Revised Fixed-base Model for Estimation of Adult Mortality in Developing Countries

This paper presents a revised Fixed-base model (RFBM) for the estimation of adult mortality in developing countries. The RFBM also derives estimates of adult mortality from the age distributions of two censuses by 5-year age-groups. Unlike the original version, RFBM does not depend on person – years lived between exact ages 0 and 5 years ( 5 0 L ) and the mid-point population of those aged 0 – 4 years ( 5 0 N ). Both 5 0 L and 5 0 N are known to be affected by under-reporting in most developing countries. The estimates of the RFBM appear to have improved and compare favourably with estimates from other methods. We recommend the use of the model, especially when there is insufficient data for direct estimation of adult mortality in developing countries, among others.

Introduction
Mortality serves as a gauge of a nation's extent of socio-economic and health development. As the level of demographic parameters (especially mortality) of a country improves; the ranking of the country on health issues increases among the committee of nations. Among the components of population change, mortality stands out because it affects some age distributions directly and others indirectly Nwogu and Okoro [1]. The focus of this present study is on mortality at age five years and above (denoted as adult mortality in most intercensal methods).
The direct estimation of adult mortality in many developing countries is difficult due to unreliable or incomplete civil registration systems. In most developing countries, because of the limitations of civil registration systems, demographic parameters are derived from censuses and survey data ─ these two sources are often defective, Nwogu and Okoro, [1]; Okoro [2]; Okoro and Nwogu [3]. Hence, the dependence on indirect techniques for the estimation of adult mortality in developing countries.
Derivation of adult mortality from two census age distributions is an alternative means explored by most developing countries to overcome or augment her ineffective civil registration systems. The intercensal survival methods still relevant are those that assume arbitrary intercensal period such as Preston integrated method [4], Preston-Bennett method [5], United Nations Synthetic Survival Ratio [6], Fixed-base model proposed by Nwogu and Okoro [1], etc. The Fixed-base model and Preston-Bennett method [5] are similar and do not make reference to life tables.
The Fixed-base model is considered for revision because it adopted a more conservative approach in the estimation of person-years lived ( ) x n L and expectation of life at age x ( ) x e but not without a limitation. The probability of surviving to exact age one ( 1 l ) an index for infant mortality is known to produce poor results according to many authors (UN [7]; kpedekpo [8]; Nwogu [9]; Okoro et al. [10]). In deriving 0 5 L used in the estimation of x n L , 1 l is one of the components used. However, it has been shown that 1 l is affected by under reporting. If it is possible to obtain an estimate that does not depend on both There are six sections to the paper. The first section contains an introduction; the second section contains a review of the Fixed-base model; the third section contains the derivation of the revised Fixed-base model; the fourth section contains empirical examples; the fifth section contains discussion; and the final section contains the conclusion.

Fixed-base Model
Nwogu and Okoro [1] proposed a Fixed-base model for estimation of adult mortality in developing countries based on the concept of variable-r method. The model proposed by Nwogu and Okoro [1] for estimation of expectation of life at age x ( is the total of person-years lived between exact ages 0 and 5 years, and is the mid-point population, and is the age-specific growth rate, Ln is log base e . For the open age interval (   ), Nwogu and Okoro [1] applied an expression by Kpedekpo [8] for the estimate of the number of person-years lived from  years and where, 9 0 l is the radix (equal to one) while 5 l and 10 l are the probabilities of surviving to exact ages 5 and 10 respectively, and other elements are as defined in the original Fixed-base model.

Estimation of open ended interval life table function (
For the open age interval (   ) in this revised version, we adopted an approach by Preston-Bennett [5] for the estimate of the number of person-years lived from  years and above )]

Algorithm for estimation of adult mortality using the revised fixed-base model
The measure of adult mortality adopted in this study is the expectation of life at age x ) ( x e for ...
is the 5-year age distribution derived from the two census

Empirical Examples
The revised model with other related models were applied to selected developing countries in this section.

Application to Nigeria female populations: 1991 -2006
The revised model was applied to census age distributions of the Nigerian female populations, from 1991 to 2006. Equations (3.6) through (3.11) were applied to the female populations to obtain the expectation of life at age x.

Application to Zambia female populations: 2000 -2010
The revised model was also applied to Zambia Female Populations, 2000 -2010.

Discussion
The RFBM was applied to census Okoro and Nwogu [15]. The data used in preparing or simulating such life tables are not from developing countries. They are mere interpolated values that put question marks on such estimates.
For Zambia, the age-specific estimates of the expectation of life of the RFBM appear very consistent with the WHO estimates from ages 5 to 20. Then, from age 35 and above, the WHO and Preston-Bennett method estimates were consistently higher than the RFBM estimates. The observed differences among the methods may be due to reasons mentioned above. It may also be that both the World Health Organization and Preston-Bennett method overestimated the life expectancy for that age range. Even at that, WHO estimates are interpolated values from existing model life tables which makes it susceptible to errors while the RFBM, FBM, and Preston-Bennett method derive their estimates directly from reported census data which may be affected by agemisreporting. Just like other inter-censal survival methods, the RFBM may be biased by net-migration, completeness of the two censuses, etc., UN [6]; UN [7]; ECA [16].

Conclusion
This paper revised the Fixed-base model for the estimation of adult mortality in developing countries. The model can derive estimates of adult mortality from the age distributions of two censuses. The estimates from the RFBM compared favourably well with estimates from other methods. World Health Organization estimates are used for assessment in this study not as a direct comparison of results because the data sources and methodology are not the same. The RFBM appear more consistent with the original version and the WHO estimates. The RFBM is a good alternative for the estimation of adult mortality in developing countries. The use of indirect techniques for estimation of most demographic parameters and socio-economic indicators should not obstruct the need to improve the civil registration systems so that mortality estimates can be derived directly from death records without going through the rigorous processes of indirect techniques.

