1 Introduction

Many results on the epidemic virus transmission have been studied by the dynamical analytic theory^[1-7]. There is a model of nonlinear differential system for its basic physical phenomena. Thus, the solving method of the nonlinear epidemic virus transmission population dynamic system is an important field. From the solution, we need to research its behaviors.

The nonlinear differential system is a very attractive investigated subject^[8-9]. Many approximate methods have been developed, such as the boundary layer method, the matched asymptotic method, and the multiple scales method. Many scholars have done a great deal of work^[10-13]. The researchers also studied a class of ecological environment system, the human immunodeficiency virus (HIV) transmission, the reaction diffusion system, and so on^[14-24]. In this paper, using a special and valid functional homotopy analysis method^[25-27], we study a class of epidemic virus transmission population dynamic system.

2 Epidemic virus transmission population dynamic system

The following is a class of epidemic virus transmission population dynamic system^[6-7]:

(1)

(2)

(3)

where x(_t) and y(_t) are the numbers of sufferer population and susceptible population in the broken area, respectively, t is the time, a, b, c, A, and B are constants, axy is the increased rate for sufferers, -bx is the decrement rate of death in Eq.(1), -axy is the decrement growth rate of susceptible population since taking the measure of epidemic prevention, -cx²y denotes the decrement rate of susceptible population since taking the measure of epidemic prevention, dx is the growth rate of infected population when they manifold in Eq.(2), and αf(x, y) and βg(x, y) are disturbed terms for the numbers of sufferer population and susceptible population, respectively.

Generally speaking, from the nonlinear epidemic virus transmission population dynamic system, we cannot find the exact solution by using the finite terms for the elementary functions. Thus, we use the homotopic mapping method and solve approximate analytic solutions for the epidemic virus transmission population dynamic system (1)-(3).

3 Functional homotopic mapping

We first introduce the functional mappings H_i [x, y, s] (i=1, 2) on ,

(4)

(5)

where s is a factitious parameter, and x and y are initial functions for the homotopic mappings. The linear operators L_i (i=1, 2) are

Obviously, from the homotopic mappings (4) and (5), H_i(x, y, 1)=0 (i=1, 2), and the epidemic virus transmission population dynamic systems (1) and (2) are the same. Thus, the solution (X(t, 1), Y(t, 1)) to H_i (x, y, s) =0(i=1, 2) as s→ 1 satisfying the condition (3) is the exact solution to the epidemic virus transmission population dynamic system (1)-(3).

Select the initial functions x and y which are the solutions to the following system:

(6)

(7)

with the initial condition (3).

It is easy to see that x and y are

(8)

(9)

The above x is denoted by Eq.(8).

Let

(10)

From the homotopic mappings (4) and (5), substituting Eq.(10) into Eqs.(4) and (5), developing the nonlinear terms in s, and equating the coefficients of the same power of s for H_i (x, y, s)=0(i=1, 2), from the coefficients in s⁰ for H_i (x, y, s)=0(i=1, 2), we have

(11)

(12)

Thus, the solutions to the system (11) and (12) are

From Eqs.(8) and (9), we have

(13)

(14)

where x₀ is denoted by Eq.(13).

Substituting Eq.(10) into Eqs.(4) and (5), from the coefficients in s¹ for H_i (x, y, s)=0(i=1, 2), we have

(15)

(16)

The solutions (x₁(t), y₁(t)) to the system (15) and (16) with the zero initial condition are

(17)

(18)

Then, from Eqs.(10), (13), (14), (17), and (18) and taking s=1, we obtain the first approximate solution (X₁ (t), Y₁(t)) to the epidemic virus transmission population dynamic system (1)-(3),

(19)

(20)

where x₀ and y₀ are denoted by Eqs.(13) and (14), respectively.

Substituting Eq.(10) into Eqs.(4) and (5), from the coefficients in s² for H_i (x, y, s)=0(i=1, 2), we have

(21)

(22)

The solutions (x₂ (t), y₂ (t)) to the system (21) and (22) with the zero initial condition (3) are

(23)

(24)

From Eqs.(19), (20), and (7) and taking s=1, we obtain the second approximate solution (X₂ (t), Y₂ (t)) to the epidemic virus transmission population dynamic system (1)-(3),

(25)

(26)

where (x₀, y₀) and (x₁, y₁) are denoted by Eqs.(13), (14), (17), and (18), respectively.

Using the same method, substituting Eq.(10) into the functional homotopic mapping Eqs.(4) and (5) and from the coefficients of sⁱ(i=3, 4, …) for H_i (x, y, s)=0(i=1, 2), we can obtain x_i, y_i (i=3, 4, …).

