Journal of Function Spaces
Volume 2021 (2021), Article ID 1891749, 10 pages
https://doi.org/10.1155/2021/1891749

Research Article

Multiobjective Programming Strategy of Small- and Medium-Sized Microenterprise Credit Based on Random Factors

Zhuoran Fan, Jilong Xu, and Yuchen Li

College of Mathematical Sciences, Ocean University of China, Qingdao, Shandong, China

Correspondence should be addressed to Zhuoran Fan; fanzhuoran@stu.ouc.edu.cn

Received 23 April 2021; Accepted 10 June 2021; Published 23 June 2021

Academic Editor: Chun Lu

Copyright © 2021 Zhuoran Fan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we select eight indicators from the aspects of an enterprise’s bill transaction information, namely, whether the enterprise’s loan is in breach of contract, effective invoice rate, total utilization rate of price and tax, negative invoice rate, strength of enterprise, coefficient of variation, flow efficiency of assets, and influence of upstream and downstream enterprises; then, we construct an evaluation index system. According to different industries, different categories, and the impact of random factors, we divide the types of enterprises into 10 categories. Then, we use three kinds of Poisson random numbers to carry out numerical simulation on the total price and tax of enterprises in different industries under the influence of COVID-19.

1. Background

When banks provide loans to small- and medium-sized and microenterprises (small- and medium-sized and microenterprises are abbreviated as SMMEs), they often judge whether to lend or not through credit risk assessment. Because of the lack of mortgage assets in SMMEs, the bank will make credit risk assessment based on the credit policy, influence, strength, and stability of supply and demand relationship of the enterprise, and then determine whether to lend, loan amount, interest rate and term, and other credit strategies. Some corporate banks have credit records, some have no credit records. However, in the face of the impact of sudden random factors on enterprises, how to give the credit strategy when the annual total credit is fixed.

2. The Selection of Credit Risk Quantitative Index

This paper analyzes the relevant data indicators of enterprises with credit records, takes into account the actual situation affecting the credit problems of SMMEs and refers to the advanced international standards, and selects eight quantitative indicators affecting the credit risk of enterprises according to China’s national conditions and the bank’s credit policy:

(1)Whether the enterprise loan is in breach of contract is an important indicator for the bank to examine whether the enterprise can bring the money. Default is 0 and nondefault is 1. means whether th enterprise is in breach of contract. means the enterprise defaults, while means that the enterprise has not breached the contract(2)Effective invoice rate: it is equal to the ratio of the number of valid invoices to the total number of invoices. is used to denote the effective invoice rate of the th enterprise, indicates the number of valid invoices for the th enterprise, and represents the total invoice number of the th enterprise. Thus, the corresponding formula of the effective invoice rate of the th enterprise is as follows:

(3)Utilization rate of total price and tax: it is equal to the ratio of the total price and tax of the valid invoice to the total price and tax of all invoices. Putting represents the utilization rate of the total price and tax of an effective invoice of the th enterprise(4)Negative rate of invoice : it is equal to the ratio of the number of invoices of the th enterprise whose value of the total invoice price and tax is “-” to the number of total invoices of the th enterprise(5): it is equal to the ratio of the difference between the total price and tax of the output and input of the th enterprise to the total price and tax of input, which indicates the strength of the enterprise. Putting represents the total price and tax of the output (sales revenue) of the th enterprise, and represents the total price and tax of the th enterprise’s input (purchased products), which uses the following corresponding formula:

(6)Coefficient of variation: it indicates the stability of supply and demand relationship of enterprises. Using represents the coefficient of variation of the th enterprise. represents the total input price and tax of the th enterprise in the th month, is the total output value tax of the th enterprise in the th month, and represents the net income of the th enterprise in the th month. If , let us take directly . The corresponding formula is follows:

where represents the average value of in 12 months of the th enterprise

(7)Liquidity efficiency of assets: it refers to the comparative relationship between current assets and current liabilities of SMMEs in the same period, that is, the short-term solvency of SMMEs

The following table shows the asset flow data of the th enterprise in 12 months, as shown in Table 1.