APPENDIX
Appendix A

Peer-review history:
The peer review history for this paper can be accessed here (Please copy paste the total link in your browser address bar) https://www.sdiarticle5.com/review-history/88889


POVERTY REDUCING EFFECTS OF PROCESSED BEEF MARKETERS: EVIDENCE FROM KADUNA, KANO AND KATSINA STATES OF NIGERIA

The objective of this study was to provide empirical information on the poverty reducing effects of processed marketers in Kaduna, Kano and Kastina States of Nigeria. The study made use of primary data, which were obtained through the use of a computer-aided personal interview (CAPI) software on tablets. Analysis of the data was done using Foster-Greer-Thorbecke's (FGT) Weighted poverty index and Tobit regression model. The results of the poverty profile of raw and processed beef cattle marketers in the study area showed that 40% and 38% of the raw and processed beef marketers respectively constituted the share of the population that cannot afford to buy a basic basket of goods (food and non-food) equivalent to the poverty line of N75, 600. The depth of poverty (poverty gap index) of poor raw beef marketers (0.10) was lower than that of the poor processed beef marketers (0.14). The degree of poverty among the poor raw beef marketers given by the estimated severity of poverty (0.04) was equally lower than that of the poor processed beef marketers (0.07). The results on poverty reducing effects of processed beef marketing activities on poverty revealed that raw beef and processed beef marketing activities significantly reduce poverty incidence by 1.7% and 3.3% for the beef cattle farmers and traders respectively. In the light of the decreasing unemployment, it is recommended that it is recommended that beef marketers should be supported with easy access to credit such as the collateral-free and interest-free loans under the MARKETMONI and TRADERMONI loan schemes of the Government Enterprise and Empowerment Programme (GEEP).

INTRODUCTION
Agriculture provides livelihoods to more than one billion people worldwide and remains the back bone of many low-income countries, accounting for 60.4 percent of employment and contributing up to twothirds of gross domestic product in some of those countries (East African Business Council (EABC, 2020). The Nigeria agricultural sector holds the key to the country's drive for economic diversification. Agricultural sector remains the largest employer of labour in the country, providing jobs for more than one third (36.4%) of the Nigerian labour force (Princewaterhouse Coopers (PWC), 2020). Livestock production accounted for 1.8% of the GDP and 6.9% to the agricultural GDP of Nigeria in 2018 (Central Bank of Nigeria (CBN), 2018). Nigeria is one of the leading livestock producers in Central and West Africa and the country's cattle population is estimated at over 16 million heads, far ahead of Niger (8.7 million), Mali (8.2 million) and Chad (7 million), (Bernard, Bonnet and Guibert, 2010). However, a more recent data puts the number of cattle heads as 19.5 million heads (Food and Agricultural Organization of the United Nation (FAOSTAT), 2017). Livestock production systems and marketing represent a potential pathway out of poverty for many smallholders' livestock farmers and tend to serve poor consumers, creating an even tighter focus on the poor (Ari et al., 2016). of cattle products especially beef and milk to meet animal protein requirement falls short of the demand. The demand-supply gap is further threatened by the reoccurring cattle rustling especially in north-west zone of Nigeria and farmers-herders conflicts across different parts of Nigeria (Shehu et al., 2017;Ajibo et al., 2018). The outcome of enhanced production and marketing of cattle and its products especially beef can potentially lead to better income and nutritional status of households thus positively impinging their living standard (Mafimisebi et al., 2014). In contrast to development approaches that focus narrowly on improving the capacities of smallholders to increase their productivity or better manage natural resources, proponents of value chain development challenges development organizations to work with diverse stakeholders to understand the performance of the value chain and identify mutually beneficial options for improving chain performance.
In order to examine these issues raised, the specific objective are to: i.
determine the poverty status of raw and processed beef marketers and ii.
determine the effects of marketing activities on the poverty status of raw and processed beef marketers.

MATERIALS AND METHOD Description of the Study Area
The study was conducted in Kaduna, Kano and Katsina states of north-west zone, Nigeria. The northwest comprises of Jigawa, Kano, Katsina, Kaduna, Zamfara, Sokoto and Kebbi states. According to the 2006 population census, the total population of the zone is estimated at 35.7 million with an average density of 103 persons per square kilometer (National Population Commission (NPC), 2006). The projected population of the zone in 2018 is about 48.3 million, based on an annual growth rate of 3.2%. The northwest zone is known for livestock production activities such as cattle, goat, sheep, poultry etc. The zone produced over half of the entire cattle in the country in the national agriculture sample survey with 9,892,240 cattle heads representing 52.4% of cattle in Nigeria (NBS/FMARD, 2012). Kaduna, Katsina and Kano States had 655,382, 2,008,592 and 546,303 cattle heads respectively (NBS/FMARD, 2012).

Sampling procedure
Two categories of respondents were used for the study. These are raw (wholesalers, retailers) and processed beef marketers (suya producers, kilishi producers, danbun nama producers). Multi-stage sampling technique was employed in the selection of the value chain actors. The selection of the sample for each actor was carried out in the following manner. Raw beef marketers: In the selection of raw beef marketers, 10 raw beef marketers (wholesalers and retailers) were randomly selected in each of the 6 LGAs to give a sample size of 60 raw beef marketers (wholesalers and retailers) Processed beef marketers: In the selection of processed beef marketers 10 processed beef marketers (suya producers, kilishi producers and danbun nama producers) to give a sample size of 60 processed beef marketers (suya producers kilishi producers, danbun nama producers) derived from a list of raw and processed beef marketers compiled with the assistance of extension agents and the beef marketers associations. The major limitation of the study is that the sample size of raw and processed beef marketers was not proportionately sampled, due to lack of information on the population of the actors in their respective market segments. The total sample size was 120. The study made use of primary data, which were obtained through the use of a computeraided personal interview (CAPI) version of survey instrument rather than a paper-based questionnaire to improve the quality of data. The survey was administered using trained enumerators under the supervision of the researcher. Two different sets of CAPI version of questionnaire were used; raw beef marketers and processed beef marketers.

