Statistical and computational analysis for corruption and poverty model using Caputo-type fractional differential equations

Abstract

Since there is a clear correlation between poverty and corruption, mathematicians have been actively researching the concept of poverty and corruption in order to develop the optimal strategy of corruption control. This work aims to develop a mathematical model for the dynamics of poverty and corruption. First, we study and analyze the indicators of corruption and poverty rates by applying the linear model along with the Eviews program during the study period. Then, we present a prediction of poverty rates for 2023 and 2024 using the results of the standard problem-free model. Next, we formulate the model in the frame of Caputo fractional derivatives. Fundamental properties, including equilibrium points, basic reproduction number, and positive solutions of the considered model are obtained using nonlinear analysis. Sufficient conditions for the existence and uniqueness of solutions are studied via using fixed point theory. Numerical analysis is performed by using modified Euler method. Moreover, results about Ulam-Hyers stability are also presented. The aforementioned results are presented graphically. In addition, a comparison with real data and simulated results is also given. Finally, we conclude the work by providing a brief conclusion.

1. Introduction

Corruption has become, at present, one of the most critical global problems. It is addressed by many academics, researchers, and governmental and international organizations because of its seriousness and based on the idea that corruption is a destructive phenomenon that affects the economy and development [1]. Corruption is one of the leading causes and determinants of poverty and an obstacle to reducing and eliminating it. It destroys all government efforts in developing countries that seek to alleviate poverty, thus weakening economic growth and diverting its path, and weakening social, economic, and political institutions, where we note the relationship between corruption and poverty in government sectors. Therefore, eradicating poverty requires first eliminating corruption [2]. Kotkin has argued that corruption refers to an absence of conscience, moral sense, and integrity [3]. There are three ways to classify corruption: participation, transaction, and commitment. Mainly, people with high incomes benefit from corruption [4]. According to Johnson, the burden of corruption is on the shoulders of people with low incomes and therefore pay part of their income to individuals with high incomes. Corruption diverts government spending from projects that benefit low-income people, such as health and education, and goes to spending on defense and armaments, where corruption is widespread [5].

Ogboru argued that corruption has normative and ethical dimensions and appears when an employee uses his government position and status for personal gain; corruption increases when the criteria for its determination are not strict. Corruption diverts the available resources into private accounts and weakens the state's ability to promote development, thus hampering efforts to combat and reduce poverty [6]. Poverty is widespread in African countries where development is low and living standards are deteriorating, and thus corruption appears in those regions where poverty exists [7].

The empirical and theoretical literature on the relationship between poverty and corruption has been searched since the late nineties, when Transparency International (TI) focused on the problem of corruption and the participation rights of individuals in the economic and political fields, see [8].

Dincer and Gunalp [9] examined the influence of corruption on poverty and income inequality in the USA, utilizing an objective gauge of corruption and various gauges of poverty and inequality. Their outcomes confirm that increased corruption leads to rising poverty and income inequality. In [10], the author attempted to evaluate the impact of corruption on Nigeria's poverty situation by employing descriptive analysis supported by empirical evidence. They have established that corruption has negatively affected the operations of poverty reduction efforts in Nigeria. Unver and Koyuncu [11] examined the effect and relationship of poverty on corruption. The study used data analysis of 154 countries from 2000 to 2013, showing that the poverty variable has a positive and statistically significant impact on corruption. Negin and his coauthors in 2010 described the correlation between the phenomenon of poverty and corruption. They investigated the Granger relationship model and used data from 97 countries from 1997 to 2006 for the Human Poverty and the Corruption Index. Their results revealed that poverty and corruption are positively related. Another study was conducted during the period from 2005 to 2009 in nine ASEAN countries by Rahayu et al. [12]. In this study, the authors discussed Granger's causal relationship between the poverty index and corruption. Their results showed that there is a one-way causal relationship destined from corruption to poverty, while the poverty variable does not affect the corruption variable. In 2018, Azwar and Saragih proved that corruption leads to a rise in poverty. They used the ARDL and ECM models from 1995 to 2017 to investigate the relationship between corruption and poverty. Their results showed that the corruption index has a positive and significant effect at 1% on the poverty variable, see [13].

In [14], the authors investigated the impact of corruption and government spending on poverty rates in Nigeria, relying on the ARDL model from 1996 to 2016. Their results showed that corruption has a positive relationship, i.e., the higher the CPI, the greater the poverty. Mathematical modeling is an enormous tool for expressing and dealing with complicated phenomena. The importance of fractional calculus (FC) and its applications [15], [16], [17] in modeling a variety of complicated phenomena in a wide range of engineering and science has recently increased. Due to the representation of memory and hereditary qualities, FC can also aid in the modeling of real phenomena and in providing a better grasp of the explanation of real-world problems than the order of ordinary integers. The idea of the fractional derivative (FD) is presented by Riemann Liouville, which depends on power law. Subsequently, the said operators were extended to some new forms which were increasingly used in the last few years for investigating epidemic models, we refer to some work as [18], [19], [20], [21], [22].

The literature appears that so few works are available on statistical and computational analysis for a corruption and poverty model. Until now, no literature has been available on the fractional model of corruption and poverty predicting the effect of corruption on poverty under the general form of Caputo fractional derivatives. Therefore, this is the primary motivation behind our work. Also, the comparison of the actual data with different fractional-order simulations is one of our unique aims in this work. So, this research paper analyzes a novel mathematical model involving the general form of Caputo fractional derivative to study the corruption and poverty transmission mechanism. We discuss the fundamental characteristics of the proposed model. Also, some qualitative analysis of the model is investigated in detail. Additionally, to simulate the results and solve the suggested model, an effective approximate technique on the basis of the fractional-order modified Euler method is developed. Throughout the approximation, this approach assesses the slope again. This method aims to compute more accurate estimates by computing slopes halfway down the line segment rather than taking approximations with the slopes specified in the function. We use actual data to simulate the results. Some recent results on corruption model, we refer to [23], [24], [25]. For some analysis results we use the tools applied in [26], [27], [28], [29], [30].

This manuscript's core structure is divided into eight sections. We present some statistics findings on poverty and corruption in the next section before moving on to the main portions of this paper. In the third section, the model is developed within the context of Caputo fractional derivatives. While section 5 is devoted to the fundamental characteristics of the proposed model, section 4 introduces various definitions and lemmas that are crucial in establishing our primary findings. The purpose of Section 6 is to present theoretical results of given model. Additionally, a numerical method for assessing the approximations of the model's solutions is provided. We demonstrate numerical simulations of our model in Section 7 that match the numbers in Table 4 numerical values. Finally, we offer a succinct conclusion to wrap up the work.

Table 4.

Nomenclatures and interpretations of model (2).

