.(2)
Take the virus transmission parameters asa=b=c=1,α=d=e=0,β=γ=5, and KS=S or KS=S5, respectively. We can obtain the simulated curves St,It, and Rt (see Figures 7–9). Also, we show the simulated curves of St,It, and Rt, respectively, where the real curve is the case as KS=S and the ′+′ curve is the case as KS=S5.Figure 4
Illustration for the curve ofSt (real curve as β=γ=0.5 and ′+′ curve as β=γ=5).Figure 5
Illustration for the curve ofIt (real curve as β=γ=0.5 and ‘+' curve as β=γ=5).Figure 6
Illustration for the curve ofRt (real curve as β=γ=0.5 and ‘+’ curve as β=γ=5).Figure 7
Illustration for the curve ofSt (real curve as KS=S, and ‘o' curve as KS=S5).Figure 8
Illustration for the curve ofIt (real curve as KS=S, and ‘o’ curve as KS=S5).Figure 9
Illustration for the curve ofRt (real curve as KS=S, and 'o' curve as KS=S5).Thus, from Figures4–9, we can use the parameters to control the virus transmission.


## 7. Conclusions

The epidemic virus transmission is a complicated phenomenon. Hence, we need to simplify the basic model and solve it by using the approximate method. The method of the generalized variational iteration is simple and valid.The generalized variational iteration is an approximate method, which differs from the general numerical method. The expansion of the solution can also be used analytically. Thus, we can further study the qualitative and quantitative behaviors of the epidemic virus transmission population for the infected and the susceptible populations in the broken area.



---

*Source: 8498190-2020-05-23.xml*