Analytical Technique Foster-Greer-Thorbecke's (FGT) Weighted Poverty Index
The Foster, Greer and Thorbecke (FGT) measures of poverty are widely used because they are consistent and additively decomposable (Foster et al., 1984). This was used to achieve part of the objective of this study. Poverty head count index, poverty gap index and squared poverty gap index was computed to measure the incidence, depth and severity of poverty among the raw beef and processed beef marketers. A relative poverty line was constructed based on the Mean Per Adult-equivalent Household Expenditure (MPAHE) of the raw beef and processed beef marketers. To account for intra-household composition in estimating household expenditure, the modified OEDC equivalence scale was used to calculate the adultequivalent household size (Organisation for Economic Co-operation and Development ( Tobit regression model Tobit regression model was employed to achieve part of objective v. (Effects of beef cattle activities on the poverty status of raw and processed beef marketers). The Tobit model is an extension of Probit model, and it is one of the approaches dealing with the problem of censored data (Johnston and Dandiro, 1997). This is a hybrid of the discrete and continuous models. The use of Tobit model is conceptually preferable to convectional linear regression models because parameter estimates from the former overcome most weaknesses of linear probability model namely: providing estimates which are asymptotically consistent and efficient (Mcdonald and Moffit, 1980). The Tobit model is expressed based on Tobin (1958) as follows: where, = observed dependent variable measuring both the probability of being poor and poverty intensity of the raw and processed beef marketers (It is discrete, when the value chain actors are not poor and continuous, when they are poor).
= latent variable indicating that poverty may or may not be directly observable. Hence, poverty is observed if and unobservable if = the key interest is on causality between raw and processed beef marketing activities and poverty. In addition to the primary variable of interest given by returns on beef cattle value chain activities, other variables that affect poverty based on theory and empirical findings were included in the model to avoid model mis-specification which will result in biased estimates. Therefore, inclusion of a large vector of control variables reduces potential omitted variable bias (Wooldridge, 2016;Verhofstadt and Maertens, 2014). Following McDonald and Moffit (1980), the Tobit model can be further disaggregated to determine the effect of a change in the value of the i th variable on a change in the probability of a household being in poverty and the expected intensity of the poverty. The expected change in the intensity of poverty with respect to a change in an explanatory variable is given by a rescaled estimate which is the marginal effect of an explanatory variable on the expected value of the intensity of the poverty calculated at the mean values of the explanatory variables: Where, = cumulative normal distribution of z (adjustment factor) z = z-score for the area under normal curve = standard deviation of the error term The explicit form of the regression model is as follows:

Model for raw and process beef marketers
Where, = dependent variable (intensity of poverty among raw beef marketers and processed beef marketers with non-poor beef marketers having a value of zero and the farther away the value is from zero, the higher the intensity of poverty) = returns on raw and processed beef marketing (N per 100 kg of raw meat and processed beef sold) = age (years) educational level (years) household size (number) dependency ratio (ratio of the number of nonworking household members to working household members) crop income (N) access to credit (N) membership of association (number of years of spent as a member of association) = non-agricultural income (N) participation in other beef cattle value chain activities (Dummy: 1 yes, 0 otherwise) Log transformed monetary variables were used in the estimated OLS and Tobit regression models. The log transformation of monetary amounts is necessary to make such variable more normally distributed (less skewed or heteroskedastic) and reduce the influence of outliers (makes estimates less sensitive to outlying observations) to ensure more reliable estimates (Wooldridge, 2016).

Results and Discussion Effects of raw and processed beef marketing Activities on Poverty Status of the marketers
Based on the food and non-food consumption expenditure of the raw and processed beef marketers as presented in table 2, the relative poverty line was estimated to be N75, 600. Using the relative poverty line of N75, 600 in the study area, the poverty headcount index which is the share of the beef value chain actors whose per-adult equivalent consumption expenditure per annum is below the poverty line is 0.40 and 0.38 for raw beef marketers and processed beef marketers respectively. This implies that 40% and 38% of the raw beef marketers and processed beef marketers respectively constituted the share of the population that cannot afford to buy a basic basket of goods (food and non-food) equivalent to the poverty line. The depth of poverty (poverty gap index) of poor raw beef marketers (0.10) was lower than that of the poor processed beef marketers (0.14). This indicates that the poor raw beef marketers are farther away from the poverty line than the poor processed beef marketers and the implication is that it is easier for raw beef marketers to move above the poverty line or move out of the poverty trap than the poor processed beef marketers. The degree of inequality (squared poverty gap index or poverty severity) among the poor raw beef marketers given by the estimated severity of poverty (0.04) was equally lower than that of the processed beef marketers (0.07). This shows that poverty tends to be less severe among raw beef marketers than the processed beef marketers. From the poverty profile of raw and processed beef marketers based on the three poverty indices, it can be deduced that the processed beef marketers were worst off in terms of poverty in comparison with the processed beef marketers.

Table 2 Distribution of poverty indices of raw and processed beef Actors Poverty headcount Poverty gap Poverty severity
Raw beef marketers 0.40 0.10 0.04 Processed beef marketers 0.38 0.14 0.07