Thus, we can obtain the nth(n=1, 2, …) approximate solutions X_n (t), Y_n (t)(n=1, 2, …) successively,

where

4 Accuracy of approximate solutions

In order to show the accuracy of the approximate solutions, we compare with the special cases. For simplicity, we set

where ε ≥ 0 is a small parameter, and take the non-dimensional functions

and the constants

The epidemic virus transmission population dynamic system is

(27)

(28)

(29)

Then, from Eq.(10), as c=0, we have

(30)

(31)

From Eqs.(30) and (31), we can obtain the first approximate solution (X₁ (t), Y₁ (t)) to the epidemic virus transmission population dynamic system (27)-(28) using the method of functional homotopic mapping,

(32)

(33)

where x₀ and y₀ are denoted by Eqs.(30) and (31), respectively.

As a reduced case from Eq.(32), we can illustrate the first-order approximate solution X₁ (t) and the exact solution X(t) (see Fig. 1).

Similarly, from Eq.(33), we can obtain the first-order approximate solution Y₁ (t) and the exact solution Y(t).

Using the functional fixed point theory of functional analysis, we can prove that^{[9, 27]} from Eq.(10), the homotopic mapping solutions

are uniformly convergent on t∈[0, T₀], where T₀ is a large enough constant. Thus, the nth(n=1, 2, …) order analytic solutions X_n (t), Y_n (t)(n=1, 2, …) are approximate to the exact analytic solution X(t), Y(t) increasingly.

For example, as ε =0, 05, from Eqs.(32) and (33), we can obtain the first-order approximate analytic solution X₁ (t) and the exact analytic solution X(t) (see Fig. 2).

Moreover, using the perturbation theory^[8-9], the nth-order approximate analytic solution (X_n, Y_n) to the epidemic virus transmission population dynamic system (27)-(29) has the following asymptotic behaviors:

where (X, Y) is the exact analytic solution to the system (27)-(29).

Therefore, we can see that the approximate analytic solution to the virus transmission population dynamic system possesses good accuracy using the functional mapping method.

5 Discussion

We can obtain the simulated curves for the approximate analytic solution and take measure to control the epidemic virus transmission population.

(ⅰ) Change the parameter a

For simplicity, taking the non-dimensional parameters b=c=d=A=B =1, ε =0.05, and the non-dimensional functions f=sin x, g=cos y, we compare the results with a=2 and a=3. Then, the curves of the first-order approximate analytic solution X₁ can be seen in Fig. 3.

From Fig. 3, we can see that the first-order approximate analytic function X₁ (t) of the sufferer population decreases with a=2 faster than that with a=3.

Similarly, we can see that the first-order approximate solution Y₁ (t) of the sufferer population decreases with a=2, which is much faster than that with a=3.

(ⅱ) Change the parameter c

For example, taking the non-dimensional parameters a=2, b=d=A= B=1, ε =0.05, and the non-dimensional functions f=sin x, g=cos y, we compare the results with c=1 and c=10. Then, the curves of the first-order approximate analytic solution X₁ can be seen in Fig. 4.

From Fig. 4, we can see that the first-order approximate analytic function X₁ (t) of sufferer population decreases as c=1, which is nearly equal to that of c=10.

(ⅲ) Change the functions f(u, v) and g(u, v)

For example, taking the non-dimensional parameters a=2, b=c=d=A= B=1, ε =0.05, we compare the results with f=sin x, g=cos y and f=exp (-y), g=exp (-x). Then, the first-order approximate analytic solution X₁ can be seen in Fig. 5.

From Fig. 5, we can see that the first-order approximate analytic function X₁ (t) of sufferer population decreases as f=sin x, g=cos y and f=exp(-y), g=exp(-x) and is steady (similarly, we can see that the first-order approximate analytic function Y₁ (t) of sufferer population decreases as f=sin x, g=cos y and f=exp(-y), g=exp(-x) and is steady). However, due to complexity for the disturbed terms, the behaviors for the first-order approximate analytic function X₁ (t) and Y₁(t) have various changes.

Of course, it is better to use the higher-order approximate analytic solution to study the behavior of the epidemic virus transmission population dynamic system.

6 Conclusions

The epidemic virus transmission is a complicated phenomenon. Hence, we need to reduce to the basic model and solve it using the approximate method. The functional homotopic mapping method is a simple and valid method.

The functional homotopic mapping is an approximate analytic method, which differs from the general numerical method. From the approximate analytic expansions of solution to the epidemic virus transmission population dynamic system, using the homotopic mapping method, we can also execute the continuous analytic operations, such as the differential operation and the integral operation. Thus, we can further study the qualitative and quantitative behaviors of the human group for the infected and the susceptible population in the broken area.