Table 1: Asset flow data of enterprises.

Month
1
2

12

Total of input price and tax

Total of output price and tax

Net income

The net income of the previous month is transferred to the next month as part of the next month’s input, which shows the liquidity of funds. The liquidity efficiency of the th enterprise asset can be expressed as follows:

where 12 represents 12 months, and refers to the number greater than 0 in of th enterprise in 12 months. In fact, is the proportion of the number of months whose value is greater than 0 to the total number of months. The larger the value indicates that the better the flow efficiency of th enterprise.

(8)Influence of upstream and downstream enterprises : the influence is expressed by the maximum number of effective cooperation between the th enterprise and upstream and downstream enterprises. In order to quantitatively describe the influence of upstream and downstream enterprises, the influence function of upstream and downstream enterprises is introduced with reference to the negative exponential function of the psychological curve [1]:

where refers to the largest number of input invoice and output invoice of the th enterprise in 12 months. Following the increase of , the influence of upstream and downstream enterprises will also increase.

In the quantitative index system affecting the credit risk of SMMEs, the first to fourth indexes reflect the reputation of enterprises, the fifth index reflects the strength of enterprises, the sixth index reflects the stability of the supply and demand relationship of enterprises, the seventh index reflects the size of the credit risk of enterprises, and the eighth index reflects the influence of enterprises and upstream and downstream enterprises.

3. Comprehensive Evaluation of Credit Risk Quantitative Index System

In order to eliminate dimension and the positive and negative effects of index, in this paper, the fuzzy membership method is used to standardize the index. Let be the th index value of the th evaluation object; be the standardized value of the th index of the th evaluation object and be the number of objects to be evaluated. Then, the positive index standardization formula (6) and the negative index standardization formula (7) can be used to standardize the index [2]:

Among the 8 indicators of the quantitative index system affecting credit risk of SMMEs, the fourth indicator (negative invoice rate) and the sixth indicator (enterprise coefficient of variation) are both negative indicators, which need to be processed with the help of formula (7), while other indicators are calculated with the help of formula (6).

The following uses the entropy weight TOPSIS method to evaluate the credit risk quantitative index system of SMMEs. On the one hand, the entropy weight method is used to determine the coefficient of the credit risk quantitative index system. On the other hand, the TOPSIS method, that is, the technology of approaching the ideal solution, is used to determine the ranking of the evaluated object SMMEs. The core idea of the TOPSIS method is to define the positive ideal solution and negative ideal solution of the decision problem, and then compare and evaluate the distance between the solution and the positive ideal solution and negative ideal solution, and finally calculate the relative closeness degree between each solution and the ideal solution, and order the advantages and disadvantages of the solution.

3.1. Entropy Weight Method Being Used to Calculate the Objective Weight of Indexes

Set as the normalized value of the th indicator in the th system, where and . For a given index , the larger the difference of , the larger the comparative effect of this index has on the system, that means the more information the index contains and transmits.

The specific steps of the entropy method to determine the index weight are as follows:

(i)Calculating the entropy value of 8 indicators such as effective invoice rate. Set as the entropy value of theth index, the solution process is as follows [3]:

where is the characteristic proportion of the th index in the th system, and . are the sum of all system observation data of the th indicator

(ii)Calculation of the coefficientof variance of the th index (iii)Determine the weight coefficients of 8 indexes

3.2. Weighting of Standardized Data

Let be the weighted value of the th index standardized data of the th SMMEs, be the normalized value of the th index observed value of the th SMMEs, and be the weight coefficient. According to the weighting method, it can be seen that

3.3. Determining the Positive and Negative Ideals of the Evaluation System

Set and as the maximum and minimum value of the th index observation data, respectively, :

It is easy to know that the positive and negative ideal solutions of the evaluation system are, respectively, and .

