Conditions | E0 | E1 | E∗ |
R10<1 and R20<1 | Yes (GAS) | No | No |
R10<1 and R20>1 | Yes (Unstable) | Yes (GAS) | No |
R10>1 and R20<1 | Yes (Unstable) | No | Yes (GAS) |
R10>1 and R20>1 | Yes (Unstable) | Yes (Unstable) | Yes (GAS) |
In this study, we constructed the credit-scoring model of P2P loans by using several machine learning and artificial neural network (ANN) methods, including logistic regression (LR), a support vector machine, a decision tree, random forest, XGBoost, LightGBM and 2-layer neural networks. This study explores several hyperparameter settings for each method by performing a grid search and cross-validation to get the most suitable credit-scoring model in terms of training time and test performance. In this study, we get and clean the open P2P loan data from Lending Club with feature engineering concepts. In order to find significant default factors, we used an XGBoost method to pre-train all data and get the feature importance. The 16 selected features can provide economic implications for research about default prediction in P2P loans. Besides, the empirical result shows that gradient-boosting decision tree methods, including XGBoost and LightGBM, outperform ANN and LR methods, which are commonly used for traditional credit scoring. Among all of the methods, XGBoost performed the best.
Citation: An-Hsing Chang, Li-Kai Yang, Rua-Huan Tsaih, Shih-Kuei Lin. Machine learning and artificial neural networks to construct P2P lending credit-scoring model: A case using Lending Club data[J]. Quantitative Finance and Economics, 2022, 6(2): 303-325. doi: 10.3934/QFE.2022013
[1] | Yan’e Wang , Zhiguo Wang, Chengxia Lei . Asymptotic profile of endemic equilibrium to a diffusive epidemic model with saturated incidence rate. Mathematical Biosciences and Engineering, 2019, 16(5): 3885-3913. doi: 10.3934/mbe.2019192 |
[2] | Yoichi Enatsu, Yukihiko Nakata . Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate. Mathematical Biosciences and Engineering, 2014, 11(4): 785-805. doi: 10.3934/mbe.2014.11.785 |
[3] | Pan Yang, Jianwen Feng, Xinchu Fu . Cluster collective behaviors via feedback pinning control induced by epidemic spread in a patchy population with dispersal. Mathematical Biosciences and Engineering, 2020, 17(5): 4718-4746. doi: 10.3934/mbe.2020259 |
[4] | Jummy F. David, Sarafa A. Iyaniwura, Michael J. Ward, Fred Brauer . A novel approach to modelling the spatial spread of airborne diseases: an epidemic model with indirect transmission. Mathematical Biosciences and Engineering, 2020, 17(4): 3294-3328. doi: 10.3934/mbe.2020188 |
[5] | Chang-Yuan Cheng, Shyan-Shiou Chen, Xingfu Zou . On an age structured population model with density-dependent dispersals between two patches. Mathematical Biosciences and Engineering, 2019, 16(5): 4976-4998. doi: 10.3934/mbe.2019251 |
[6] | Yicang Zhou, Zhien Ma . Global stability of a class of discrete age-structured SIS models with immigration. Mathematical Biosciences and Engineering, 2009, 6(2): 409-425. doi: 10.3934/mbe.2009.6.409 |
[7] | Zongwei Ma, Hongying Shu . Viral infection dynamics in a spatial heterogeneous environment with cell-free and cell-to-cell transmissions. Mathematical Biosciences and Engineering, 2020, 17(3): 2569-2591. doi: 10.3934/mbe.2020141 |
[8] | Yujie Sheng, Jing-An Cui, Songbai Guo . The modeling and analysis of the COVID-19 pandemic with vaccination and isolation: a case study of Italy. Mathematical Biosciences and Engineering, 2023, 20(3): 5966-5992. doi: 10.3934/mbe.2023258 |
[9] | Zhen Jin, Guiquan Sun, Huaiping Zhu . Epidemic models for complex networks with demographics. Mathematical Biosciences and Engineering, 2014, 11(6): 1295-1317. doi: 10.3934/mbe.2014.11.1295 |
[10] | Qian Yan, Xianning Liu . Dynamics of an epidemic model with general incidence rate dependent on a class of disease-related contact functions. Mathematical Biosciences and Engineering, 2023, 20(12): 20795-20808. doi: 10.3934/mbe.2023920 |
In this study, we constructed the credit-scoring model of P2P loans by using several machine learning and artificial neural network (ANN) methods, including logistic regression (LR), a support vector machine, a decision tree, random forest, XGBoost, LightGBM and 2-layer neural networks. This study explores several hyperparameter settings for each method by performing a grid search and cross-validation to get the most suitable credit-scoring model in terms of training time and test performance. In this study, we get and clean the open P2P loan data from Lending Club with feature engineering concepts. In order to find significant default factors, we used an XGBoost method to pre-train all data and get the feature importance. The 16 selected features can provide economic implications for research about default prediction in P2P loans. Besides, the empirical result shows that gradient-boosting decision tree methods, including XGBoost and LightGBM, outperform ANN and LR methods, which are commonly used for traditional credit scoring. Among all of the methods, XGBoost performed the best.
Population migration is a common phenomenon. With the migration of population, infectious diseases can easily spread from one area to another, so it is meaningful to consider population migration when studying the spread of infectious diseases [1,2,3,4,5,6,7].
Wang and Mulone [2] established an SIS infectious disease model with standard incidence based on two patches. It is proved that the basic reproduction number is the threshold of the uniform persistence and disappearance of the disease. The dispersal rate of the population will make the infectious disease persist or disappear in all patches. There will be no the phenomena that infectious diseases persists in one patch but disappears in the other.
Sun et al. [3] put forward an SIS epidemic model with media effect in a two patches setting. Under the assumption that the migration matrix is irreducible, it is proved that if the basic reproduction number is greater than 1 then the system persists and solutions converge to an endemic equilibrium and that if the basic reproduction number is less than 1 then solutions tend to an equilibrium without disease.
Gao et al. [5,6] studied an SIS multi-patch model with variable transmission coefficients. Their results show that the basic reproduction number R0 is a threshold parameter of the disease dynamics.
All the patch models referenced above assume that the migration matrix is irreducible. The studies in which the migration matrix is reducible are few. Therefore, based on the case of two patches, we consider that the individuals can only migrate from one patch to the other. In this case, the migration matrix is reducible. It can characterize the phenomenon that individuals migrate in one direction between two regions, such as, from the rural patch to the urban one [8] and from a small community hospital to a large teaching hospital [4].