Effect of raw beef marketing activities on poverty status of the marketers
The maximum likelihood estimates of the Tobit regression in Table 3 show the determinants of poverty status of raw beef marketers with particular interest on the effect of returns on beef value chain activities (BCA). Other explanatory variables are included in the model to avoid omitted variable bias.
The log-likelihood ratio of 124.08 was significant at 1% probability level and this indicates the joint significance of the independent variables included in the model. The estimated coefficient of returns on BCA was negative and significant at 5% probability level which indicates that raw beef marketers returns on BCA significantly decreases the intensity of poverty among raw beef marketers by 1.7% ceteris paribus. This implies that an increase in returns realized from beef cattle value chain activities will decrease the likelihood and intensity of poverty among the raw beef marketers. This corroborates Ari et al. (2016) who noted that livestock production systems and marketing represent a potential pathway out of poverty for many smallholders' livestock value chain actors.
The other variables that had significant effect on poverty status of the beef marketers are dependency ratio, access to credit, membership of association, non-agricultural income and other value chain activities. The coefficient of dependency ratio was positive and significant at 10% probability level, which indicates that a unit increase in the dependency ratio tends to increase the likelihood and intensity of poverty among the raw beef marketers. This is likely because most dependants particularly the aged and children contribute less to raw income generation through raw beef marketing and other income sources. This study is in line with the findings of Ezeh et al. (2019) who found that number of dependants had a positive influence on the likelihood of poverty among beekeepers in Kaduna state. Access to credit had a negative and significant relationship with poverty status of raw beef marketers at 1% probability level. This implies that with more credit obtained, the likelihood and intensity of poverty will tend to decrease among the raw beef marketers. This is plausible as access to credit can enable raw beef marketers to address cash constraints in beef marketing and ultimately increase their income generation. This finding is in line with that of Onwumere et al. (2017) who stated that access to credit by farm households had significant negative relationship with poverty status at 10% level of significance indicating that the probability of being poor reduces with an additional unit of credit acquired. The coefficient of membership of association was positive and significant at 10% probability level, which is against a priori expectation of poverty reducing effect of social capital formation through membership of associations. This result implies that raw beef marketers who belong to an association are more vulnerable to poverty than raw beef marketers who do not belong to an association. A plausible explanation for this unexpected result is that raw beef marketers who are members of cooperative may be contributing their time and monetary resources with no commensurate benefits such as information, credit facilities, capacity building etc. from the association which results in their membership being counterproductive. This result is not in consonance with the finding of Ojo et al. (2020) who showed that membership of cooperative was negative and at 10% probability level. The coefficient of non-agricultural income was negative and significant at 5% probability level. This implies that the prevalence of poverty among raw beef marketers will be reduced as non-agricultural income increases. This likely arises as nonagricultural income complements the raw beef marketers' sources of income by availing the marketers' additional resources for investment and consumption purposes, which creates pathway out of poverty. The negative relationship between nonagricultural income earning and poverty was equally reported by Akpan et al. (2016). The estimated coefficient of other chain activities was negative and significant at 1% probability level. This implies that as the participation of raw beef marketers in other beef activities increases such as integrating forward into processed beef marketing, the likelihood and intensity of poverty among the raw beef marketers will tend to decrease. Explanatory variables such as age, education, household size and crop income were not significantly related to the poverty status of raw beef marketers in the study area. 124.08*** Note: *,** and *** indicates significance at 10, 5 and 1% probability levels respectively, BCA = beef cattle value chain activities, M.E = marginal effect on the censored expected value (all observations)

Effect of processed beef marketing activities on the poverty status of the marketers
The maximum likelihood estimates of the Tobit regression in Table 4 show the determinants of poverty status of processed beef marketers with particular interest on the effect of returns on beef value chain activities (BCA). Other explanatory variables are included in the model to avoid omitted variable bias. The log-likelihood ratio of 95.22 was significant at 1% probability level and this indicates the joint significance of the independent variables included in the model. The estimated coefficient of returns on BCA was negative and significant at 5% probability level. This implies that opportunities to increase returns realized from beef cattle value chain activities will decrease the likelihood and intensity of poverty among the processed beef marketers by 3.3% ceteris paribus.
The other variables that had significant effect on poverty status of the beef cattle farmers are education, dependency ratio, access to credit and other value chain activities. The coefficient of education was negative and significant at 1% probability level. This implies that a unit increase in the education of processed beef marketers tends to decrease the likelihood and intensity of of poverty among processed beef marketers. This likely arises because education enhances the human capital of the processed beef marketers, which improves their productivity and economic opportunities for income generation. The result is consistent with the poverty reducing effect of education reported in various empirical studies (Ojo et al., 2020;Onwumere et al., 2017). The coefficient of dependency ratio was significant at 1% probability level and had a positive relationship with poverty. This implies that an increase in the dependency ratio tends to increase the likelihood and intensity of poverty among processed beef marketers. This could be attributed to an increase in food and non-food consumption expenditure associated with an increase in the number of dependants. This result is consistent with Ezeh et al. (2019) who reported poverty increasing effect of dependency ratio. The coefficient of access to credit was negative and significant at 5% probability level. This implies that an increase in credit at the disposal of the processed beef marketers tends to decrease their likelihood and intensity of poverty. The reason for this is that access to credit makes it possible for the processed beef marketers to meet the day-to-day financial needs of managing their business profitably towards increasing their returns on investment and improving their wellbeing. This finding agrees with that of Onwumere et al. (2017) who reported that amount of credit obtained had negative relationship with the intensity of poverty among artisanal fish farming households. The coefficient of other chain activities was negative and significant at 10% probability level. This implies that processed beef marketers participation in other beef cattle value chain activities will have the tendency of reducing their likelihood and intensity. This is plausible only when increased income generation results from integrating into other beef cattle activities and subsequently improving the processed beef marketers' ability to meet their food and non-food consumption needs. Explanatory variables such as age, household size, crop income, membership of association and non-agricultural income were not significantly related to the poverty status of processed beef marketers in the study area 95.22*** Note: *,** and *** indicates significance at 10, 5 and 1% probability levels respectively, BCA = beef activities, M.E = marginal effect on the censored expected value (all observations)

CONCLUSION AND RECOMMENDATION
Based on the findings of the study, it can be concluded that raw and processed beef marketing activities significantly reduces the poverty incidence of raw and processed beef marketers by 1.7% and 3.3% respectively. This implies that beef marketing activities is a potential pathway out of poverty for many raw and processed beef marketers. Arising from the access to credit for both raw and processed beef marketers, it is recommended that beef marketers should be supported with easy access to credit such as the collateral-free and interest-free loans under the MARKETMONI and TRADERMONI loan schemes of the Government Enterprise and Empowerment Programme (GEEP). This is necessary as better access to credit through the government loan schemes and other sources will help to enhance the activities of the raw and processed beef marketers and offer opportunities for increasing the scale of operation and vertical integration.