Nomenclature	Description	Numerical values
Λ	Birth rate into the susceptible population	0.097
$\mathcal{N}$	Total population
μ	Death removal rate	0.00099
π1	Corruption transmission probability per contact	0.0011
π2	Poverty transmission probability per contact	0.0098
τ1	Effort rate against corruption	0.0001
τ2	Effort rate against poverty	0.0001
π1(1 − τ1)	Effective corruption contact rate	0.00109989
π2(1 − τ2)	Effective poverty contact rate	0.00979902
γ1	Rate at which corrupt individuals become poor	0.0058
γ2	Rate at which poverty individuals become corrupt	0.0701
ω	Proportion of corrupt individuals prosecuted and jailed	0.00019
$\frac{1}{\psi }$	Average period prosecuted individuals spend in prison	7.01 × 10−4
ψ(1 − ϕ)	Transition rate from jailed to corrupt	0.000700606739
ψϕ	Transition rate from jailed to honest	0.000000393261
βC	Proportion of corrupt individuals that progress to the $\mathcal{H}$ compartment	0.000000701
βP	Proportion of poverty individuals that progress to the $\mathcal{H}$ compartment	0.0000013
σ	Proportion of susceptible individuals that progress to the $\mathcal{H}$ compartment	0.000833

2. Statistical results

We present the empirical findings and some statistical discussions. First, we study and analyze the descriptive data on corruption and poverty. Next, we investigate the effect of corruption on poverty and poverty prediction.

2.1. Descriptive data analysis

Table 1 shows the classification of the corruption perception index for (2003–2022). The spread of corruption in Yemen was noticeably large according to the methodology adopted until 2012, when it scored less than 3 points, so it is considered among the countries with the most widespread corruption. It is also clear that the points are close to zero, which means that corruption is spreading continuously, as it decreased from (2.6) in 2003 to (2.1) degrees in 2011. We also find the spread of the corruption index during the period (2012–2022), as it decreased by eight points during the period, i.e., from 23 in 2012 to 16 in 2022. Moreover, Yemen was ranked among the lagging centers for fighting corruption, as it ranked 176 out of 180 countries in 2022. This indicates an increase in corruption rates in state agencies and institutions in the future, see Fig. 1.

Table 1.

Corruption perceptions index and poverty rate in Yemen for (2003–2022).

Years	Corruption Score	Corruption Rank	Number of countries	Poverty rate
2003	42.97	2.6	88	133
2004	43.00	2.4	112	146
2005	42.50	2.7	103	159
2006	41.50	2.6	111	163
2007	44.50	2.5	131	180
2008	47.60	2.3	141	180
2009	48.80	2.1	154	180
2010	49.00	2.2	154	178
2011	49.85	2.1	154	183
2012	47.20	23	156	176
2013	47.80	18	167	176
2014	48.60	19	161	175
2015	52.00	18	154	198
2016	54.00	14	170	176
2017	56.89	16	175	198
2018	57.95	14	176	180
2019	58.22	15	177	198
2020	58.70	15	176	180
2021	58.88	16	174	180
2022	58.90	16	176	180

Figure 1.

Corruption score for the period (2003–2022).

Moreover, Table 1 displays the evolution of poverty rates. We find that the poverty rate increased from 42.97 in 2003 until it reached 49.85 in 2011, then it decreased in 2013 to 47.80 and rose until it reached 58.90 in 2022. In general, we find that the poverty rate increased by 37% during the period (2013–2022); see Fig. 2.

Figure 2.

The poverty rate from 2003 to 2022.

2.2. Forecasting strategy

In this portion, we investigate the conditions and assumptions of simple linear regression for the independent variable and the dependent variable. Indeed, we transformed the independent variable (corruption) through the fractional rank to a normal distribution function (Idf Normal), with a mean of 10.2 and a standard deviation of 7.5. Whereas the dependent variable (poverty) was transformed with an average (50.44) and a standard deviation (6.10); thus, the data became normally distributed, as in Table 2.

Table 2.

Jarque-Bera test for normal distribution.

Variable	The Poverty	The Corruption
Jarque-Bera	0.532581	0.404221
Probability	0.766216	0.817004

We find that the probability value of Jarque-Bera is >0.05, which means that the data follow a normal distribution. In order to know the relationship between the corruption and poverty index, we find the correlation coefficient is (0.885), which is a strong positive correlation and statistically significant at 0.01. Next, we estimate the linear regression model by applying the least squares between corruption and poverty, considering corruption as an explanatory variable and the poverty rate as a dependent variable. The results of the estimation showed ($\beta =0.19,Std-Error=0.02,t=7.68>2,Prob=0.0000<0.01$), while the coefficient of the determination reached ($R^2=57.3\text{\%}$), which means that the corruption variable explains 57% of the variance in poverty. As for the significance of the regression test, it is reached $(F-statistic=59.07,andProb=0.00000<0.01)$. Therefore, we accept the research hypothesis that the regression is significant, and therefore there is an effect of the corruption variable on poverty. The regression results indicate that whenever the corruption variable increases by one unit, the poverty rate increases by 0.194662. Here some statistical data are given in Table 3.

Table 3.

Results of Linear regression between corruption and poverty in Yemen.

Dependent Variable: POVERTY
Sample: 2003 2022
Method: Least Squares
Included observations: 20
Variable	Coefficient	Std. Error	t-Statistic	Prob
C	21.14641	3.871329	5.462313	0.0060
CORRUPTION	0.194662	0.025326	7.686284	0.0000
R-squared	0.586473	Mean dependent var	50.44300
Adjusted R-squared	0.573500	SD. dependent var	6.104812
Sum squared resid	165.3617	Schwarz criterion	5.249853
SE. of regression	3.030967	Akaike info criterion	5.150280
Log-likelihood	-49.5028	Hannan-Quinn criteria	5.169717
F-statistic	59.07896	Durbin-Watson stat	1.978165
Prob (F-statistic)	0.000000

Fig. 3 shows the forecast of the poverty rate from 2023 to 2024. To this end, we use a standard problem-free linear regression model, where there is no residual regression problem because it reached $(Jarque-Bera=0.77458andProb=0.67889>0.05)$ and was normally distributed. Also, there is no heterogeneity problem for the variance of random errors of the regression model, where the probability value for the Harvey test of Heteroskedasticity is $0.2175>0.05$. Additionally, Fig. 3 exhibits an increase in poverty rates for 2023 and 2024.

Figure 3.

Forecasting the poverty rate for the years 2023 and 2024.

3. Mathematical analysis of model

The model (1) suggested by Eguda et al. [26] is given by

$$\{\begin{array}{c}\hfill {\mathcal{S}}^{\prime }=\mathrm{\Lambda }-{\pi }_1(1-{\tau }_1)\frac{\mathcal{CS}}{\mathcal{N}}-{\pi }_2(1-{\tau }_2)\frac{\mathcal{PS}}{\mathcal{N}}-(\sigma +\mu )\mathcal{S},\hfill \\ \hfill {\mathcal{C}}^{\prime }={\pi }_1(1-{\tau }_1)\frac{\mathcal{CS}}{\mathcal{N}}+{\gamma }_2\frac{\mathcal{PS}}{\mathcal{N}}+\psi (1-\varphi )\mathcal{I}\hfill \\ \hfill -({\gamma }_1+\omega +{\beta }_c+\mu )\mathcal{C},\hfill \\ \hfill {\mathcal{P}}^{\prime }={\pi }_2(1-{\tau }_2)\frac{\mathcal{PS}}{\mathcal{N}}+{\gamma }_1\mathcal{C}-{\gamma }_2\frac{\mathcal{PS}}{\mathcal{N}}-({\beta }_p+\mu )\mathcal{P},\hfill \\ \hfill {\mathcal{I}}^{\prime }=\omega \mathcal{C}-(\psi (1-\varphi )+\psi \varphi +\mu )\mathcal{I},\hfill \\ \hfill {\mathcal{H}}^{\prime }={\beta }_c\mathcal{C}+{\beta }_p\mathcal{P}+\psi \varphi \mathcal{I}+\sigma \mathcal{S}-\mu \mathcal{H}.\hfill \end{array}$$ (1)

• Susceptible individuals $\mathcal{S}$ are the individuals who have never participated in any corrupt actions that effectively affect the economic development of the country.