It is well known that the incidence rate plays an important role in the modeling of infectious disease. Considering the saturation phenomenon for numerous infected individuals, Capasso and Serio [9] first introduce a nonlinear bounded function g(I) to form the interaction term g(I)S in 1978. It can characterize the behavioral changes of individuals, such as wearing masks or reducing their social activities and direct contact with others with the increase of infectious individuals. After that, the saturation incidence rate has attracted much attention and various nonlinear types of incidence rate are employed. The most commonly used types are Holling type Ⅱ λSI1+αI [10,11,12] and βSI1+αS [13], Monod-Haldane type λSI1+αI2 [14], Beddington-DeAngelis type λSI1+αS+βI [15,16,17] and Crowley-Martin type λSI(1+αS)(1+βI) [18,19].
In this paper, we consider infectious disease transmission models with saturation incidence rate. The rest of this paper is organized as follows: In Section 2, we establish a two-patch SIS model with saturating contact rate and one-directing population dispersal. We discuss the existence of disease-free equilibrium, boundary equilibrium and endemic equilibrium and prove the global asymptotic stability of the equilibriums in Section 3. In Section 4, we perform simulations to illustrate the results and analyze the effect of the contact rate and population migration on epidemic transmission. Finally, we discuss in Section 5.
In the two patches, the population is divided into two states: susceptible and infective. Thus we can establish a two-patch SIS model with saturating contact rate and one-directing population dispersal
{dS1(t)dt=A1−d1S1−β1S1I11+α1I1−mS1+γ1I1,dS2(t)dt=A2−d2S2−β2S2I21+α2I2+mS1+γ2I2,dI1(t)dt=β1S1I11+α1I1−d1I1−mI1−γ1I1,dI2(t)dt=β2S2I21+α2I2−d2I2+mI1−γ2I2, | (2.1) |
where Si is the number of susceptible population in patch i (i=1,2), Ii is the number of infective population in patch i (i=1,2), Ai is the recruitment into patch i (i=1,2), di is the natural mortality rate, γi is the recovery rate of an infective individual in patch i (i=1,2), m is the migration rate form patch 1 to patch 2. Since the individuals can migrate from the first patch to the second, patch 1 is the source patch and patch 2 is the sink patch. The initial conditions is
Si(0)>0, Ii(0)≥0, i=1,2, I1(0)+I2(0)>0. | (2.2) |
Denote the population in patch i by Ni. Then Ni=Si+Ii. From system (2.1), the differential equations governing the evolution of N1 and N2 are
{dN1(t)dt=A1−(d1+m)N1,dN2(t)dt=A2−d2N2+mN1. | (2.3) |
Obviously, system (2.3) has a unique equilibrium (N∗1,N∗2)=(A1d1+m,A2d2+mA1d2(d1+m)) which is globally asymptotically stable for (2.3). So (2.1) is equivalent the following system
{dN1(t)dt=A1−(d1+−m)N1,dN2(t)dt=A2−d2N2+mN1,dI1(t)dt=β1(N1−I1)I11+α1I1−d1I1−mI1−γ1I1,dI2(t)dt=β2(N2−I2)I21+α2I2−d2I2+mI1−γ2I2. | (2.4) |
Because limt→∞Ni(t)→N∗i (i=1,2), system (2.4) leads to the following limit system
{dI1(t)dt=β1(N∗1−I1)I11+α1I1−d1I1−mI1−γ1I1,dI2(t)dt=β2(N∗2−I2)I21+α2I2−d2I2+mI1−γ2I2. | (2.5) |
Let Ω={(I1,I2)|0≤I1≤N∗1,0≤I2≤N∗2}. Then Ω is invariant region for system (2.5).
Define the basic reproduction number in the two patches respectively by R10=β1A1(d1+γ1+m)2=β1d1+γ1+mN∗1, R20=β2(d2+γ2)N∗2. The basic reproduction number R10 gives the expected secondary infections in the source patch produced by a primary infected individual in the source patch when the population is supposed to be in the disease-free equilibrium. The basic reproduction number R20 gives the expected secondary infections in the sink patch produced by a primary infected individual in the sink patch when the population is supposed to be in the disease-free equilibrium. Then we have the following theorem.
Theorem 3.1. For the system (2.5), we have
(i) The disease-free equilibrium E0:=(0,0) always exists;
(ii) The boundary equilibrium E1:=(0,β2N∗2−d2−γ2(d2+γ2)α2+β2) exists if R20>1;
(iii) There is a unique epidemic equilibrium E∗ if R10>1.
Proof. (ⅰ) can be easily proved.
Let
β1(N∗1−I1)I11+α1I1−d1I1−mI1−γ1I1=0, | (3.1) |
β2(N∗2−I2)I21+α2I2−d2I2+mI1−γ2I2=0. | (3.2) |
From Eq (3.1), we can have I1=0 always satisfies Eq (3.1). When I1=0, from Eq (3.2), we have
I2=β2N∗2−d2−γ2(d2+γ2)α2+β2. |
If R20>1, then I2=β2N∗2−d2−γ2(d2+γ2)α2+β2>0. So The boundary equilibrium E1:=(0,β2N∗2−d2−γ2(d2+γ2)α2+β2) exists if R20>1. The conclusion (ⅱ) is proved.
If R10>1, Eq (3.1) has a positive solution I∗1=β1N∗1−(d1+m+γ1)(d1+m+γ1)α1+β1. Solve Eq (3.2), we have
I2=(β2N∗2−d2−γ2+mα2I1)±√(β2N∗2−d2−γ2+mα2I1)2+4[(d2+γ2)α2+β2]mI12[(d2+γ2)α2+β2]. | (3.3) |
Substituting I∗1 into Eq (3.3), we have
I∗2=(β2N∗2−d2−γ2+mα2I∗1)±√(β2N∗2−d2−γ2+mα2I∗1)2+4[(d2+γ2)α2+β2]mI∗12[(d2+γ2)α2+β2]. |
Since I∗2≥0 is meaning only, we take
I∗2=(β2N∗2−d2−γ2+mα2I∗1)+√(β2N∗2−d2−γ2+mα2I∗1)2+4[(d2+γ2)α2+β2]mI∗12[(d2+γ2)α2+β2]. |
So if R10>1, there is a unique epidemic equilibrium E∗=(I∗1,I∗2), where I∗1=β1N∗1−(d1+m+γ1)(d1+m+γ1)α1+β1 and I∗2=(β2N∗2−d2−γ2+mα2I∗1)+√(β2N∗2−d2−γ2+mα2I∗1)2+4[(d2+γ2)α2+β2]mI∗12[(d2+γ2)α2+β2]. The conclusion (ⅲ) is proved.