A Peridynamic Differential Operator-Based Model for Quantifying Spatial Non-Local Transport Behavior of Pollutants in Heterogeneous Media

: Modeling pollutant transport in heterogeneous media is an important task of hydrology. Pollutant transport in a non-homogeneous environment typically exhibits non-local transport dynamics, whose efﬁcient characterization requires a parsimonious model with the non-local feature. This study encapsulates the non-local transport characteristic of pollutants into the peridynamic differential operator (PDDO) and develops a PDDO-based model for quantifying the observed pollutant non-local transport behavior. The simulation results show that the proposed model can describe pollutant non-local transport behavior in various heterogeneous media. The non-local nature of pollutant transport can be adjusted by pre-deﬁned weight function w ( | ξ | ) and horizon H x . Applications show that the PDDO-based model can better capture pollutant non-local transport behavior than the classical advection–diffusion equation (ADE) model, especially for quantifying the tail of the experimental data late. Analyses further reveal that the PDDO-based model can characterize both normal (Fickian) and anomalous (Lévy) diffusion regimes.

Introduction
The efficient modeling of pollutant transport in a natural environment (surface and subsurface water) is one of the major issues of computational hydrology [1][2][3][4][5]. Pollutant transport in heterogeneous media exhibits complex dynamics, due to the multi-scale heterogeneity of the media, the complex exchange processes between water and tracer, and complex turbulence in the natural environment [5][6][7]. A key problem is how to accurately characterize pollutant transport with complex dynamics in heterogeneous media. Thus, many models which are capable of well describing the complex processes of pollutant transport in heterogeneous media have attracted extensive attention in the last two decades, especially the non-local models [8][9][10][11]. It is also noteworthy that although the physical heterogeneity of the porous medium is essential, many studies have focused on the effects of spatially variable geochemical heterogeneities because the heterogeneous soil cannot be treated as uniform and homogeneous in the natural subsurface environment. The spatially variable geochemical heterogeneities affect pollutant transport significantly [12][13][14][15]. Additionally, many researchers have investigated the transport behavior of molecular size pollutants as well as suspended particle pollutants and found that the transport behavior between the solute type and suspended particle type of pollutants is very different [15,16]. In this study, we focus on the non-local pollutant transport in heterogeneous media, which is usually characterized by an apparent early arrival feature of the pollutant plume and can pose a high risk to the environment [17]. The dynamics of pollutant non-local transport in the heterogeneous systems are not fully understood, motivating this study.
The non-local transport dynamics of pollutant particles may be related to the preferential flow paths in heterogeneous systems [18]. As shown in Figure 1, the pollutant particles may experience a sudden entrainment process, and they can travel considerably far along the "preferential flow paths". Considering the non-local transport behaviors of pollutants, various non-local models have been proposed [19][20][21]. The main discrepancy between the non-local and classical local models is that non-local models usually refer to the integro-differential expression to characterize the spatial interaction natures in heterogeneous systems. The non-local transport behavior of pollutants exists at all scales: from the macroscale in field experiments to laboratory experiments, especially on the micro-scale. Hence, the existing models in prediction are sometimes acceptable in practice, but not always. The measured concentration flux can differ by one to two orders of magnitude from predictions [22].  However, despite considerable efforts over the last two decades, the capability to accurately characterize pollutant transport behavior in the heterogeneous system remains inadequate. One of the major obstacles is the poor understanding and modeling of non-local transport behaviors of pollutants in a wide range of conditions. The advection-dispersion equation (ADE) model provides a classical framework for characterizing pollutant transport and makes an important contribution to the following theoretical development. However, a growing number of researchers have found that the pollutant moving through non-homogeneous media (surface or subsurface water) cannot be described by using the classical ADE model. This study aims to quantify pollutant non-local transport behaviors in a wide range of conditions and proposes a peridynamic differential operator (PDDO) based model for quantifying the observed non-local transport behavior. PDDO was first built by Madenci et al. [23,24], who proposed the PDDO by introducing the classical peridynamic (PD) theory [25]. The PDDO can describe any order of partial derivatives of the spatial and temporal functions by using the orthogonality property of the PD functions. The PDDO is expressed in terms of only integration over the domain of the pre-defined horizon. Hence, PDDO converts the local differentiation equation to its non-local integration form in a unified manner, and it is not prone to the singularities that arise from the presence of discontinuities in modeling [24,26]. Here, this study introduces a PDDO into the ADE model to describe the dynamics of pollutant non-local transport. Considering various pre-defined weight functions and horizon domains, the proposed PDDO-based model can characterize pollutant non-local transport behavior in various conditions. The classical local model is extended to a general region scale model, even on a global scale, as shown in Figure 2.
The literature reviewed above takes into account the basic concept maps of the pollutant non-local transport and lays the foundation for the wide application of the PDDO-based model. The purpose of this study is to answer the following three questions using the proposed PDDO-based model. First, can the PDDO-based model bridge the pollutant transport's local and regional scales? Second, how do the pre-defined weight function and horizon domains of the PDDO-based model affect the dynamics of the pollutant non-local transport process? Third, can the PDDO-based model accurately characterize the spatial correlation of pollutant non-local transport?
The rest of this work is organized as follows. Section 2 reviews the PDDO theory and proposes a PDDO-based model. Section 3 gives an analytical solution and numerical algorithm of the PDDO-based model. One-dimensional and two-dimensional results of the PDDO-based model with different weight functions and horizon domains are calculated, and the applications of the proposed non-local model are also shown in this section. Section 4 discusses the diffusion regime of the PDDO-based model with various PD functions, and the limitations of the PDDObased model are also stated here. Conclusions are finally drawn in Section 5.