• Corrupt individuals $\mathcal{C}$ are people who are regularly engaged with degenerate practices and are fit for affecting a susceptible individual to become corrupt.

• Poverty Individuals $\mathcal{P}$ are the people who come up short on standard measures or material belongings and, furthermore, the resources to fulfil their fundamental necessities.

• Imprisoned individuals $(\mathcal{I})$ are the individuals who have been sentenced for corrupt practices and detained for a particular time frame during which they can't be associated with any corrupt demonstrations and can't impact others.

• Honest Individuals $\mathcal{H}$ are the people who can never be corrupt regardless of their psychological state. According to the report [27], the total population of the country is $\mathcal{N}=21$ million. Since the exact amount or density is not known so we take $\mathcal{S}=2$ million, $\mathcal{C}=0.5$ million, $\mathcal{P}=12$ million, $\mathcal{I}=0.5$ million, and $\mathcal{H}=6$ million. We describe the nomenclatures of the model in Table 4.

We update the model (1) from traditional order to Caputo fractional derivative in (2) as

$$\begin{array}{c}\hfill {}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{S}=\mathrm{\Lambda }-{\pi }_1(1-{\tau }_1)\frac{\mathcal{CS}}{\mathcal{N}}-{\pi }_2(1-{\tau }_2)\frac{\mathcal{PS}}{\mathcal{N}}-(\sigma +\mu )\mathcal{S},\hfill \\ \hfill {}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{C}={\pi }_1(1-{\tau }_1)\frac{\mathcal{CS}}{\mathcal{N}}+{\gamma }_2\frac{\mathcal{PS}}{\mathcal{N}}+\psi (1-\varphi )\mathcal{I}\hfill \\ \hfill -({\gamma }_1+\omega +{\beta }_c+\mu )\mathcal{C},\hfill \\ \hfill {}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{P}={\pi }_2(1-{\tau }_2)\frac{\mathcal{PS}}{\mathcal{N}}+{\gamma }_1\mathcal{C}-{\gamma }_2\frac{\mathcal{PS}}{\mathcal{N}}-({\beta }_p+\mu )\mathcal{P},\hfill \\ \hfill {}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{I}=\omega \mathcal{C}-(\psi (1-\varphi )+\psi \varphi +\mu )\mathcal{I},\hfill \\ \hfill {}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{H}={\beta }_c\mathcal{C}+{\beta }_p\mathcal{P}+\psi \varphi \mathcal{I}+\sigma \mathcal{S}-\mu \mathcal{H},\hfill \end{array}$$ (2)

under the initial values as

$$\mathcal{S}(0)={\mathcal{S}}_0>0,\mathcal{C}(0)={\mathcal{C}}_0>0,\mathcal{P}(0)={\mathcal{P}}_0>0,$$

$$\mathcal{I}(0)={\mathcal{I}}_0>0,\mathcal{H}(0)={\mathcal{H}}_0>0,$$

where ${}^C\mathcal{D}_{0^{+}}^{\varrho }$ denotes fractional order differentiation with $0<\varrho \le 1$.

4. Background results

Consider the Banach space denoted by $C=C(\mho ,{\mathbb{R}}^5)$, where $\mho =[0,T]$, and $\mathrm{\Psi }=(\mathcal{S},\mathcal{C},\mathcal{P},\mathcal{I},\mathcal{H})$, then the norm is given by

$$\Vert \mathrm{\Psi }\Vert =\underset{\vartheta \in \mho }{\sup }\{|\mathrm{\Psi }(\vartheta )|:\mathrm{\Psi }\in C\},$$

where

$$|\mathrm{\Psi }(\vartheta )|=|\mathcal{S}(\vartheta )|+|\mathcal{C}(\vartheta )|+|\mathcal{P}(\vartheta )|+|\mathcal{I}(\vartheta )|+|\mathcal{H}(\vartheta )|,$$

and $\mathcal{S},\mathcal{C},\mathcal{P},\mathcal{I},\mathcal{H}\in C$. Fractional order derivative of Caputo type is defined as [16]

$${}^C\mathcal{D}_{0^{+}}^{\varrho }\varkappa (\vartheta )=\frac{1}{\mathrm{\Gamma }(n-\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{n-\varrho -1}{\varkappa }^{(n)}(\kappa )d\kappa ,\vartheta >0,$$

where $n-1<\varrho \le n$, and $n=[\varrho ]+1$.

The Mittag-Leffler function (MLF) is defined by

$$E_{\varrho ,\sigma }(\mathcal{Z})=\sum _{i=0}^{\infty }\frac{{\mathcal{Z}}^i}{\mathrm{\Gamma }(i\varrho +\sigma )},\text{Re}(\varrho )>0,\mathcal{Z}\in \mathbb{C}.$$

The correspondent fractional integral is given by ([16])

$${\mathcal{I}}_{0^{+}}^{\varrho }\varkappa (\vartheta )=\frac{1}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}\varkappa (\kappa )d\kappa .$$

Lemma 4.1 [16] —

Let $n+1<\varrho \le n$ , $n\in \mathbb{N}$ and $\varkappa :\mho \to \mathbb{R}$ be a function. Then

$${\mathcal{I}}_{0^{+}}^{\varrho }{}^C\mathcal{D}_{0^{+}}^{\varrho }\varkappa (\vartheta )=\varkappa (\vartheta )-\sum _{j=0}^n\frac{{\varkappa }^{\left(j\right)}(0)}{j!}{\vartheta }^j.$$

Lemma 4.2 [28] —

For $0<\varrho \le 1$ , the generalized Taylor formula for $\varkappa (\vartheta )$ can be written as

$$\varkappa (\vartheta )=\sum _{i=0}^n\frac{{\vartheta }^i}{\mathrm{\Gamma }(i\varrho +1)}{}^C\mathcal{D}_{0^{+}}^{i\varrho }\varkappa (0)+\frac{1}{\mathrm{\Gamma }((n+1)\varrho +1)}{}^C\mathcal{D}_{0^{+}}^{(n+1)\varrho }\varkappa (\zeta )$$

such that $\zeta \in [0,\vartheta ]$ , $\vartheta \in (0,a]$ .