This completes the proof of the theorem.
From the above analysis, we have the following theorem.
Theorem 3.2. For the system (2.5), we have
(i) If R10<1 and R20<1, there is the disease-free equilibrium E0 only;
(ii) If R10<1 and R20>1, there are the disease-free equilibrium E0 and the boundary equilibrium E1;
(iii) If R10>1 and R20<1, there are the disease-free equilibrium E0 and the epidemic equilibrium E∗;
(iv) If R10>1 and R20>1, there are the disease-free equilibrium E0, the boundary equilibrium E1 and the epidemic equilibrium E∗.
Remark 3.1. Define the basic reproduction number R0 of the system (2.5) by the spectral radius of the next generation matrix [20], we have
R0=ρ(β1d1+γ1+mN∗10−mβ2N∗2(d2+γ2)(d1+γ1+m)β2(d2+γ2)N∗2), |
where ρ(A) denotes the spectral radius of a matrix A. So from the above analysis, we know that R0=max{R10,R20}.
The next, we shall discuss the local stability of the disease-free equilibrium firstly. Then we discuss the global asymptotical stability.
Theorem 3.3. For the system (2.5), we have
(i) If R10<1 and R20<1, the disease-free equilibrium E0 is locally asymptotically stable;
(ii) If R10>1 or R20>1, the disease-free equilibrium E0 is unstable.
Proof. The linearized system of (2.5) at the equilibrium E0 is
{dI1(t)dt=(β1N∗1−d1−m−γ1)I1,dI2(t)dt=(β2N∗2−d2−γ2)I2+mI1. | (3.4) |
The associated characteristic equation of the linearized system of (3.4) at the equilibrium E0 is
F(λ)=|λ−(β1N∗1−(d1+m+γ1))0−mλ−(β2N∗2−d2−γ2)|=0 | (3.5) |
It is easy to see that the two eigenvalues of characteristic Eq (3.5) are
λ1=β1N∗1−(d1+m+γ1)=(R10−1)(d1+m+γ1) |
and
λ2=β2N∗2−d2−γ2=(R20−1)(d2+γ2). |
So, when R10<1 and R20<1, the disease-free equilibrium E0 is locally asymptotically stable; However, if R10>1 or R20>1, the disease-free equilibrium E0 is unstable.
Remark 3.2. From Theorem 3.3, we know that for the system (2.5), if R0<1 the disease-free equilibrium E0 is locally asymptotically stable; if R0>1, the disease-free equilibrium E0 is unstable.
Theorem 3.4. For the system (2.5), if R10<1 and R20<1, the disease-free equilibrium E0 is globally asymptotically stable.
Proof. Since Ii1+αiIi≤Ii for i=1,2, from system (2.5), we can obtain that
{dI1(t)dt≤(β1N∗1−d1−m−γ1)I1,dI2(t)dt≤(β2N∗2−d2−γ2)I2+mI1. | (3.6) |
Define an auxiliary linear system using the right hand side of (3.6) as follows
{dI1(t)dt=(β1N∗1−d1−m−γ1)I1,dI2(t)dt=(β2N∗2−d2−γ2)I2+mI1. |
It can be rewritten as
(I1I2)′=(β1N∗1−d1−m−γ10mβ2N∗2−d2−γ2)(I1I2). | (3.7) |
if R10<1 and R20<1, we can solve (3.7) and know that limt→∞I1(t)=0 and limt→∞I2(t)=0. By the comparison principle [21], we can conclude that when R10<1 and R20<1, all non-negative solutions of (2.5) satisfy limt→∞Ii(t)=0 for i=1,2. So the disease-free equilibrium E0 is globally asymptotically stable.
In this subsection, we will discuss the local stability of the boundary equilibrium firstly. Then discuss the global asymptotical stability.
Theorem 3.5. For the system (2.5), if R10<1 and R20>1, the boundary equilibrium E1 is globally asymptotically stable.
Proof. The Jacobian matrix at the boundary equilibrium E1 of system (2.5) is
J=(β1N∗1−d1−m−γ10m(d2+γ2)(1−R20)−(β2α2+α22)((d2+γ2)(R20−1)(d2+γ2)α2+β2)2(1+α2(d2+γ2)(R20−1)(d2+γ2)α2+β2)2). |
The two eigenvalues of the Jacobian matrix are
λ1=β1N∗1−(d1+m+γ1)=(R01−1)(d1+m+γ1) |
and
λ2=(d2+γ2)(1−R02)−(β2α2+α22)((d2+γ2)(R2−1)(d2+γ2)α2+β2)2(1+α2(d2+γ2)(R2−1)(d2+γ2)α2+β2)2. |
So, when R10<1 and R20>1, λ1<0 and λ2<0. That is the boundary equilibrium E1 is locally asymptotically stable.
For every (I1(0),I2(0))∈Ω, assume the solution of the system (2.5) with initial value (I1(0),I2(0)) is (I1(t),I2(t)). Since
dI1(t)dt=(β1+(d1+γ1)α1+mα1)(β1N∗1−d1−m−γ1β1+(d1+γ1)α1+mα1−I1)I11+α1I1, |
if R10<1, dI1(t)dt<0, then I1(t) is positive and decreasing and limt→∞I1(t)=0. So for sufficiently small positive number ϵ1, there exists a T, such that I1(T)=ϵ1 and I1(t)<ϵ1 when t>T.
The following, we prove that for any ϵ>0, there exists a T∗>T such that |I2(T∗)−β2N∗2−d2−γ2β2+(d2+γ2)α2|<ϵ. And because E1=(0,β2N∗2−d2−γ2β2+(d2+γ2)α2) is locally asymptotically stable, we have E1 is globally asymptotically stable.