Model Development
Pollutant transport simulation is one of the core tasks of computational hydrology, and various physical models have been developed and applied in the past decades [21,[27][28][29]. Mathematical modeling and understanding pollutant transport requires determining spatial derivatives of the concentration. However, many studies have found that pollutants exhibit non-local transport behavior, especially in non-homogeneous environments, which cannot be well characterized by the classical local definition of spatial derivatives in modeling [17,30]. Considering the drawbacks of the differentiation processes, the integration processes are more suitable for characterizing non-local pollutant transport. Integration is a typical non-local process because it depends on the entire range of integration. Hence, the PDDO is introduced in this study to characterize the pollutant non-local transport behavior.
The most widely used model ADE is always applied to characterize the pollutant transport behavior, which, however, cannot well describe the non-local transport behavior of pollutants in heterogeneous media [21,31]. The classical ADE types model can be expressed as follows [32][33][34][35][36]:  [37]. The above ADE-type models have been widely used in many studies to describe the pollutant transport process. However, the observed non-local pollutant transport cannot be precisely described by the R-ADE model. The non-local transport that exists in various systems proposes, as do many studies, the following fractional ADE (F-ADE) model to characterize the observed non-local transport behavior of pollutants: where ∂ β /∂x β is the fractional operator. In this study, the Riemann-Liouville fractional derivative is considered in the simulation [38].
The F-ADE model provides a robust tool to describe the non-local pollutant transport along the preferential flow path, as shown in Figure 1. The basic form of space F-ADE replaces the integer-order derivative in space with a fractional one β (1<β ≤2). When β = 2, the F-ADE model reduces to the classic ADE. However, various environmental factors can affect pollutant transport in natural conditions. The F-ADE model cannot capture the non-locality of the bedload transport at various spatial scales due to the mathematical definition of the fractional operator. This theoretical deficiency limits researchers from further understanding the pollutant non-local transport behavior. Large deviations can arise from the lack of comprehensive quantification of the pollutant non-local transport process. Hence, we will propose a non-local model for quantifying pollutant non-local transport behavior with various conditions.

A Brief Review of PDDO
Unlike the classical methods (such as finite difference method, finite element method, etc.) for solving the advection-diffusion equation (ADE) type models to characterize the pollutant transport, the PDDO can be used to reconstruct the integral form of the target differential equation [24,26]. Hence, PDDO is an alternative tool to describe the solute nonlocal transport. Although the differentiation process is usually more direct than integration in analytical mathematics, the reverse is true in computational mathematics, especially in the presence of a jump discontinuity or a singularity. Integration is a non-local process because it depends on the entire range of integration. Considering non-local pollutant transport, the PDDO is an efficient tool for describing the pollutant non-local transport behavior along the preferential paths.
The PDDO can be obtained by using the Taylor series expansion of a function f (x + ξ) as follows: where R(x) represents the remainder.
Here, a orthogonality property of PD functions, g p N (ξ), is introduced, which satisfies the following equations [23]: and where w q (ξ) is the weight functions, and it can be used to measure the strength of the influence of the integral points in the defined horizon H x .
Assuming that the value of the remainder, R(x), is negligibly small, and multiplying each term in Equation (3) by the proposed PD functions g p N (ξ) and integrating over the family of point, x, defined as H It is worth noting that each point has its family members in the range of the horizon H x and occupies an infinitesimally small entity. Hence, each family's properties (such as size, shape, etc.) can be very different, and they significantly affect the strength of non-locality. The strength of the interaction between the family points in each family group is specified by the proposed weight functions w q (ξ).
Based on the orthogonal nature of the function g p N (ξ), the spatial derivatives can be expressed as follows: Equation (7) gives a integration expressions of the spatial derivatives in the horizon H x . In the regular PDDO, the shape of the horizon is specified as a sphere (3 dimensions), cylinder (2 dimensions), and segment (1 dimension) of the modeling. Further, the charac-teristic internal length parameter (radius of the defined sphere, circle, or line), δ, referred to as the "horizon", is constant.

PDDO-Based Model
As described above, pollutants may exhibit non-local transport behavior in heterogeneous media, and the classical ADE types model (1) cannot well characterize the non-local transport behavior. Considering the drawbacks of the model (1), the PDDO is introduced here to characterize the non-local transport behavior of pollutants. Figure 2 plots the conceptual map of the defined horizon H x , and the local field of the model (1) is drawn for comparison. As shown in Figure 2, the horizon H x determines the non-local transport zone of pollutants, which also indicates the length of the preferential path in the media. A longer preferential path implies a stronger non-local transport behavior of pollutant particles, resulting in a larger horizon H x .
By introducing the PDDO into Equation (1), the following PDDO-based non-local transport model can be obtained: the parameters v, D, and R are same as Equation (1), and the PD functions g 1 N (ξ) and g 2 N (ξ) are determined by using Equations (4) and (5).
Since g p N (ξ) is zero outside of the defined horizon H x , Equation (8) can be written as follows: where ( * ) denotes the convolution operator: and Thus, Equation (9) is the PDDO-based model for characterizing the non-local transport behavior of pollutants. The analytical solution and numerical algorithm for solving the PDDO-based model (9) will be further given in the following section. Horizon:

Numerical Algorithm of the PDDO-Based Model
As described above, the analytical form of the PDDO-based model cannot be obtained in most cases. Hence, the numerical algorithm is herein developed to solve the proposed PDDO-based model.
The one-dimensional case of the PD functions g p N (ξ) can be set as follows [39]: Considering the same weigh functions w 1 (|ξ|) and w 2 (|ξ|), for all terms, the Equation (4) can be expressed as follows: Equation (13) can be written as follows: and where and Inserting a into Equation (12), g p 2 (ξ) can be obtained. The implicit discrete form of Equation (8) can be expressed as follows: where N i is the number of the family point of the i-th space point, ∆x and ∆t are space interval and time interval, respectively. Equation (19) gives an implicit form of the proposed PDDO-model. Compared with the discrete form of the classical ADE models by using the finite difference method, the discrete form of the PDDO-based model also indicates that the target point is not only affected by the nearby points, but also affected by all family points in the defined near-field H x .

Analytical Solution of PDDO Model
PDDO-based model (9) gives a powerful tool to solve the non-local transport behavior of pollutants, and the PD functions g p N (ξ) and horizon H x determine the strength of the non-local property of pollutant transport. The horizon H x and weight function w(ξ) are pre-defined according to the environment state before the simulation. In the simulation, the PD functions g p N (ξ) should be analytically or numerically determined according to the pre-defined horizon H x and weight function w(ξ), which are the core tasks in modeling.
In order to analytically solve the proposed PDDO-based model (9), taking the Fourier to transform Equation (9), one obtains where k is the Fourier variable. Collating Equation (20) yields Considering the initial point source conditionĉ(k, 0) = 1, the analytical solution of Equation (21) Taking inverse Fourier transform of Equation (22) yields Equation (23) gives a general form of the analytical solution of the proposed PDDObased model (9). In order to obtain an explicit expression of the analytic solution (23), it is necessary to obtain the form of the function g p 2 (ξ) in the Fourier space. However, it is impossible to obtain an analytical form of the function g p 2 (ξ) in the Fourier space in most cases. Therefore, we will use the numerical algorithm to solve the PDDO-based model (9), as described in the section methodology.