3.4. Calculating the Euclidean Distance between the Evaluation System and the Ideal Solution

Let be the Euclidean distance between the weighted value of the th enterprise and the positive ideal solution and be the Euclidean distance between the weighted value of the th enterprise and the negative ideal solution. Then

3.5. Calculating the Relative Closeness Evaluation Result

Set as the relative closeness of all the indexes and the ideal solution of the th enterprise, then

where .

Determine the development status of the evaluated index by calculating the closeness. The greater the relative closeness , the closer the evaluated index is to the ideal solution, and the better the development status.

4. Banks’ Credit Strategies for SMMEs under Random Factors

Let be the amount of the bank’s loan to the th SMMEs and be the interest rate of the bank’s loan to the th SMMEs. Whether the bank gives loans to th SMMEs, we use the function

The production, operation, and economic benefits of enterprises may be affected by some unexpected factors, and the size of the impact is related to different industries and different types of enterprises. For example, when COVID-19 became widespread, the demand for medical services and products produced by healthcare companies increased rapidly. With the help of relevant state policies, the total credit amount of banks to such healthcare companies and health enterprises will increase. At the same time, in order to avoid the rapid transmission of COVID-19, the state often needs to cut off some transmission routes. For example, during the outbreak of COVID-19, the state issued policies to close some self-employed small- and medium-sized enterprises, so as to reduce the movement of people and avoid cross-infection caused by too many people. In this regard, banks will reduce the total amount of credit to such self-employed SMMEs to avoid credit risk.

According to different industries, different categories, and the size of the impact, we classify enterprises as follows: self-employed enterprises, trade and transportation industry, literature and art advertising industry, manufacturing industry, service industry, financial investment industry, medical and health industry, high-tech enterprises, catering industry, and other industries.

In order to visually show the impact of credit risk and possible sudden factors on each enterprise, we carry out the numerical fluctuation of the total input price tax and the total output price tax of 10 types of enterprises. According to the actual impact of COVID-19 on society, the total input price and tax and the total output price and tax of the medical and health industry should be increased, while the total input price and tax and the total output price and tax of the individual business should be reduced. The concrete method is to add random numbers (Poisson random numbers) that are divided into three types for simulation.

The first category is to increase the total input and output tax of the medical and health industry, and the total input and output tax are, respectively, and . By adding random number (0~100%), the total input price and tax and the total sales tax after the influence are, respectively and . In MATLAB software, the function is used to achieve this [4, 5].

In the second category, for self-employed enterprises, the total input price and tax and the total output price and tax of the catering industry are reduced. The original total input price and tax and the total output price and tax are, respectively, and . By adding random number (-100%~0), the total input price and tax and the total output price and tax are and . This is achieved with the help of the function.

In the third category, the influence of other industries is relatively small, and the random number (-50% ~50%) is added and fluctuates randomly, and the original total of the input price and tax and the total of the output price and tax are and , respectively. The total input price and tax and output price and tax are and . This is achieved with the help of the function .

5. Multiobjective Planning Strategy of SMME’s Credit under Random Factors

When the COVID-19 outbreak occurred, the demand for services and products provided by medical and health enterprises also increased rapidly, and the resulting enterprise profits also increased, so the ability of enterprises to repay loans increased. It is a pity that the profit of the self-employed enterprise is reduced or stagnated, and the ability to repay the loan is weakened. Due to the impact of unexpected factors, the repayment ability is weakened and the bank’s income is affected.