Since
dI2(t)dt=(β2+(d2+γ2)α2)(β2N∗2−d2−γ2β2+(d2+γ2)α2−I2)I21+α2I2+mI1, |
if I2(T)<β2N∗2−d2−γ2β2+(d2+γ2)α2, then dI2(t)dt>0 for t>T. So I2(t) is increasing and there exists T∗1 such that |I2(T∗1)−β2N∗2−d2−γ2β2+(d2+γ2)α2|<ϵ;
if I2(T)>β2N∗2−d2−γ2β2+(d2+γ2)α2, there are two cases:
ⅰ) I2(t) is decreasing for t>T. In this case, there exists T∗2>T, such that |I2(T∗2)−β2N∗2−d2−γ2β2+(d2+γ2)α2|<ϵ;
ⅱ) There exists T1>T, such that dI2(T1)dt>0. That is
(β2+(d2+γ2)α2)(β2N∗2−d2−γ2β2+(d2+γ2)α2−I2(T1))I2(T1)1+α2I2(T1)+mI1(T1)>0. |
Since I1(t)<ϵ1 for t>T, we have
(β2+(d2+γ2)α2)(β2N∗2−d2−γ2β2+(d2+γ2)α2−I2(T1))I2(T1)1+α2I2(T1)+mϵ1>0. |
Since I2(T)>β2N∗2−d2−γ2β2+(d2+γ2)α2, we have
(β2+(d2+γ2)α2)(β2N∗2−d2−γ2β2+(d2+γ2)α2−I2(T1))α2+β2+(d2+γ2)α2β2N∗2−d2−γ2+mϵ1>0. |
So
I2(T1)−β2N∗2−d2−γ2β2+(d2+γ2)α2<α2+β2+(d2+γ2)α2β2N∗2−d2−γ2(d2+γ2)α2+β2mϵ1. |
If only ϵ1<(d2+γ2)α2+β2(α2+β2+(d2+γ2)α2β2N∗2−d2−γ2)mϵ, then |I2(T∗2)−β2N∗2−d2−γ2β2+(d2+γ2)α2|<ϵ. It is completed.
In this subsection, we will discuss the local stability of the epidemic equilibrium firstly and then discuss the global asymptotical stability.
Theorem 3.6. For the system (2.5), if R10>1, the epidemic equilibrium E∗ is locally asymptotically stable.
Proof. The Jacobian matrix at the epidemic equilibrium E∗ of system (2.5) is
J=(−β1I1∗1+α1I∗1+β1(N∗1−I∗1)(1+α1I∗1)2−d1−m−γ10mβ2N∗2−2β2I∗2−2β2α2I∗22−(d2+γ2)(1+α2I∗2)2(1+α2I∗2)2)=((d1+m+γ1)1−R10−(β1α1d1+m+γ1+α21)I∗21(1+α1I∗1)20mβ2N∗2−2β2I∗2−2β2α2I∗22−(d2+γ2)(1+α2I∗2)2(1+α2I∗2)2). |
The two eigenvalues of the Jacobian matrix are
λ1=(d1+m+γ1)1−R10−(β1α1d1+m+γ1+α21)I∗21(1+α1I∗1)2 |
and
λ2=β2N∗2−2β2I∗2−2β2α2I∗22−(d2+γ2)(1+α2I∗2)2(1+α2I∗2)2. |
It is easy to see that if R10>1, λ1<0. The next, we need only prove the second eigenvalue λ2<0 if R10>1. Since (1+α2I∗2)2>0, we need only prove β2N∗2−2β2I∗2−2β2α2I∗22−(d2+γ2)(1+α2I∗2)2<0. Let
G(I∗2)=β2N∗2−2β2I∗2−2β2α2I∗22−(d2+γ2)(1+α2I∗2)2. |
Since I∗2≥(β2N∗2−d2−γ2+mα1I∗1)(d2+γ2)α2+β2, so
G(I∗2)=β2N∗2−2β2I∗2−2β2α2I∗22−(d2+γ2)(1+α2I∗2)2≤β2N∗2−(β2+(d2+γ2)α2)(β2N∗2−d2−γ2+mα1I∗1)(d2+γ2)α2+β2−d2−γ2=−mα1I∗1<0. |
This completes the proof.
Theorem 3.7. For the systeml (2.5), if R10>1, the epidemic equilibrium E∗ is globally asymptotically stable.
Proof. Since E∗ is stable when R10>1, we need only prove E∗ is globally attractive.
Consider the equation
dI1(t)dt=I1(β1(N∗1−I1)1+α1I1−d1−m−γ1). |
Let
f1(I1)=β1(N∗1−I1)1+α1I1−d1−m−γ1. |
Then f′1(I1)=β1−1−N∗1α1(1+α1I1)2<0. So f1(I1) is a monotonic decreasing function for all I1>0. Furthermore, f1(0)>0, f1(N∗1)<0 and f1(I∗1)=0 when R10>1. That means if I1∈(0,I∗1), f1(I1)>0 and dI1(t)dt>0; if I1∈(I∗1,N∗1), f1(I1)<0 and dI1(t)dt<0. Hence limt→∞I1(t)=I∗1. By Eq (3.3), limt→∞I2(t)=I∗2. Thus E∗ is globally asymptotically stable.
The results about the existence and stability of equilibria are summarized in Table 1.
Conditions | E0 | E1 | E∗ |
R10<1 and R20<1 | Yes (GAS) | No | No |
R10<1 and R20>1 | Yes (Unstable) | Yes (GAS) | No |
R10>1 and R20<1 | Yes (Unstable) | No | Yes (GAS) |
R10>1 and R20>1 | Yes (Unstable) | Yes (Unstable) | Yes (GAS) |
Remark 3.3. From Theorems 3.5 and 3.7, we know that for the system (2.5), if R0>1, the infectious disease is uniformly persistent. However, the infectious disease is not always uniformly persistent in every patch. If R0>1, but R10<1, the disease is uniformly persistent in the sink patch, but is extinct in the source patch. If R10>1, the disease is always uniformly persistent in every patch. This is a different conclusion resulted by the reducible migration matrix.
Remark 3.4. From Theorem 3.5, in the case that R10<1 and R20>1, the infection does not persist in the source patch but is able to persist in the sink patch. So, in the early stage of the spread of infectious disease, the sink patch should assess the reproduction number R20 reasonably and take control measures timely to prevent the epidemic.
In this section, we carry on numerical simulations to verify the theoretical conclusions, reveal the influence of the migration rate form patch 1 to patch 2 on the basic reproduction number, the transmission scale and transmission speed, and discuss the influence of the parameters α1 and α2 that measure the inhibitory effect on the basic reproduction number, the transmission scale and transmission speed.