One-Dimensional Results of the PDDO-Based Model
PDDO-based model gives an alternative tool to describe the non-local transport behavior of pollutants, and the non-local property of pollutants are determined by the weight function w(|ξ|) and w 2 (|ξ|) and horizon H x . To investigate the effect of the weight function on the pollutant transport process, the following power-law form of the weight function is considered: Figure 3 shows the weight function with different parameter k, and an exponential function exp(−|ξ| 2 /δ 6 ) is also drawn for comparison. The result shows that the strength of point influence in the horizon increases as k decreases, and a smaller k implies a stronger non-local nature of pollutant transport. The exponential form of the weight function exhibits the weakest non-local property to characterize pollutant transport.    Figure 4a, the PDDO-based model gives a significant leading edge characteristic in the early time, which means that the pollutants described by the non-local model can reach the observation site much earlier than described by the local model. Furthermore, the early arrival characteristic of pollutants becomes more evident with the enhancement of the non-local nature, and this is mainly because the pollutants are easier to reach the observation point through the preference paths. Figure 4b shows the snapshots of the PDDO-based model, and the results indicate that the non-local transport behavior of pollutants exhibits trailing edge characteristics. Compared with the ADE model, the snapshots of the PDDO-based model show a significantly asymmetric characteristic. The difference between the PDDO model and the ADE model is that the non-local nature is considered in the PDDO model. The concentration of the pollutant is not only locally relevant but also correlates with regions within the non-local area. Therefore, the introduction of the non-local nature leads to pollutant particles with early arrival features in BTCs and a significant trailing edge in the spatial snapshots.

Two-Dimensional Results of the PDDO-Based Model
Equation (9) gives a non-local description of one-dimensional pollutant transport. To further investigate the non-local transport behavior of pollutants in heterogeneous media, here we give the following two-dimensional form of the PDDO-based model. We assume that the advection process occurs only in the x-direction. One obtains ·(c(x + ξ 1 , y + ξ 2 , t) − c(x, y, t))dV x (25) where g 10 2 , g 20 2 , and g 02 2 are PD functions; see Madenci et al. [24] for the detailed derivation. The last subsection discusses the weight function for characterizing the pollutant non-local transport. Here, we compared two different horizon ranges H x in describing pollutant transport. Figure 5 shows the simulation results of the two-dimensional PDDObased model (25); two different H x are considered, i.e., H x = 0.6 Ω and H x = 0.05 Ω. The results indicate that the plume gives a symmetric (Gaussian) shape in a smaller H x . This is mainly because the PDDO-based model can regress to a local ADE form, and the analytical solution of the ADE is Gaussian distribution in space.
It is worth noting that the domain of the PDDO-based model is divided into a finite number of cells, and the interior points can be assigned a symmetric family in space, while the points near the boundary have non-symmetric families. Therefore, the points near the boundary have their own PD function g p N (ξ). As shown in Figure 5a, the large horizon H x give a large number of non-symmetric families of the points, and the plumes show significantly asymmetric features.