Lemma 4.3 [16] —

Let $0<\varrho \le 1$ and $g\in C(\mho ,\mathbb{R})$ . The solution of

$${}^C\mathcal{D}_{0^{+}}^{\varrho }\varkappa (\vartheta )=g(\vartheta )\varkappa (0)={\varkappa }_0$$

is given by

$$\varkappa (\vartheta )={\varkappa }_0+\frac{1}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}\varkappa (\kappa )d\kappa .$$

5. Model fundamental properties

In this section, we discuss the essential results of solutions of model (2), which are necessary for the proof of stability results.

5.1. Non-negative solution

It has been perceived that in many poverty models, the system attains its poverty-free state when corruption disappears from the population. We now give the existence condition for the non-negative solution of the model (2).

Let ${\mathbb{R}}_{+}^5=\{\varkappa \in {\mathbb{R}}^5:\varkappa \ge 0\}$ and let $\varkappa (\vartheta )={(\mathcal{S}(\vartheta ),\mathcal{C}(\vartheta ),\mathcal{P}(\vartheta ),\mathcal{I}(\vartheta ),\mathcal{H}(\vartheta ))}^T$. To prove the non-negative solution of model (2), we shall need the next generalized mean value theorem:

Theorem 5.1 [28] —

Let $h(\vartheta )\in C[0,T]$ and ${}^C\mathcal{D}_{0^{+}}^{\varrho }\in C(0,T]$ for $0<\varrho \le 1$ . Then for $\vartheta \in [0,T]$ , we have

$$\frac{h(\vartheta )-h(0)}{\vartheta }=\frac{1}{\mathrm{\Gamma }(\varrho )}{}^C\mathcal{D}_{0^{+}}^{\varrho }h(c),c\in [0,\vartheta ].$$

Corollary 5.2 [28] —

• 1. Let ${}^C\mathcal{D}_{0^{+}}^{\varrho }h(\vartheta )\ge 0$ , for all $\vartheta \in (0,T)$ . Then h is nondecreasing on $[0,T]$ .
• 2. Let ${}^C\mathcal{D}_{0^{+}}^{\varrho }h(\vartheta )\ge 0$ , for all $\vartheta \in (0,T)$ . Then h is nonincreasing on $[0,T]$ .

Theorem 5.3

A unique solution

$$\varkappa (\vartheta )={(\mathcal{S}(\vartheta ),\mathcal{C}(\vartheta ),\mathcal{P}(\vartheta ),\mathcal{I}(\vartheta ),\mathcal{H}(\vartheta ))}^T$$

for $\vartheta >0$ of the model (2) exists and will remains in ${\mathbb{R}}_{+}^5$ . Moreover, $\varkappa (\vartheta )$ is definitely positive.

Proof

First, all the right-hand sides of the model (2) are locally Lipschitzian and completely continuous on $\mathbb{R}$. The solution $\varkappa (\vartheta )={(\mathcal{S}(\vartheta ),\mathcal{C}(\vartheta ),\mathcal{P}(\vartheta ),\mathcal{I}(\vartheta ),\mathcal{H}(\vartheta ))}^T$ with $\varkappa (0)={(\mathcal{S}(0),\mathcal{C}(0),\mathcal{P}(0),\mathcal{I}(0),\mathcal{H}(0))}^T$ exists and is unique on interval $[0,\infty )$. From above we get (3) as

$$\begin{array}{c}\hfill {{}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{S}|}_{\mathcal{S}=0}=\mathrm{\Lambda }\ge 0\hfill \\ \hfill {{}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{C}|}_{\mathcal{C}=0}={\gamma }_2\frac{\mathcal{PS}}{\mathcal{N}}+\psi (1-\varphi )\mathcal{I}\ge 0,\hfill \\ \hfill {{}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{P}|}_{\mathcal{P}=0}={\gamma }_1\mathcal{C}\ge 0,\hfill \\ \hfill {{}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{I}|}_{\mathcal{I}=0}=\omega \mathcal{C}\ge 0,\hfill \\ \hfill {{}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{H}|}_{\mathcal{H}=0}={\beta }_c\mathcal{C}+{\beta }_p\mathcal{P}+\psi \varphi \mathcal{I}+\sigma \mathcal{S}\ge 0.\hfill \end{array}$$ (3)

As per Corollary 5.2, we deduced that the solution would be in ${\mathbb{R}}_{+}^5$ for all for $\vartheta >0$.

Finally, to affirm the boundedness of the model, we have $\mathcal{N}=\mathcal{S}+\mathcal{C}+\mathcal{P}+\mathcal{I}+\mathcal{H}$. Then, by applying the Caputo operator along with using model (2), we get

$${}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{N}={}^C\mathcal{D}_{0^{+}}^{\varrho }[\mathcal{S}+\mathcal{C}+\mathcal{P}+\mathcal{I}+\mathcal{H}]=\mathrm{\Lambda }-\mu \mathcal{S}-\mu \mathcal{C}-\mu \mathcal{P}-\mu \mathcal{I}-\mu \mathcal{H}=\mathrm{\Lambda }-\mu \mathcal{N}.$$

It follows that

$$\mathcal{N}\left(\vartheta \right)=\mathcal{N}\left(0\right)E_{\varrho ,1}(-\mu {\vartheta }^{\varrho })+\mathrm{\Lambda }{\vartheta }^{\varrho }{E}_{\varrho ,\varrho +1}(-\mu {\vartheta }^{\varrho }),$$

where, $E_{\varrho ,\sigma }$ is Mittag-Leffler function which has an asymptotic behavior [17], hence, we get

$$\mathcal{N}\left(\vartheta \right)\to \frac{\mathrm{\Lambda }}{\mu },\vartheta \to \infty .$$

Then, when $\vartheta \to \infty$ we get the solution $\mathcal{N}\left(\vartheta \right)\subseteq [0,\frac{\mathrm{\Lambda }}{\mu }]$.

Therefore, the region

$$\mathcal{D}={\{(\mathcal{S},\mathcal{C},\mathcal{P},\mathcal{I},\mathcal{H})\in {\mathbb{R}}_{+}^5:\mathcal{N}\left(\vartheta \right)\le \frac{\mathrm{\Lambda }}{\mu }\}|}_{\vartheta \to \infty }$$

attracts all solutions in ${\mathbb{R}}_{+}^5$. □

5.2. Equilibrium points and stability analysis

The poverty and corruption-free equilibrium (FE) is a state where society is free of poverty and corruption. For the equilibrium points of the model (2), we have

$${}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{S}={}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{C}={}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{P}={}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{I}={}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{H}=0.$$

The FE point is denoted as ${\mathcal{PC}}_0=({\mathcal{S}}^0,{\mathcal{C}}^0,{\mathcal{P}}^0,{\mathcal{I}}^0,{\mathcal{H}}^0)$ given as

$${\mathcal{PC}}_0=(\frac{\mathrm{\Lambda }}{\mu +\sigma },0,0,0,\frac{\mathrm{\Lambda }\sigma }{\mu (\mu +\sigma )}),$$

where

$${\mathcal{S}}^0=\frac{\mathrm{\Lambda }}{\mu +\sigma },{\mathcal{H}}^0=\frac{\mathrm{\Lambda }\sigma }{\mu (\mu +\sigma )}.$$