To numerically illustrate the theoretical results, we need to choose some parameter values (see Table 2).
Parameter | Description | Value |
A1 | the recruitment rate of the population in patch 1 | 0.018 (Figures 1 and 2) |
0.03 (Figure 3) | ||
A2 | the recruitment rate of the population in patch 2 | 0.0005 (Figure 1) |
0.004 (Figures 2 and 3) | ||
β1 | the transmission rate in patch 1 | 0.00001 |
β2 | the transmission rate in patch 2 | 0.00005 |
α1 | the parameter that measure the inhibitory effect in patch 1 | 0.02 |
α2 | the parameter that measure the inhibitory effect in patch 1 | 0.02 |
d1 | the death rate in patch 1 | 0.0003 |
d2 | the death rate in patch 2 | 0.0003 |
γ1 | the death rate in patch 1 | 0.0001 |
γ2 | the death rate in patch 2 | 0.0001 |
m | the migration rate form the patch 1 to the patch 2 | 0.00005 |
We verify the theoretical conclusions firstly. Denote the density of the infective individuals in patch 1 by i1(t)=I1(t)N∗1. Denote the density of the infective individuals in patch 2 by i2(t)=I2(t)N∗2. Figure 1 shows the evolution of the density of infective individuals in the two patches when R10=0.8889 and R20=0.7500. As predicted by the analytic calculation, the infectious disease in the two patches will disappear eventually. Figure 2 shows the evolution of the density of infective individuals in the two patches when R10=0.8889 and R20=1.8750. We can see the infectious disease will be endemic in patch 2 and the infectious disease in patch 1 will disappear eventually. Figure 3 shows the evolution of the density of infective individuals in the two patches when R10=1.4815 and R20=2.2917. We can see the infectious disease will be endemic in the two patches. And we can see the infectious disease will be endemic in the two patches if R10>1 from the subfigures (c) and (d) of Figure 4.
Second, we reveal the influence of the migration rate m on the transmission in Figure 5. With the increasing of m, the density of infective individuals in patch 1 i1 is decreasing, however the density of infective individuals in patch 2 i2 is increasing.
Third, we reveal the parameters α1 and α2 on the transmission scale and transmission speed. We can see that when α1 is increasing, the density of infective individuals in patch 1 i1 is decreasing from Figure 6 and When α2 is increasing, the density of infective individuals in patch 2 i1 is decreasing from Figure 7.
Many scholars have studied infectious disease transmission with population migration [1,2,3,4,5,6,7], assuming that the migration matrix is irreducible, and found that the propagation dynamics of infectious diseases is determined by the basic reproduction number of the system. When the basic reproduction number is less than 1, the infectious disease eventually becomes extinct; when the basic reproduction number is larger than 1, the infectious disease is epidemic eventually. Since the migration matrix is irreducible, all patches are a connected whole. In all patches, infectious diseases are either extinct or epidemic. That is there is not the phenomenon that infectious diseases are extinct in some patches but epidemic in the others.
Because the studies about the spread of infectious diseases with reducible migration matrix are rare, in this paper, we proposed a two-patch SIS model with saturating contact rate and one-directing population dispersal, discussed the global asymptotic stability of the disease-free equilibrium, the boundary equilibrium and the endemic equilibrium respectively, and revealed the influence of saturating contact rate and migration rate on basic reproduction number and the transmission scale. We have the following main conclusions:
1) If R10>1 then the system tends to a global endemic equilibrium in which infected individuals are present in both patches provided initially there were infected individuals in the source patch; If R10<1 and R20>1 then the system converges to an equilibrium with infected individuals only in the sink patch; If R10<1 and R20<1 then the system converges to the disease-free equilibrium.
2) When migration rate is increasing, the density of infective individuals in the source patch is decreasing; but the density of infective individuals in the sink patch is increasing;
3) With the increasing of the parameter αi (i=1,2) in saturating contact rate, the density of infective individuals in patch i (i=1,2) is decreasing.
The similar conclusions can be obtained for the two patch SI model
{dS1(t)dt=A1−d1S1−β1S1I11+α1I1−mS1,dS2(t)dt=A2−d2S2−β2S2I21+α2I2+mS1,dI1(t)dt=β1S1I11+α1I1−d1I1−mI1,dI2(t)dt=β2S2I21+α2I2−d2I2+mI1. |
We can generalize the current model in many aspects to increase realism. For instance, the infection rate can be given by β(I)SI. We can give the properties on function β(I) such that β(I) is decreasing and tends to 0 when I tends to infinity. The mortality rates of the susceptible and infected individuals are the same in the current model. In fact, the disease-induced death rate can not be neglected sometimes. So the disease-induced death rate can be considered. It is also significant to consider heterogeneous number of contacts for each individual on complex network. There are many paper on this topic [22,23]. One can investigate the multi-patch epidemic model with reducible migration matrix.
This work is supported by the National Natural Sciences Foundation of China (Nos.12001501, 12071445, 11571324, 61603351), Shanxi Province Science Foundation for Youths (201901D211216), the Fund for Shanxi '1331KIRT'.
All authors declare no conflicts of interest in this paper.