5.1. Determination of Objective Function

From the front, we can see that means the comprehensive evaluation score of the th enterprise out of enterprises. Let

Let the th enterprise repayment for the bank loan ratio be . Taking it here

Thus, the amount of the loan that the th enterprise can repay is . To establish the objective function

On the other hand, the smaller the bank’s lending risk, the better. indicates the unit capital risk of the th SMMEs, and represents the investment risk brought by the capital flow of the th SMMEs, and establishes an objective function for this purpose:

On the other hand, the bank loan amount should take into account the business strength of the enterprise. This paper uses the sample variance index of loan amount and total input price and tax to describe the balance of credit amount, and establishes the objective function for this purpose:

5.2. Determination of Constraints

(i)The loan limit of the established bank to the enterprise determined to be loaned is 10-100 (ten thousand), so

(ii)The annual loan interest rate of the bank to the enterprise determined to lend is 4%~15%. Thus

(iii)The balance of a bank’s investment in enterprises. It’s represented by . The demand for services and products provided by medical and health enterprises is increasing rapidly. Therefore, the total input price and tax of such enterprises should also be increased, and the amount of bank loans to such enterprises should be increased. When the number of self-employed enterprises decreases or stagnates, the total input value and tax should also be reduced, and the amount of bank loans to such enterprises should be reduced. The upper and lower limits of the total balance of input price and tax for medical and health input are adjusted to 0.8 and 2. Considering that an individual business cannot be given a loan completely, the upper and lower limits of the total balance of input price and tax of an individual business are adjusted to 0.3 and 1, the upper and lower limits of other industries remain at 0.5 and 1.5. Set = “medical enterprise code”; = “individual enterprise code”; and = “all other enterprise codes”. We agreed that

(iv)Whether the bank loans to the enterprise and the loan amount is consistent, let be a very small positive number and be a very large positive number. The values of 1 and 0 of , respectively, indicate that the bank loans to the th enterprise and does not lend to the th enterprise. In order to ensure the consistency of bank loans to the enterprise and the loan amount, there are constraints

(v)Total amount of loan. Assuming that the total amount of loan is 100 million when the bank loans to enterprises, the unit here takes 10000 yuan. We have

6. Example Checking

This paper verifies the multiobjective planning strategy of SMMEs under the influence of COVID-19 by using the related data. The original data of this paper comes from the data of competition question C for CUMCM-2020 (China University mathematical modeling competition), which can be downloaded publicly [6] (http://www.mcm.edu.cn/html_cn/node/10405905647c52abfd6377c0311632b5.html).

Firstly, the Poisson random number is considered, and with the help of the TOPSIS evaluation method, the scores and ranking comparison table of 302 enterprises before and after the introduction of random distribution are obtained [7–17]. The scores and ranking of the top 20 enterprises with enterprise number before and after the introduction of random distribution are shown in Table 2.

Table 2: Comparison of scores and rankings of the top 20 enterprises with enterprise numbers before and after the introduction of random distribution.

The score results of entropy weight method before introducing a random distribution
The score results of entropy weight method after introducing a random distribution

Enterprise numbers
Score
Ranking
Enterprise numbers
Score
Ranking

1
0.03470546
133
1
0.01051545
258

2
0.03473871
132
2
0.01009714
271

3
0.26589653
77
3
0.2664342
52

4
0.29959602
8
4
0.36079418
3

5
0.26882309
21
5
0.31805003
22

6
0.2663194
61
6
0.04038756
92

7
0.26669382
46
7
0.01158391
189

8
0.26632932
59
8
0.0171994
112

9
0.26681615
40
9
0.01643889
114

10
0.26611135
72
10
0.11524284
72

11
0.26909606
19
11
0.32030139
17

12
0.2667195
44
12
0.31590827
32

13
0.26702299
31
13
0.01160196
187

14
0.26627581
63
14
0.11280042
75

15
0.26699131
33
15
0.31609219
31

16
0.27680573
15
16
0.32686684
12

17
0.2469376
90
17
0.29235498
47

18
0.26674657
42
18
0.31711769
26

19
0.26660041
50
19
0.31528929
40

20
0.24583911
93
20
0.24091041
58

It can be seen from Table 2 that, after the introduction of random distribution, the ranking of enterprises with enterprise labels ranging from 1 to 20 changed correspondingly—some changed greatly, while some changed less—indicating that our model has good practicability.