Theorem 5.4

The reproduction number ${\mathcal{R}}_{\mathcal{PC}}$ of the model (2) is

$${\mathcal{R}}_{\mathcal{PC}}=\frac{\mathrm{\Lambda }{\pi }_2(1-{\tau }_2)}{N(\mu +{\beta }_p)(\mu +\sigma )}+\frac{\mathrm{\Lambda }{\pi }_1(\mu +\psi -(\mu +\psi ){\tau }_1)}{N(\sigma +\mu )k_1}.$$

Proof

The next-generation operator method is used here to obtain the ${\mathcal{R}}_{PC}$ as follows:

$$\mathcal{F}=\left(\begin{array}{ccc}\hfill \frac{{\pi }_1(1-{\tau }_1){\mathcal{S}}^0}{\mathcal{N}}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \frac{{\pi }_2(1-{\tau }_2){\mathcal{S}}^0}{\mathcal{N}}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),$$

$$\mathcal{V}=\left(\begin{array}{ccc}\hfill {\gamma }_1+\omega +{\beta }_c+\mu \hfill & \hfill 0\hfill & \hfill -\psi (1-\varphi )\hfill \\ \hfill -{\gamma }_1\hfill & \hfill {\beta }_p+\mu \hfill & \hfill 0\hfill \\ \hfill -\omega \hfill & \hfill 0\hfill & \hfill \psi (1-\varphi )+\psi \varphi +\mu \hfill \end{array}\right).$$

Next, we have to find the generation matrix as ${\mathcal{FV}}^{-1}$

$${\mathcal{V}}^{-1}=\left(\begin{array}{ccc}\hfill \frac{\mu +\psi }{k_1}\hfill & \hfill 0\hfill & \hfill \frac{\psi -\varphi \psi }{k_1}\hfill \\ \hfill \frac{\mu {\gamma }_1+\psi {\gamma }_1}{k_2}\hfill & \hfill \frac{1}{\mu +{\beta }_p}\hfill & \hfill \frac{\psi {\gamma }_1-\varphi \psi {\gamma }_1}{k_2}\hfill \\ \hfill \frac{\omega }{k_1}\hfill & \hfill 0\hfill & \hfill \frac{\mu +\omega +{\gamma }_1+{\beta }_c}{k_1}\hfill \end{array}\right)$$

where

$$k_1={\mu }^2+\mu \psi +\mu \omega +\mu {\gamma }_1+\psi {\gamma }_1+\mu {\beta }_c+\psi {\beta }_c+\varphi \psi \omega ,k_2={\mu }^3+{\mu }^2\psi +{\mu }^2\omega +{\mu }^2{\gamma }_1+{\mu }^2{\beta }_c+{\mu }^2{\beta }_p+\mu \psi {\gamma }_1+\mu \psi {\beta }_c+\mu \psi {\beta }_p+\mu \omega {\beta }_p+\mu {\gamma }_1{\beta }_p+\psi {\gamma }_1{\beta }_p+\mu {\beta }_c{\beta }_p+\psi {\beta }_c{\beta }_p+\mu \varphi \psi \omega +\varphi \psi \omega {\beta }_p$$

and

$${\mathcal{FV}}^{-1}=\left(\begin{array}{ccc}\hfill -\frac{Q_1}{N(\mu +\sigma )k_1}\hfill & \hfill 0\hfill & \hfill -\frac{Q_2}{N(\mu +\sigma )k_1}\hfill \\ \hfill -\frac{Q_3}{N(\mu +\sigma )k_2}\hfill & \hfill -\frac{\mathrm{\Lambda }{\pi }_2({\tau }_2-1)}{N(\mu +\sigma )(\mu +{\beta }_p)}\hfill & \hfill -\frac{Q_4}{N(\mu +\sigma )k_2}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right).$$

So that

$$|{\mathcal{FV}}^{-1}-\lambda I|=\left|\begin{array}{ccc}\hfill -\frac{Q_1}{N(\mu +\sigma )k_1}-\lambda \hfill & \hfill 0\hfill & \hfill -\frac{Q_2}{N(\mu +\sigma )k_1}\hfill \\ \hfill -\frac{Q_3}{N(\mu +\sigma )k_2}\hfill & \hfill -\frac{\mathrm{\Lambda }{\pi }_2({\tau }_2-1)}{N(\mu +\sigma )(\mu +{\beta }_p)}-\lambda \hfill & \hfill -\frac{Q_4}{N(\mu +\sigma )k_2}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -\lambda \hfill \end{array}\right|=0$$

where $Q_1=\mathrm{\Lambda }{\pi }_1({\tau }_1-1)(\mu +\psi )$, $Q_2=\mathrm{\Lambda }{\pi }_1(\psi -\varphi \psi )({\tau }_1-1)$, $Q_3=\mathrm{\Lambda }{\pi }_2({\tau }_2-1)(\mu {\gamma }_1+\psi {\gamma }_1)$, $Q_4=\mathrm{\Lambda }{\pi }_2({\tau }_2-1)(\psi {\gamma }_1-\varphi \psi {\gamma }_1)$. Thus, we get ${\lambda }_1=\frac{\mathrm{\Lambda }{\pi }_2(1-{\tau }_2)}{N(\mu +{\beta }_p)(\mu +\sigma )},{\lambda }_2=0$, and ${\lambda }_3=\frac{\mathrm{\Lambda }{\pi }_1(\mu +\psi -(\mu +\psi ){\tau }_1)}{N(\sigma +\mu )k_1}$, from which the basic reproduction number of our model (2) is

$$\rho ({\mathcal{FV}}^{-1})={\mathcal{R}}_{\mathcal{PC}}=\frac{\mathrm{\Lambda }{\pi }_2(1-{\tau }_2)}{N(\mu +{\beta }_p)(\mu +\sigma )}+\frac{\mathrm{\Lambda }{\pi }_1(\mu +\psi -(\mu +\psi ){\tau }_1)}{N(\sigma +\mu )k_1}={\mathcal{R}}_{\mathcal{P}}+{\mathcal{R}}_{\mathcal{C}},$$

$${\mathcal{R}}_{\mathcal{P}}=\frac{\mathrm{\Lambda }{\pi }_2(1-{\tau }_2)}{N(\mu +{\beta }_p)(\mu +\sigma )},{\mathcal{R}}_{\mathcal{C}}=\frac{\mathrm{\Lambda }{\pi }_1(\mu +\psi -(\mu +\psi ){\tau }_1)}{N(\sigma +\mu )k_1},$$

where ${\mathcal{R}}_{\mathcal{C}}$ is the minimum number of cases of corruption caused by a corrupt individual among a population living in poverty, and ${\mathcal{R}}_{\mathcal{P}}$ is the minimum number of poor individuals caused by one poor individual from a corrupt population. □

The local and global stability of poverty and corruption FE is described by the following two theorems (see [26]).

Theorem 5.5

The poverty and corruption-FE ${\mathcal{PC}}_0$ of the model (2) is locally asymptotically stable if ${\mathcal{R}}_{\mathcal{C}}<1$ and ${\mathcal{R}}_{\mathcal{P}}<1$ .