[1] | Aldrich JH, Nelson FD (1984) Quantitative Applications in the Social Sciences: Linear Probability, Logit, and Probit Models, Thousand Oaks, CA: SAGE Publications. https://doi.org/10.4135/9781412984744 |
[2] | Alexander VE, Clifford CC (1996) Categorical Variables in Developmental Research: Methods of Analysis, Elsevier. https://doi.org/10.1016/B978-012724965-0/50003-1 |
[3] |
Arya S, Eckel C, Wichman C (2013) Anatomy of the Credit Score. J Econ Behav Organ 95: 175–185. https://doi.org/10.1016/j.jebo.2011.05.005 doi: 10.1016/j.jebo.2011.05.005
![]() |
[4] |
Baesens B, Van Gestel T, Viaene S, et al. (2003) Benchmarking state-of-the-art classification algorithms for credit scoring. J Oper Res Soc 54: 627–635. https://doi.org/10.1057/palgrave.jors.2601545 doi: 10.1057/palgrave.jors.2601545
![]() |
[5] |
Baesens B, Gestel TV, Stepanova M, et al. (2004) Neural Network Survival Analysis for Personal Loan Data. J Oper Res Soc 56: 1089–1098. https://doi.org/10.1057/palgrave.jors.2601990 doi: 10.1057/palgrave.jors.2601990
![]() |
[6] | Bishop CM (2006) Pattern Recognition and Machine Learning, Springer. https://doi.org/10.1007/978-0-387-45528-0_5 |
[7] | Bolton C (2010) Logistic Regression and its Application in Credit Scoring, University of Pretoria. Available from: http://hdl.handle.net/2263/27333. |
[8] |
Breiman L (1996) Bagging Predictors. Mach Learn 24: 123–140. https://doi.org/10.1007/BF00058655 doi: 10.1007/BF00058655
![]() |
[9] | Breiman L, Friedman J, Stone CJ, et al. (1984) Classification and Regression Trees, Taylor & Francis. https://doi.org/10.1201/9781315139470 |
[10] | Brown M, Grundy M, Lin D, et al. (1999) Knowledge-Base Analysis of Microarray Gene Expression Data Using Support Vector Machines, University of California in Santa Cruz. https://doi.org/10.1073/pnas.97.1.262 |
[11] | Byanjankar A, Heikkilä M, Mezei J (2015) Predicting credit risk in peer-to-peer lending: A neural network approach. In 2015 IEEE symposium series on computational intelligence, IEEE, 719–725. https://doi.org/10.1109/SSCI.2015.109 |
[12] |
Cao A, He H, Chen Z, et al. (2018) Performance evaluation of machine learning approaches for credit scoring. Int J Econ Financ Manage Sci 6: 255–260. https://doi.org/10.11648/j.ijefm.20180606.12 doi: 10.11648/j.ijefm.20180606.12
![]() |
[13] | Chen S, Wang Q, Liu S (2019) Credit risk prediction in peer-to-peer lending with ensemble learning framework. In 2019 Chinese Control And Decision Conference (CCDC), IEEE, 4373–4377. https://doi.org/10.1109/CCDC.2019.8832412 |
[14] | Chen TQ, Guestrin C (2016) XGBoost: A Scalable Tree Boosting System. KDD'16 Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 785–794. https://doi.org/10.1145/2939672.2939785 |
[15] |
Cortes C, Vapnik V (1995) Support-vector networks. Mach Learn 20: 273–297. https://doi.org/10.1007/BF00994018 doi: 10.1007/BF00994018
![]() |
[16] | Crouhy M, Galai D, Mark R (2014) The Essentials of Risk Management, 2nd Edition. McGraw-Hill. Available from: https://www.mhprofessional.com/9780071818513-usa-the-essentials-of-risk-management-second-edition-group. |
[17] |
Cybenko G (1989) Approximation by Superpositions of a Sigmoidal Function Mathematics of Control. Signals Syst 2: 303–314. https://doi.org/10.1007/BF02551274 doi: 10.1007/BF02551274
![]() |
[18] |
Duan J (2019) Financial system modeling using deep neural networks (DNNs) for effective risk assessment and prediction. J Franklin Inst 356: 4716–4731. https://doi.org/10.1016/j.jfranklin.2019.01.046 doi: 10.1016/j.jfranklin.2019.01.046
![]() |
[19] |
Duchi J, Hazan E, Singer Y (2011) Adaptive Subgradient Methods for Online Learning and Stochastic Optimization. J Mach Learn Rese 12: 2121–2159. https://doi.org/10.5555/1953048.2021068 doi: 10.5555/1953048.2021068
![]() |
[20] | Elrahman SMA, Abraham A (2013) A Review of Class Imbalance Problem. J Network Innov Comput 1: 332–340. http://ias04.softcomputing.net/jnic2.pdf |
[21] |
Everett CR (2015) Group Membership, Relationship Banking and Loan Default Risk: the Case of Online Social Lending. Bank Financ Rev 7: 15–54. https://doi.org/10.2139/ssrn.1114428 doi: 10.2139/ssrn.1114428
![]() |
[22] |
Friedman JH (2001) Greedy Function Approximation: A Gradient Boosting Machine. Ann Stat 29: 1189–1232. https://doi.org/10.1214/aos/1013203451 doi: 10.1214/aos/1013203451
![]() |
[23] |
Genuer R, Poggi JM, Tuleau-Malot C (2010) Variable selection Using Random Forests. Pattern Recogn Lett 31: 2225–2236. https://doi.org/10.1016/j.patrec.2010.03.014 doi: 10.1016/j.patrec.2010.03.014
![]() |
[24] | Glorot X, Bengio Y (2010) Understanding the Difficulty of Training Deep Feedforward Neural Networks. J Mach Learn Res 9: 249–256. http://proceedings.mlr.press/v9/glorot10a.html |
[25] | Guyon I, ElNoeeff A (2003) An Introduction to Variable and Feature Selection. J Mach Learn Res 3: 1157–1182. https://www.jmlr.org/papers/v3/guyon03a.html |
[26] | Hastie T, Tibshirani R, Friedman JH (2009) The elements of statistical learning: data mining, inference, and prediction, Springer. https://doi.org/10.1007/978-0-387-84858-7 |
[27] | He KM, Zhang XY, Ren SQ, et al. (2015) Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification. IEEE international conference on computer vision. https://doi.org/10.1109/ICCV.2015.123 |
[28] | Ho TK (1995) Random Decision Forest, Proceeding of the 3rd International Conference on Document Analysis and Recognition, 278–282. https://doi.org/10.1109/ICDAR.1995.598994 |