After the introduction of random distribution, the changes in scores and rankings.

of the top 20 enterprises among the 302 enterprises are shown in Table 3.

Table 3: Changes in scores and rankings of the top 20 companies after the introduction of random distribution.

The score results of entropy weight method before introducing a random distribution
The score results of entropy weight method after introducing a random distribution

Enterprise numbers
Score
Ranking
Enterprise numbers
Score
Ranking

206
0.65353762
1
206
0.74570025
1

30
0.51530701
2
30
0.55732306
2

4
0.36079418
3
237
0.33358854
3

107
0.34892084
4
235
0.31445308
4

92
0.34436155
5
89
0.31144136
5

89
0.34363768
6
107
0.30352414
6

76
0.34281575
7
220
0.30110706
7

220
0.34233996
8
4
0.29959602
8

122
0.33108545
9
92
0.2928879
9

26
0.32893502
10
76
0.29120591
10

62
0.32855488
11
122
0.28217424
11

16
0.32686684
12
26
0.28128101
12

38
0.32600357
13
38
0.28008684
13

45
0.32299682
14
62
0.27817557
14

110
0.32166147
15
16
0.27680573
15

33
0.32044747
16
45
0.27234886
16

11
0.32030139
17
33
0.27128483
17

53
0.3194912
18
110
0.27112156
18

111
0.31946501
19
11
0.26909606
19

63
0.31856213
20
63
0.26895159
20

As can be seen from Table 3, after the introduction of random distribution, the number of the top 20 enterprises is basically still in the top 20, indicating that our comprehensive evaluation method is relatively good and the ranking distribution is relatively stable.

From Table 4, we can see the ranking changes of enterprises in the case of occurrence of emergent factors and absence of emergent factors. It can be found that under the influence of COVID-19, the rating and ranking of enterprises in the medical and health industry have increased, indicating that under the influence of COVID-19, such enterprises have a good credit situation and a low credit risk. However, the decline in the score and ranking of self-employed enterprises indicates that under the influence of COVID-19, the credit situation of such enterprises is poor and the credit risk is high, which is in line with the actual situation. It indicates that our TOPSIS evaluation method is effective and can be better applied to the situation when random factors occur.

Table 4: Changes in medical and individual business scores and rankings after the introduction of random distribution.

The score results of entropy weight method before introducing a random distribution
The score results of entropy weight method after introducing a random distribution

Enterprise numbers
Score
Ranking
Enterprise numbers
Score
Ranking

E195 (medical)
0.266610
48
E195 (medical)
0.314380
44

E398 (medical)
0.014558
189
E398 (medical)
0.012037
166

E420 (medical)
0.014021
227
E420 (medical)
0.013773
184

E373 (individual)
0.014911
162
E373 (individual)
0.009712
279

E124 (individual)
0.034705
133
E124 (individual)
0.010516
258

E125 (individual)
0.034739
132
E125 (individual)
0.010516
271

When the total annual credit of the bank is 100 million yuan, we use the data given in the attached table of question C to establish the multiobjective programming model of 302 enterprises.

The multiobjective function includes the following: , , and

The constraint conditions are as follows:

The software programming of Lingo is used to solve the above multiobjective function programming model [18]. In the three target functions, let be the scale coefficient of the th objective function, which satisfies

Set three different proportional coefficients and get different results of different loan amounts, which are analyzed in the following list (see Table 5).

Table 5: Analysis of different results obtained by different scale coefficients of the three objective functions.

Plan
1
2
3
4
5
6
7
8

0.7
0.6
0.6
0.5
0.6
0.5
0.4
0.3

0.2
0.2
0.1
0.25
0.15
0.15
0.15
0.15

0.1
0.2
0.3
0.25
0.25
0.35
0.45
0.55

The number of different values of the loan amount
7
9
9
8
11
17
19
20

According to the analysis in Table 5, the balance of credit amount is important for banks. Finally, the eighth plan is selected to obtain the specific credit plan for 302 SMMEs, as shown in Table 6 below. The loan amount of enterprises not listed in Table 6 is 100,000 yuan.