Theorem 5.6

The poverty and corruption-FE ${\mathcal{PC}}_0$ of the model (2) is globally asymptotically stable if ${\mathcal{R}}_{\mathcal{C}}<1$ and ${\mathcal{R}}_{\mathcal{P}}<1$ .

6. Theoretical results of the model (2)

The existence of the solution to the model is checked by converting the model into a system of fractional differential equations. Indeed, the right sides of the model (2) can be expressed in (4) as

$${\phi }_1=\mathrm{\Lambda }-{\pi }_1(1-{\tau }_1)\frac{\mathcal{CS}}{N}-{\pi }_2(1-{\tau }_2)\frac{\mathcal{PS}}{N}-(\sigma +\mu )\mathcal{S},{\phi }_2={\pi }_1(1-{\tau }_1)\frac{\mathcal{CS}}{N}+{\gamma }_2\frac{\mathcal{PS}}{N}+\psi (1-\varphi )\mathcal{I}-({\gamma }_1+\omega +{\beta }_c+\mu )\mathcal{C},{\phi }_3={\pi }_2(1-{\tau }_2)\frac{\mathcal{PS}}{N}+{\gamma }_1\mathcal{C}-{\gamma }_2\frac{\mathcal{PS}}{N}-({\beta }_p+\mu )\mathcal{P},{\phi }_4=\omega \mathcal{C}-(\psi (1-\varphi )+\psi \varphi +\mu )\mathcal{I},{\phi }_5={\beta }_c\mathcal{C}+{\beta }_p\mathcal{P}+\psi \varphi \mathcal{I}+\sigma \mathcal{S}-\mu \mathcal{H}.$$ (4)

Using (4), the model (2) in simple way can be described in (5) as

$${}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{V}(\vartheta )=\mathrm{\Psi }(\vartheta ,\mathcal{V}(\vartheta )),\mathcal{V}(0)={\mathcal{V}}_0.$$ (5)

According to Lemma 4.3, the system (5) is equivalent to the following integral equation (6)

$$\mathcal{V}(\vartheta )={\mathcal{V}}_0+\frac{1}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}\mathrm{\Psi }(\kappa ,\mathcal{V}(\kappa ))d\kappa ,$$ (6)

where

$$\{\begin{array}{c}\hfill \mathcal{V}(\vartheta )={(\mathcal{S},\mathcal{C},\mathcal{P},\mathcal{I},\mathcal{H})}^T,\hfill \\ \hfill {\mathcal{V}}_0={({\mathcal{S}}_0,{\mathcal{C}}_0,{\mathcal{P}}_0,{\mathcal{I}}_0,{\mathcal{H}}_0)}^T,\hfill \\ \hfill \mathrm{\Psi }(\vartheta ,\mathcal{V}(\vartheta ))={({\phi }_1,{\phi }_2,{\phi }_3,{\phi }_4,{\phi }_5)}^T\hfill \end{array}$$

and ${\phi }_i$ $(i=1,2,...,5)$ are defined as in (4).

Theorem 6.1

Suppose

• $(H_1)$
  $\mathrm{\Psi }:\mho \times \mathbb{R}\to \mathbb{R}$and ∃ $K_{\mathrm{\Psi }}>0$, such that

  $$|\mathrm{\Psi }(\vartheta ,\mathcal{V}(\vartheta ))-\mathrm{\Psi }(\vartheta ,{\mathcal{V}}^{\mathrm{★}}(\vartheta ))|\le K_{\mathrm{\Psi }}|\mathcal{V}(\vartheta )-{\mathcal{V}}^{\mathrm{★}}(\vartheta )|,for\vartheta \in \mho and\mathcal{V},{\mathcal{V}}^{\mathrm{★}}\in \mathbb{R}.$$
• $(H_2)$ $\frac{T^{\varrho }{K}_{\mathrm{\Psi }}}{\mathrm{\Gamma }(\varrho +1)}<1$ ,

then the model (2) has a unique solution.

Proof

Let C be a Banach space and $\mathrm{\Phi }:C\to C$ be an operator defined by $\mathrm{\Phi }\mathcal{V}(\vartheta )=\mathcal{V}(\vartheta )$ where $\mathcal{V}(\vartheta )$ is defined by (6) in (7) as

$$\mathrm{\Phi }\mathcal{V}(\vartheta )={\mathcal{V}}_0+\frac{1}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}\mathrm{\Psi }(\kappa ,\mathcal{V}(\kappa ))d\kappa .$$ (7)

Let $\mathcal{V},{\mathcal{V}}^{\mathrm{★}}\in C$ and $\vartheta \in \mho$. Then by (7) and (H₁), we have

$$\Vert \mathrm{\Phi }\mathcal{V}-\mathrm{\Phi }{\mathcal{V}}^{\mathrm{★}}\Vert =\underset{\vartheta \in \mho }{\max }|\mathrm{\Phi }\mathcal{V}(\vartheta )-\mathrm{\Phi }{\mathcal{V}}^{\mathrm{★}}(\vartheta )|\le \underset{\vartheta \in \mho }{\max }\frac{1}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}\times |\mathrm{\Psi }(\kappa ,\mathcal{V}(\kappa ))-\mathrm{\Psi }(\kappa ,{\mathcal{V}}^{\mathrm{★}}(\kappa ))|d\kappa \le \frac{K_{\mathrm{\Psi }}}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}\underset{\kappa \in \mho }{\max }|\mathcal{V}(\kappa )-{\mathcal{V}}^{\mathrm{★}}(\kappa )|d\kappa \le \frac{T^{\varrho }{K}_{\mathrm{\Psi }}}{\mathrm{\Gamma }(\varrho +1)}\Vert \mathcal{V}-{\mathcal{V}}^{\mathrm{★}}\Vert .$$

Due to (H₂), Φ is contraction. As a result the model (2) has a unique solution. □

6.1. HU stability

In this portion, we derive the HU-type stability of the system (5), which leads us to the HU-type stability of the model (2). Consider a small perturbation $\sigma \in C$ such that $|\sigma (\vartheta )|\le ϵ$ for $\vartheta \in \mho$ with $\sigma (0)=0$.

Lemma 6.2

The solution of problem (8)

$${}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{V}(\vartheta )=\mathrm{\Psi }(\vartheta ,\mathcal{V}(\vartheta ))+\sigma (\vartheta ),\mathcal{V}(0)={\mathcal{V}}_0$$ (8)

obeys the inequity (9) given by

$$|\mathcal{V}(\vartheta )-({\mathcal{V}}_0+\frac{1}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}\mathrm{\Psi }(\kappa ,\mathcal{V}(\kappa ))d\kappa )|\le \frac{T^{\varrho }ϵ}{\mathrm{\Gamma }(\varrho +1)}.$$ (9)

Proof

As per Lemma 4.3, the system (8) has a solution

$$\mathcal{V}(\vartheta )={\mathcal{V}}_0+\frac{1}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}[\mathrm{\Psi }(\kappa ,\mathcal{V}(\kappa ))+\sigma (\kappa )]d\kappa .$$

Hence

$$|\mathcal{V}(\vartheta )-({\mathcal{V}}_0+\frac{1}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}\mathrm{\Psi }(\kappa ,\mathcal{V}(\kappa ))d\kappa )|\le \frac{1}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}|\sigma (\kappa )|d\kappa \le \frac{T^{\varrho }ϵ}{\mathrm{\Gamma }(\varrho +1)}.$$

 □

Theorem 6.3

Under hypotheses of Theorem 6.1 along with (9) , the solution of (6) is HU stable and consequently, the model (2) is HU stable.