[29] |
Ho TK (1998) The Random Subspace Method for Constructing Decision Forests. IEEE T Pattern Anal 20: 832–844. https://doi.org/10.1109/34.709601 doi: 10.1109/34.709601
![]() |
[30] | Hochreiter S, Bengio Y, Frasconi P, et al. (2001) Gradient Flow in Recurrent Nets: the Difficulty of Learning Long-Term Dependencies. In A Field Guide to Dynamical Recurrent Networks, IEEE, 237–243. https://doi.org/10.1109/9780470544037.ch14. |
[31] | Hsu CW, Chang CC, Lin CJ (2003) A Practical Guide to Support Vector Classification. National Taiwan University, 1–12. Available from: https://www.csie.ntu.edu.tw/~cjlin/papers/guide/guide.pdf. |
[32] |
Ioffe S, Szegedy C (2015) Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. International conference on machine learning, 448–456. https://doi.org/10.48550/arXiv.1502.03167 doi: 10.48550/arXiv.1502.03167
![]() |
[33] | Iyer R, Khwaja AI, Luttmer EF, et al. (2009) Screening in New Credit Markets: Can Individual Lenders Infer Borrower Creditworthiness in Peer-to-Peer Lending? AFA 2011 Denver Meetings Paper. https://doi.org/10.2139/ssrn.1570115 |
[34] |
Kang H (2013) The Prevention and Handling of the Missing Data. Korean J Anesthesiol 64: 402–406. https://doi.org/10.4097/kjae.2013.64.5.402 doi: 10.4097/kjae.2013.64.5.402
![]() |
[35] | Ke GL, Meng Q, Finley T, et al. (2017) LightGBM: A highly Efficient Gradient Boosting Decision Tree, Neural Information Processing Systems, 3149–3157. Available from: https://proceedings.neurips.cc/paper/2017/file/6449f44a102fde848669bdd9eb6b76fa-Paper.pdf. |
[36] | Keogh E, Mueen A (2017) Curse of Dimensionality. Encyclopedia of Machine Learning and Data Mining, Boston: Springer. https://doi.org/10.1007/978-1-4899-7687-1_192 |
[37] | Kingma DP, Ba JL (2015) Adam: a Method for Stochastic Optimization. International Conference on Learning Representations, 1–13. https://doi.org/10.48550/arXiv.1412.6980 |
[38] | Krizhevsky A, Sutskever I, Hinton GE (2012) Imagenet Classification with Deep Convolutional Neural Networks, Advances in Neural Information Processing Systems, 1097–1105. https://doi.org/10.1145/3065386 |
[39] | Lantz B (2013) Machine Learning with R. Packt Publishing Limited. Available from: https://edu.kpfu.ru/pluginfile.php/278552/mod_resource/content/1/MachineLearningR__Brett_Lantz.pdf. |
[40] | Li LH, Sharma AK, Ahmad R, Chen RC (2021) Predicting the Default Borrowers in P2P Platform Using Machine Learning Models. In International Conference on Artificial Intelligence and Sustainable Computing. https://doi.org/10.1007/978-3-030-82322-1_20 |
[41] | Lin HT, Lin CJ (2003) A Study on Sigmoid Kernels for SVM and the Training of Non-PSD Kernels by SMO-type Methods. National Taiwan University. Available from: https://www.csie.ntu.edu.tw/~cjlin/papers/tanh.pdf |
[42] | Lu L, Shin YJ, Su YH, et al. (2019) Dying ReLU and Initialization: Theory and Numerical Examples. arXiv preprint arXiv: 1903.06733. https://doi.org/10.4208/cicp.OA-2020-0165 |
[43] | Maas AL, Hannun AY, Ng AY (2013) Rectifier Nonlinearities Improve Neural Network Acoustic Models. ICML Workshop on Deep Learning for Audio, Speech, and Language Processing. Available from: https://ai.stanford.edu/~amaas/papers/relu_hybrid_icml2013_final.pdf. |
[44] | Madasamy K, Ramaswami M (2017) Data Imbalance and Classifiers: Impact and Solutions from a Big Data Perspective. Int J Comput Intell Res 13: 2267–2281. Available from: https://www.ripublication.com/ijcir17/ijcirv13n9_09.pdf. |
[45] |
McCulloch WS, Pitts W (1943) A Logical Calculus of the Ideas Immanent in Nervous Activity. Bull Math Biophys 5: 115–133. https://doi.org/10.2307/2268029 doi: 10.2307/2268029
![]() |
[46] | Mester LJ (1997) What's the Point of Credit Scoring? Bus Rev 3: 3–16. Available from: https://www.philadelphiafed.org/-/media/frbp/assets/economy/articles/business-review/1997/september-october/brso97lm.pdf. |
[47] | Mijwel MM (2018) Artificial Neural Networks Advantages and Disadvantages. Available from: https://www.linkedin.com/pulse/artificial-neural-networks-advantages-disadvantages-maad-m-mijwel/. |
[48] | Mills KG, McCarthy B (2016) The State of Small Business Lending: Innovation and Technology and the Implications for Regulation. HBS Working Paper No. 17-042. https://doi.org/10.2139/ssrn.2877201 |
[49] |
Mountcastle VB (1957) Modality and Topographic Properties of Single Neurons of Cat's Somatic Sensory Cortex. J Neurophysiol 20: 408–434. https://doi.org/10.1152/jn.1957.20.4.408 doi: 10.1152/jn.1957.20.4.408
![]() |
[50] |
Ohlson JA (1980) Financial Ratios and the Probabilistic Prediction of Bankruptcy. J Account Res 18: 109–131. https://doi.org/10.2307/2490395 doi: 10.2307/2490395
![]() |
[51] | Patro SGK, Sahu KK (2015) Normalization: A Preprocessing Stage. https://doi.org/10.17148/IARJSET.2015.2305 |
[52] |
Pontil M, Verri A (1998) Support Vector Machines for 3D Object Recognition. IEEE Trans PAMI 20: 637–646. https://doi.org/10.1109/34.683777 doi: 10.1109/34.683777
![]() |
[53] |
Qian N (1999) On the Momentum Term in Gradient Descent Learning Algorithms. Neural Networks 12: 145–151. https://doi.org/10.1016/S0893-6080(98)00116-6 doi: 10.1016/S0893-6080(98)00116-6
![]() |
[54] |
Quinlan JR (1987) Simplifying Decision Trees. Int J Man-Mach Stud 27: 221–234. https://doi.org/10.1016/S0020-7373(87)80053-6 doi: 10.1016/S0020-7373(87)80053-6
![]() |