Applications
Here, we check the applicability of the PDDO-based model (9) in quantifying pollutant non-local transport documented in the literature [17]. To explore the dynamics of tracers transport in the alluvial setting with heterogeneity, Yin et al. [17] built a two-dimensional, different alluvial setting with various hydrofacies structures generated by using the T-PROGS, and simulated the conservative tracer transport by using the Monte-Carlo method. The detailed information can be found in the literature [17].
Hence, Yin et al. [17] gives an ideal set of data for investigating the non-local pollutant transport in heterogeneous media. As shown in Figure 6, four snapshots of the tracers are documented in the studies, i.e., 27 days, 132 days, 224 days, and 328 days. The experimental data show that the trailing edge becomes stronger with time evolution, which indicates that the non-local transport behavior enhances with time evolution.
We used the first snapshot (27 days) to determine the parameters v, D, and R for the PDDO-based and F-ADE models and then predicted the later snapshots by adjusting the parameters H x and β. Considering the parameters v and D of the R-ADE model have the same dimensions as those of the PDDO-based model, the parameters v and D in the R-ADE model were set to be the same as those of the PDDO-based model, andonly the parameter R was adjusted. The results of the ADE and F-ADE models are calculated for comparison. The weight function of the PDDO are pre-defined as w(|ξ|) = δ 2 /|ξ| 2 in the simulations. Figure 6 plots the fitting results of the PDDO-based model for each snapshot. The results show that the PDDO-based and F-ADE models can well capture the non-local transport behavior, while the ADE model cannot capture the observation results. Moreover, the F-ADE model can capture the trailing edge of the snapshot well, but it is not accurate for the peak position. This phenomenon may be due to the F-ADE model being a global non-local model, which assumes that the pollutant transport process is affected by the entire space, leading it to overestimate the non-local transport behavior of the pollutant. The main discrepancy between the PDDO-based model and the ADE model is that the PDDO-based model can well capture the trailing edge feature of the experimental data. The ADE model is a local definition, which cannot capture the observed non-local pollutant transport behavior. Additionally, the experimental data show significant asymmetric features in the snapshots, and the ADE can describe the symmetric transport behavior of the pollutants. Table 1 lists the best-fitting parameters of the PDDO-based and ADE models, respectively. The results show that the horizon H x of the PDDO-based model increases with time evolution, which is consistent with the experimental data. (d) Figure 6. Comparison between the documented snapshots (symbols) and the best-fit results using the ADE (1) and PDDO-based (9) models at four times (t = 27, 132, 224, and 328 days) along the 300-m-long heterogeneous media. The weight function set as k = 2 for Eq. (24), and the horizon set as H x = n · ∆x.
We used the first snapshot (27 days) to determine the parameters v, D, and R for the PDDO-based and F-ADE models and then predicted the later snapshots by adjusting the parameters H x and β . Considering the parameters v and D of the R-ADE model have the same dimensions as those of the PDDO-based model, the parameters v and D in the R-ADE model were set to be the same as those of the PDDO-based model, only the parameter R was adjusted. The results of the ADE and F-ADE models are calculated for comparison. The weight function of the PDDO are pre-defined as w(|ξ|) = δ 2 /|ξ| 2 in the simulations. Figure. 6 plots the fitting results of the PDDO-based model for each snapshot. The results show that the PDDO-based and F-ADE models can well capture the non-local transport behavior, while the ADE model cannot capture the observation results. Moreover, the F-ADE model can capture the trailing edge of the snapshot well, but it is not accurate for the peak position. This phenomenon may be since the F-ADE model is a global non-local model, which assumes that the pollutant transport process is affected by the entire space, leading it to overestimate the non-local transport behavior of the pollutant. The main discrepancy between the PDDO-based model and the ADE model is that the PDDO-based model can well capture the trailing edge feature of the experimental data. The ADE model is a local definition, which cannot capture the observed non-local pollutant transport behavior. Besides, the experimental data show significant asymmetric features in the snapshots, and the ADE can describe the symmetric transport behavior of the pollutants. Table. (1) lists the best fitting parameters of the PDDO-based and ADE models, respectively. The results show that the near-filed H x of the PDDO-based model increases with time evolution, which is consistent with the experimental data. Figure 6. Comparison between the documented snapshots (symbols) and the best-fit results using the ADE (1) and PDDO-based (9) models at four times (t = 27, 132, 224, and 328 days) along the 300 m-long heterogeneous media. The weight function set as k = 2 for Equation (24), and the horizon set as H x = n · ∆x.  The non-local pollutant transport that exists in various systems can be attributed to very different mechanisms. For example, in open water (e.g., rivers), the turbulence and the heterogeneous riverbed that will have an accelerated effect on the pollutant transport lead to non-local transport behavior. The early arrivals and trailing edges of pollutants (as shown in Figure 4) can be observed for highly permeable preferential paths in non-homogeneous media, especially in the media with fractures [18]. The ADE model does not consider the long-distance spatial correlation (i.e., the non-local nature in space), resulting in its inability to effectively capture the transport behavior of pollutants in non-homogeneous media. However, the F-ADE model assumes that pollutant particles can jump an infinite distance in one movement, which is inconsistent with the actual situation. The FADE model is a global domain model that may often overestimate the extent of the non-local transport of pollutants. More specifically, in non-homogeneous domains, highly developed spatial continuity causes the preferential flow path, producing high-velocity regions, and the pollutant tracers have a relatively high probability in the non-local jump. Due to the adjustability of the weight function and horizon zone, the PDDO-based model can describe pollutant non-local transport in various conditions. Considering the diversity of pollutant non-local transport mechanisms, the PDDO-based model is a generalized model that can describe the transport of pollutants in aquifers, the transport of pollutants in surface waters, etc.

Diffusion Regime of the PDDO-Based Model
The theoretical issue of pollutant diffusion has long been central when using fluid mechanics to explore pollutant transport. A standard tool to explore the stochastic motion of Brownian tracers is the mean squared displacement (MSD) of the tracers. Similar to Brownian motion, this subsection aims to evaluate the PDDO-based model by comparing the MSD, which is the criteria for diffusion states in quantifying pollutant transport. The MSD is herein defined as follows [40]: Equation (26) gives an analytical form of the MSD of particles. Here, we use the following equation to calculate the MSD of particles in the framework of the PDDO-based model [31,40]: where c(x, t) is the solution of the model (9) in the case of v = 0 with an instantaneous point source c(x, 0) = δ(x). Based on the analytical solution, (23) of model (9), and set v = 0, yields Considering a general form ofĝ 2 N (k) as followŝ Inserting Equation (29) into Equation (28), one obtains where C 1 = D · C/R First, we consider that α = 2, taking inverse Fourier transform of Equation (30) yields Inserting Equation (31) into Equation (27), one obtains Equation (32) gives that the MSD R MSD (t) is linear with time, and the Fickian diffusion regime is obtained in the case of α = 2.
Furthermore, considering that 1 < α < 2, Equation (30) is the characteristic function of a centered and symmetric Lévy distribution. Therefore, the c(x, t) has the following power-law asymptotic: Due to Equation (33) being a Lévy distribution, the mean squared displacement diverges: Hence, the analytical result shows that the MSD is nonlinear with time in the case of 1 < α < 2, and the anomalous diffusion regime occurs. Based on the above analysis of the MSD of the PDDO-based model, the proposed non-local model can capture both normal (Fickian) and anomalous diffusion regimes of pollutants. The PDDO-based model bridges the local and regional scales by considering various PD functions.
It is also noteworthy that this study has three main limitations of the PDDO-based model (9) in quantifying pollutant non-local transport behavior. First, the weight function w(|ξ|) is chosen as empirical, which should be further explored for various conditions. Second, the horizon H x is determined by fitting the experimental data, and the quantitative relationship between H x and media needs to be established in the future. Third, the nonlocal property leads to the calculation time of the PDDO-based model being much higher than that of the local model. Fast algorithms for solving the PDDO-based model need to be further investigated.

Conclusions
This study proposes a non-local model for quantifying pollutant transport in a heterogeneous environment. The non-local property of pollutant transport is encapsulated in the PDDO. Three conclusions can be drawn from this study.
(1) PDDO gives an efficient tool to describe the non-local transport behavior of pollutants, and the non-local transport feature of pollutants can be adjusted by considering different weight functions w(|ξ|) and horizon H x .
(2) The analyses of the MSD for the PDDO-based model give that the PDDO model can characterize both the normal (Fickian) and anomalous transport behavior of pollutants. The PDDO-based model bridges the local scale and region scale by considering various PD functions.
(3) Applications show that the PDDO-based model can efficiently characterize the observed non-local transport behavior of pollutants, and the non-local feature of pollutant transport enhances with time evolution.

Data Availability Statement:
The data and codes that support the findings of this study are available from the corresponding author upon reasonable request.