Table 6: List of the specific amount of 100 million yuan loan from the bank to 302 SMMEs (yuan).

Enterprise number
Loan amount

133
880344.8

147
548104

149
864340.1

166
532681.5

167
867680.4

171
150709.4

173
726597.9

178
836082.4

179
646818.7

182
425921

187
402967.4

190
414155.8

191
735308.3

193
338920

198
732913.7

205
313454.6

211
798931.5

242
184068.4

1
1000000

2
1000000

212
1000000

296
1000000

39
1000000

58
1000000

100
1000000

101
1000000

103
1000000

109
1000000

124
1000000

125
1000000

126
1000000

127
1000000

128
1000000

129
1000000

130
1000000

131
1000000

132
1000000

134
1000000

135
1000000

136
1000000

137
1000000

138
1000000

229
1000000

3
1000000

139
1000000

140
1000000

141
1000000

142
1000000

145
1000000

148
1000000

150
1000000

153
1000000

154
1000000

155
1000000

156
1000000

157
1000000

158
1000000

159
1000000

160
1000000

161
1000000

162
1000000

163
1000000

164
1000000

165
1000000

250
1000000

4
100000

169
1000000

170
1000000

174
1000000

175
1000000

176
1000000

177
1000000

181
1000000

183
1000000

184
1000000

185
1000000

186
1000000

189
1000000

196
1000000

197
1000000

199
1000000

200
1000000

201
1000000

202
1000000

203
1000000

204
1000000

258
1000000

5
100000

It can be found that during the COVID-19 epidemic, due to the rapid increase in the demand for the services and products provided by the medical and health enterprises, the investment of the finally obtained bank in this industry also increased, the sales of the self-employed industry decreased or stagnated, and the investment of the finally obtained bank in this industry also decreased or stagnated. The investment of banks in other industries is also adjusted accordingly to maintain the survival and operation of the industry, which is more consistent with the actual situation and demonstrates the effectiveness and practicability of our model.

7. Sensitivity Analysis of the Model

The sensitivity of the model is used to analyze the sensitivity and stability of the model. In order to test the stability and effectiveness of the multiobjective programming strategy model, sensitivity analysis is carried out.

In the case of COVID-19, 3 random Poisson numbers were added to simulate the total value of the total tax and total sales tax in each industry. In order to fully demonstrate the sensitivity of the model, we changed the number of algorithm runs and record the average scores of the 302 comprehensive evaluation scores obtained by each algorithm; then, the average score of the comprehensive evaluation of the 20 algorithms was obtained and drawn. The results are shown in Figure 1.

Figure 1: Sensitivity inspection chart.

By checking whether the average value of the comprehensive evaluation score is stable and centralized when the algorithm runs for 20 times, we can verify whether the model is stable. One can find that the average value of the comprehensive evaluation score of 20 times is relatively centralized and stable, floating in a certain range. It can be seen that our model is stable and practical.

In this paper, the TOPSIS evaluation method is firstly used to get the score and ranking comparison of 302 enterprises before and after the introduction of a random number, and the ranking results are analyzed. Then, considering that different industries each have a different ability to repay loans when affected by COVID-19, the ratio factor that can repay bank loans is introduced, and considering the floating amount of loans in different industries under national conditions and policies, the objective function and constraints of the multiobjective credit optimization model are modified, and the multiobjective credit optimization model of enterprises influenced by COVID-19 is established. When the total amount is 100 million, the corresponding credit decision is made. Finally, sensitivity analysis is carried out to test the stability and effectiveness of the multiobjective programming strategy model.

Data Availability

The original data of this paper comes from the data of competition question C for CUMCM-2020 (China University mathematical modeling competition), which can be downloaded publicly. Download from the following website: http://www.mcm.edu.cn/html_cn/node/10405905647c52abfd6377c0311632b5.html. The later data used to support the findings of this study are included within the supplementary information file(s).