[55] | Quinlan JR (1992) C4.5: Programs for Machine Learning. San Mateo: Morgan Kaufmann Publishers Inc. Available from: https://www.elsevier.com/books/c45/quinlan/978-0-08-050058-4. |
[56] |
Rajan U, Seru A, Vig V (2015) The Failure of Models that Predict Failure: Distance, Incentives, and Defaults. J Financ Econ 115: 237–260. https://doi.org/10.1016/j.jfineco.2014.09.012 doi: 10.1016/j.jfineco.2014.09.012
![]() |
[57] |
Rosenblatt F (1958) The Perceptron: A Probabilistic Model for Information Storage and Organization in the Brain. Psychol Rev 65: 386–408. https://doi.org/10.1037/h0042519 doi: 10.1037/h0042519
![]() |
[58] | Ruder S (2017) An Overview of Gradient Descent Optimization Algorithms. arXiv preprint arXiv: 1609.04747. https://doi.org/10.6919/ICJE.202102_7(2).0058 |
[59] |
Rumelhart DE, Hinton GE, Williams RJ (1986) Learning Representations by Back-Propagating Errors. Nature 323: 533–536. https://doi.org/10.1038/323533a0 doi: 10.1038/323533a0
![]() |
[60] | Samitsu A (2017) The Structure of P2P Lending and Legal Arrangements: Focusing on P2P Lending Regulation in the UK. IMES Discussion Paper Series, No. 17-J-3. Available from: https://www.boj.or.jp/en/research/wps_rev/lab/lab17e06.htm/ |
[61] | Serrano-Cinca C, Gutierrez-Nieto B, López-Palacios L (2015) Determinants of Default in P2P Lending. PloS One 10: e0139427. https://doi.org/10.1371/journal.pone.0139427 |
[62] |
Serrano-Cinca C, Gutiérrez-Nieto B (2016) The use of profit scoring as an alternative to credit scoring systems in peer-to-peer (P2P) lending. Deci Support Syst 89: 113–122. https://doi.org/10.1016/j.dss.2016.06.014 doi: 10.1016/j.dss.2016.06.014
![]() |
[63] |
Shannon C (1948) A Mathematical Theory of Communication. Bell Syst Tech J 27: 379–423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x doi: 10.1002/j.1538-7305.1948.tb01338.x
![]() |
[64] | Shelke MS, Deshmukh PR, Shandilya VK (2017) A Review on Imbalanced Data Handling using Undersampling and Oversampling Technique. Int J Recent Trends Eng Res. https://doi.org/10.23883/IJRTER.2017.3168.0UWXM |
[65] | Singh S, Gupta P (2014) Comparative Study Id3, Cart and C4.5 Decision Tree Algorithm: A Survey. Int J Adv Inf Sci Technol (IJAIST) 27: 97–103. https://doi.org/10.15693/ijaist/2014.v3i7.47-52 |
[66] | Srivastava N, Hinton G, Krizhevsky A, et al. (2014) Dropout: A Simple Way to Prevent Neural Networks from Overfitting. J Mach Learn Res 15: 1929–1958. https://doi.org/https://jmlr.org/papers/v15/srivastava14a.html |
[67] |
Thomas LC (2000) A Survey of Credit and Behavioural Scoring: Forecasting Financial Risk of Lending to Consumers. Int J Forecast 16: 149–172. https://doi.org/10.1016/S0169-2070(00)00034-0 doi: 10.1016/S0169-2070(00)00034-0
![]() |
[68] |
Tibshirani R (1996) Regression shrinkage and selection via the lasso. J Royal Stat Soc (Methodological) 58: 267–288. https://doi.org/10.1111/j.2517-6161.1996.tb02080.x doi: 10.1111/j.2517-6161.1996.tb02080.x
![]() |
[69] | Tieleman T, Hinton G (2012) Lecture 6.5—RMSProp, COURSERA: Neural Networks for Machine Learning. |
[70] | Verified Market Research (2021) Global Peer to Peer (P2P) Lending Market Size by Type, by End User, by Geographic Scope and Forecast. Available from: https://www.verifiedmarketresearch.com/product/peer-to-peer-p2p-lending-market/. |
[71] |
Wang Z, Cui P, Li FT, et al. (2014) A Data-Driven Study of Image Feature Extraction and Fusion. Inf Sci 281: 536–558. https://doi.org/10.1016/j.ins.2014.02.030 doi: 10.1016/j.ins.2014.02.030
![]() |
Conditions | E0 | E1 | E∗ |
R10<1 and R20<1 | Yes (GAS) | No | No |
R10<1 and R20>1 | Yes (Unstable) | Yes (GAS) | No |
R10>1 and R20<1 | Yes (Unstable) | No | Yes (GAS) |
R10>1 and R20>1 | Yes (Unstable) | Yes (Unstable) | Yes (GAS) |
Parameter | Description | Value |
A1 | the recruitment rate of the population in patch 1 | 0.018 (Figures 1 and 2) |
0.03 (Figure 3) | ||
A2 | the recruitment rate of the population in patch 2 | 0.0005 (Figure 1) |
0.004 (Figures 2 and 3) | ||
β1 | the transmission rate in patch 1 | 0.00001 |
β2 | the transmission rate in patch 2 | 0.00005 |
α1 | the parameter that measure the inhibitory effect in patch 1 | 0.02 |
α2 | the parameter that measure the inhibitory effect in patch 1 | 0.02 |
d1 | the death rate in patch 1 | 0.0003 |
d2 | the death rate in patch 2 | 0.0003 |
γ1 | the death rate in patch 1 | 0.0001 |
γ2 | the death rate in patch 2 | 0.0001 |
m | the migration rate form the patch 1 to the patch 2 | 0.00005 |
Conditions | E0 | E1 | E∗ |
R10<1 and R20<1 | Yes (GAS) | No | No |
R10<1 and R20>1 | Yes (Unstable) | Yes (GAS) | No |
R10>1 and R20<1 | Yes (Unstable) | No | Yes (GAS) |
R10>1 and R20>1 | Yes (Unstable) | Yes (Unstable) | Yes (GAS) |
Parameter | Description | Value |
A1 | the recruitment rate of the population in patch 1 | 0.018 (Figures 1 and 2) |
0.03 (Figure 3) | ||
A2 | the recruitment rate of the population in patch 2 | 0.0005 (Figure 1) |
0.004 (Figures 2 and 3) | ||
β1 | the transmission rate in patch 1 | 0.00001 |
β2 | the transmission rate in patch 2 | 0.00005 |
α1 | the parameter that measure the inhibitory effect in patch 1 | 0.02 |
α2 | the parameter that measure the inhibitory effect in patch 1 | 0.02 |
d1 | the death rate in patch 1 | 0.0003 |
d2 | the death rate in patch 2 | 0.0003 |
γ1 | the death rate in patch 1 | 0.0001 |
γ2 | the death rate in patch 2 | 0.0001 |
m | the migration rate form the patch 1 to the patch 2 | 0.00005 |