Proof

Let $\mathcal{V}\in C$ be a unique solution and $\mathcal{U}$ be any solution of (6). Then

$$|\mathcal{V}(\vartheta )-\mathcal{U}(\vartheta )|=|\mathcal{V}(\vartheta )-({\mathcal{V}}_0+\frac{1}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}\mathrm{\Psi }(\kappa ,\mathcal{U}(\kappa ))d\kappa )|\le |\mathcal{V}(\vartheta )-({\mathcal{V}}_0+\frac{1}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}\mathrm{\Psi }(\kappa ,\mathcal{V}(\kappa ))d\kappa )|+|({\mathcal{V}}_0+\frac{1}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}\mathrm{\Psi }(\kappa ,\mathcal{U}(\kappa ))d\kappa )-({\mathcal{V}}_0+\frac{1}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}\mathrm{\Psi }(\kappa ,\mathcal{U}(\kappa ))d\kappa )|\le \frac{T^{\varrho }ϵ}{\mathrm{\Gamma }(\varrho +1)}+\frac{1}{\mathrm{\Gamma }(\varrho )}\underset{0}{\overset{\vartheta }{\int }}{(\vartheta -\kappa )}^{\varrho -1}\times |\mathrm{\Psi }(\kappa ,\mathcal{V}(\kappa ))-\mathrm{\Psi }(\kappa ,\mathcal{U}(\kappa ))|d\kappa \le \frac{T^{\varrho }ϵ}{\mathrm{\Gamma }(\varrho +1)}+\frac{T^{\varrho }{K}_{\mathrm{\Psi }}}{\mathrm{\Gamma }(\varrho +1)}\Vert \mathcal{V}-\mathcal{U}\Vert ,$$

which implies $(1-\frac{T^{\varrho }{K}_{\mathrm{\Psi }}}{\mathrm{\Gamma }(\varrho +1)})\Vert \mathcal{V}-\mathcal{U}\Vert \le \frac{T^{\varrho }ϵ}{\mathrm{\Gamma }(\varrho +1)}$. Since $\frac{T^{\varrho }{K}_{\mathrm{\Psi }}}{\mathrm{\Gamma }(\varrho +1)}<1$, we get

$$\Vert \mathcal{V}-\mathcal{U}\Vert \le C_Tϵ,$$

where $C_T=\frac{T^{\varrho }}{\mathrm{\Gamma }(\varrho +1)-T^{\varrho }{K}_{\mathrm{\Psi }}}$. Hence the required results about HU stability. □

6.2. Numerical approach

In this portion, we extend the numerical Euler method [15] for our given model (2). In another meaning, we have to evaluate the approximate solutions of model (2) under the Caputo derivative. The model (2) can be written in (10) as

$$\begin{array}{c}\hfill {}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{S}(\vartheta )={\phi }_1(\mathcal{S},\mathcal{C},\mathcal{P},\mathcal{I},\mathcal{H}),\hfill \\ \hfill {}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{C}(\vartheta )={\phi }_2(\mathcal{S},\mathcal{C},\mathcal{P},\mathcal{I},\mathcal{H}),\hfill \\ \hfill {}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{P}(\vartheta )={\phi }_3(\mathcal{S},\mathcal{C},\mathcal{P},\mathcal{I},\mathcal{H}),\hfill \\ \hfill {}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{I}(\vartheta )={\phi }_4(\mathcal{S},\mathcal{C},\mathcal{P},\mathcal{I},\mathcal{H}),\hfill \\ \hfill {}^C\mathcal{D}_{0^{+}}^{\varrho }\mathcal{H}(\vartheta )={\phi }_5(\mathcal{S},\mathcal{C},\mathcal{P},\mathcal{I},\mathcal{H}),\hfill \end{array}$$ (10)

where ${\phi }_i$ ($i=1,2,...,5$) are defined by (4). Now, we subdivide the interval ℧ into ℓ subintervals $[{\vartheta }_s,{\vartheta }_{s+1}]$ with uniform width $h=T/r$ using the nodes ${\vartheta }_s=sh$; for $s=0,1,...,r$. Let $\mathcal{S}(\vartheta ),\mathcal{C}(\vartheta ),\mathcal{P}(\vartheta ),\mathcal{I}(\vartheta ),and\mathcal{H}(\vartheta )$ and their derivatives for the Caputo operator are continuous ℧. For the model (10), we apply the modified Euler's method about $\vartheta ={\vartheta }_0=0$, and for each value ϑ take value a, the term for ${\vartheta }_1$ is obtained in (11) as

$$\{\begin{array}{c}\hfill \mathcal{S}({\vartheta }_1)=\mathcal{S}({\vartheta }_0)+{\phi }_{1,{\vartheta }_0}\frac{{\vartheta }^{\varrho }}{\mathrm{\Gamma }(\varrho +1)}+{{}^C\mathcal{D}_{0^{+}}^{2\varrho }\mathcal{S}(\vartheta )|}_{\vartheta =a}\frac{{\vartheta }^{2\varrho }}{\mathrm{\Gamma }(2\varrho +1)},\hfill \\ \hfill \mathcal{C}({\vartheta }_1)=\mathcal{C}({\vartheta }_0)+{\phi }_{2,{\vartheta }_0}\frac{{\vartheta }^{\varrho }}{\mathrm{\Gamma }(\varrho +1)}+{{}^C\mathcal{D}_{0^{+}}^{2\varrho }\mathcal{C}(\vartheta )|}_{\vartheta =a}\frac{{\vartheta }^{2\varrho }}{\mathrm{\Gamma }(2\varrho +1)},\hfill \\ \hfill \mathcal{P}({\vartheta }_1)=\mathcal{P}({\vartheta }_0)+{\phi }_{3,{\vartheta }_0}\frac{{\vartheta }^{\varrho }}{\mathrm{\Gamma }(\varrho +1)}+{{}^C\mathcal{D}_{0^{+}}^{2\varrho }\mathcal{P}(\vartheta )|}_{\vartheta =a}\frac{{\vartheta }^{2\varrho }}{\mathrm{\Gamma }(2\varrho +1)},\hfill \\ \hfill \mathcal{I}({\vartheta }_1)=\mathcal{I}({\vartheta }_0)+{\phi }_{4,{\vartheta }_0}\frac{{\vartheta }^{\varrho }}{\mathrm{\Gamma }(\varrho +1)}+{{}^C\mathcal{D}_{0^{+}}^{2\varrho }\mathcal{I}(\vartheta )|}_{\vartheta =a}\frac{{\vartheta }^{2\varrho }}{\mathrm{\Gamma }(2\varrho +1)},\hfill \\ \hfill \mathcal{H}({\vartheta }_1)=\mathcal{H}({\vartheta }_0)+{\phi }_{5,{\vartheta }_0}\frac{{\vartheta }^{\varrho }}{\mathrm{\Gamma }(\varrho +1)}+{{}^C\mathcal{D}_{0^{+}}^{2\varrho }\mathcal{H}(\vartheta )|}_{\vartheta =a}\frac{{\vartheta }^{2\varrho }}{\mathrm{\Gamma }(2\varrho +1)},\hfill \end{array}$$ (11)