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Natural Science Foundation of Shandong Province of China (No. ZR2020QA078) and the National Natural Science Foundation of China (No. 12005110).

Supplementary Materials

We have uploaded three attachments. Annex 1.1: raw data on 302 enterprises without credit records which comes from the data of competition question C for CUMCM-2020. Annex 2: in this paper, we get the relevant data of 302 enterprises before and after the introduction of random distribution. Annex 3: Ligon Program Source Code. (Supplementary Materials)

References

L. Zhang and X. Wang, “Analysis on dairy farmers’ income based on Logit model,” Economic Theory and Business Management, vol. 9, pp. 101–108, 2012.

H. Yang and P. Fu, “Application of fuzzy comprehension evaluation based on entropy weight,” Journal of North China Electric Power University, vol. 32, no. 5, pp. 104–107, 2005.

G. Li, G. Chi, and Y. Cheng, “Evaluation model and empirical study of human all-round development based on entropy weight and TOPSIS,” Journal of Systems Engineering, vol. 26, no. 3, pp. 400–407, 2011.

J. Zhuo, The Application of MATLAB in Mathematical Modeling, Beijing University of Aeronautics and Astronautics Press, Beijing, 2019.

C. Gong and Z. Wang, Common Algorithm Assembly of MATLAB Language, Publishing House of Electronics Industry, Beijing, 2010.

“The data of competition question C for CUMCM-2020 (China University mathematical modeling competition),” http://www.mcm.edu.cn/html_cn/node/10405905647c52abfd6377c0311632b5.html.

S. Si and X. Sun, Mathematical Modeling Algorithms and Applications, National Defense Industry Press, Beijing, 2011.

M. Liu and C. Bai, “Optimal harvesting of a stochastic mutualism model with regime-switching,” Applied Mathematics and Computation, vol. 373, p. 125040, 2020.

C. Lu, “Dynamics of a stochastic Markovian switching predator-prey model with infinite memory and general Lévy jumps,” Mathematics and Computers in Simulation, vol. 181, pp. 316–332, 2021.

B. Han, D. Jiang, B. Zhou, T. Hayat, and A. Alsaedi, “Stationary distribution and probability density function of a stochastic SIRSI epidemic model with saturation incidence rate and logistic growth,” Chaos, Solitons & Fractals, vol. 142, article 110519, 2021.

C. Lu and X. Ding, “Periodic solutions and stationary distribution for a stochastic predator-prey system with impulsive perturbations,” Applied Mathematics and Computation, vol. 350, pp. 313–322, 2019.

C. Lu, G. Sun, and Y. Zhang, “Stationary distribution and extinction of a multi-stage HIV model with nonlinear stochastic perturbation,” Journal of Applied Mathematics and Computing, 2021.

M. Song, W. Zuo, D. Jiang, and T. Hayat, “Stationary distribution and ergodicity of a stochastic cholera model with multiple pathways of transmission,” Journal of the Franklin Institute, vol. 357, no. 15, pp. 10773–10798, 2020.

S. Zhang, T. Zhang, and S. Yuan, “Dynamics of a stochastic predator-prey model with habitat complexity and prey aggregation,” Ecological Complexity, vol. 45, article 100889, 2021.

M. Gao and D. Jiang, “Stationary distribution of a stochastic food chain chemostat model with general response functions,” Applied Mathematics Letters, vol. 91, pp. 151–157, 2019.

X. Zhang and Q. Yang, “Threshold behavior in a stochastic SVIR model with general incidence rates,” Applied Mathematics Letters, vol. 121, article 107403, 2021.

Q. Liu and D. Jiang, “Influence of the fear factor on the dynamics of a stochastic predator-prey model,” Applied Mathematics Letters, vol. 112, article 106756, 2021.

J. Xie and Y. Xue, Optimized Modeling with Lindo/Lingo Software, Tsinghua University Press, Beijing, 2005.