where ${\phi }_{i,{\vartheta }_0}:={\phi }_i(\mathcal{S}({\vartheta }_0),\mathcal{C}({\vartheta }_0),\mathcal{P}({\vartheta }_0),\mathcal{I}({\vartheta }_0),\mathcal{H}({\vartheta }_0))$ $(i=1,2,...,5)$. Allow the progression to estimate h is picked small enough; then, at that point, we might disregard the second-order term, including $h^{2\varrho }$, it follows from (11) that

$$\{\begin{array}{c}\hfill \mathcal{S}({\vartheta }_1)=\mathcal{S}({\vartheta }_0)+{\phi }_{1,{\vartheta }_0}\frac{{\vartheta }^{\varrho }}{\mathrm{\Gamma }(\varrho +1)},\hfill \\ \hfill \mathcal{C}({\vartheta }_1)=\mathcal{C}({\vartheta }_0)+{\phi }_{2,{\vartheta }_0}\frac{{\vartheta }^{\varrho }}{\mathrm{\Gamma }(\varrho +1)},\hfill \\ \hfill \mathcal{P}({\vartheta }_1)=\mathcal{P}({\vartheta }_0)+{\phi }_{3,{\vartheta }_0}\frac{{\vartheta }^{\varrho }}{\mathrm{\Gamma }(\varrho +1)},\hfill \\ \hfill \mathcal{I}({\vartheta }_1)=\mathcal{I}({\vartheta }_0)+{\phi }_{4,{\vartheta }_0}\frac{{\vartheta }^{\varrho }}{\mathrm{\Gamma }(\varrho +1)},\hfill \\ \hfill \mathcal{H}({\vartheta }_1)=\mathcal{H}({\vartheta }_0)+{\phi }_{5,{\vartheta }_0}\frac{{\vartheta }^{\varrho }}{\mathrm{\Gamma }(\varrho +1)}.\hfill \end{array}$$

Continuing on aforementioned manner, an overall formula at ${\vartheta }_{s+1}={\vartheta }_s+h$ is laid out as

$$\{\begin{array}{c}\hfill \mathcal{S}({\vartheta }_{s+1})=\mathcal{S}({\vartheta }_s)+{\phi }_{1,{\vartheta }_s}\frac{h^{\varrho }}{\mathrm{\Gamma }(\varrho +1)},\hfill \\ \hfill \mathcal{C}({\vartheta }_{s+1})=\mathcal{C}({\vartheta }_s)+{\phi }_{2,{\vartheta }_s}\frac{h^{\varrho }}{\mathrm{\Gamma }(\varrho +1)},\hfill \\ \hfill \mathcal{P}({\vartheta }_{s+1})=\mathcal{P}({\vartheta }_s)+{\phi }_{3,{\vartheta }_s}\frac{h^{\varrho }}{\mathrm{\Gamma }(\varrho +1)},\hfill \\ \hfill \mathcal{I}({\vartheta }_{s+1})=\mathcal{I}({\vartheta }_s)+{\phi }_{4,{\vartheta }_s}\frac{h^{\varrho }}{\mathrm{\Gamma }(\varrho +1)},\hfill \\ \hfill \mathcal{H}({\vartheta }_{s+1})=\mathcal{H}({\vartheta }_s)+{\phi }_{5,{\vartheta }_s}\frac{h^{\varrho }}{\mathrm{\Gamma }(\varrho +1)},\hfill \end{array}$$

where $s=0,1,2,...,r-1$.

7. Simulations and discussion

We simulate our model corresponding to the numerical values in Table 4 and using the aforesaid numerical scheme in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, respectively, against various fractional orders.

Figure 4.

Graphical presentation of the susceptible individuals' density in the proposed model corresponding to different fractional orders.

Figure 5.

Graphical presentation of the corrupt individuals' density in the proposed model corresponding to different fractional orders.

Figure 6.

Graphical presentation of the Poverty/poor individuals density in the proposed model corresponding to different fractional orders.

Figure 7.

Graphical presentation of the Prosecuted/Jailed individuals density in the proposed model corresponding to different fractional orders.

Figure 8.

Graphical presentation of the Honest Individuals density in the proposed model corresponding to different fractional orders.

During the fierce wars and terrors and bad economic conditions as well as line order situation, the country of Yemen has been driven to a very worst case. Due to these facts in the last eighteen years, the density of susceptible people shows an increase in their numbers in Fig. 4. In the same vein, the corrupt individuals' density has also increased, as in Fig. 5. Further, the density of poverty has rapidly risen, as in Fig. 6. Since the due line order situation, the density of prosecuted/Jailed individuals has declined during the past eighteen years, as in Fig. 7. This whole situation affected the honest people density which also went on to decrease, as presented in Fig. 8. Further, since these dynamics show the current scenario of the country, its line order situation, and economic condition, all are going on disturbance very rapidly.

Real and simulated results are compared in Figure 9, Figure 10 which show close resemblance.

Figure 9.

Comparison of real and simulated data for Corrupt individuals at different fractional orders.

Figure 10.

Comparison of real and simulated data for Poverty/poor individuals density proposed model corresponding to different fractional orders.

8. Conclusion

We have provided a brief overview of the two most important problems in the world and their seriousness and relationship in the literature review. After that, the corruption index data were collected from Transparency International and the poverty rates in Yemen were collected from household budget surveys and annual books issued by the Central Bureau of Statistics from 2003 to 2022. The indicators of corruption and poverty rate have been studied and analyzed using the linear model during the study period by applying for the Eviews program. Then, we have already presented a prediction of poverty rates for the years (2023 and 2024), using the results of the standard problem-free model. On the mathematical side, we have developed a poverty and corruption model in the frame of Caputo fractional derivatives. The equilibrium points, the basic reproduction number, and the positive solution of the considered model have been obtained. Then, the uniqueness and UH stability of the solution to the proposed model were investigated via Banach's fixed point theorem. The statistical results of this paper may help decision-makers combat poverty, and it is necessary to start fighting corruption because of its adverse effects on growth, government expenditures, and increasing poverty. Numerous results like the existence theory, numerical interpretation and comparison with real data simulated values have been demonstrated. In the future, we will extend the said work in more generalized form.

CRediT authorship contribution statement

Mansour A. Abdulwasaa: Investigation, Conceptualization. Sunil V. Kawale: Methodology, Data curation. Mohammed S. Abdo: Visualization, Validation, Software. M. Daher Albalwi: Visualization, Validation, Resources. Kamal Shah: Methodology, Investigation, Formal analysis. Bahaaeldin Abdalla: Writing – review & editing, Visualization. Thabet Abdeljawad: Writing – review & editing, Visualization.

Declaration of Competing Interest

No conflict of interest exists.

Data availability

No data was used for the research described in the article.