EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 1213-1227 ISSN 1307-5543 – ejpam.com Published by New York Business Global Analyzing the Impact of Control Strategies on Visceral Leishmaniasis: A Mathematical Modeling Perspective Fathelrhman EL Guma1,2, Mohamed A. Abdoon3,4, Ahmad Qazza5, Rania Saadeh5,∗, Mohammed Ali Arishi6, Abdoelnaser M. Degoot7 1 Department of Mathematics, Faculty of Science and Arts, Al Baha University, Baljurashi 1988, Saudi Arabia 2 Department of Statistical Study, Alsalam University, Alfula, Sudan 3 Department of Basic Sciences, Common First Year Deanship, King Saud University, P.O. Box 1142, Riyadh 12373, Saudi Arabia 4 Department of Mathematics, Faculty of Science, Bakht Al-Ruda University, Sudan 5 Department of Mathematics, Zarqa University, Zarqa 13110, Jordan 6 Ministry of Health, King Khalid Hospital in Tabuk, Faculty of Laboratory , Department of Histopathology, 71411, Saudi Arabia 7 African Institute for Mathematical Sciences, Kigali, Rwanda Abstract. Visceral Leishmaniasis remains a significant public health challenge in eastern Sudan despite extensive control efforts. This study employs MCMC Bayesian inference techniques to fit a mathematical model for studying visceral Leishmaniasis transmission dynamics with interventions. The research focuses on evaluation of control programs to adapt to the dynamic nature of visceral Leishmaniasis transmission. We analyze 22 years of cumulative visceral Leishmaniasis cases from eastern Sudan, an endemic region for visceral Leishmaniasis, and assessed the effectiveness of interventions implemented thus far. The results reveal that visceral Leishmaniasis is prevalent, with reported cases representing less than 20% of community infections. Improved surveillance and diagnostics are necessary for accu- rately estimating the disease burden. The effectiveness of intervention strategies for vector control for reducing VL transmission in the region is found to be limited. Furthermore, the model predicts visceral Leishmaniasis cases will continue to increase in the near future, underscoring the need adaptable initiatives to reduce the disease burden. 2020 Mathematics Subject Classifications: AMS classification codes Key Words and Phrases: Visceral Leishmaniasis, mathematical analysis, disease dynamics, parameters estimation, reproductive number, interventions. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5121 Email addresses: fathyelrhman@gmail.com (F. EL Guma), Moh.abdoon@gmail.com ( M. A. Abdoon), aqazza@zu.edu.jo (A. Qazza), rsaadeh@zu.edu.jo (R. Saadeh), Mohmmedam@moh.gov.sa (M. A. Arishi), degoot@aims.ac.za (A. M. Degoot) https://www.ejpam.com 1213 © 2024 EJPAM All rights reserved. R. Saadeh et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1213-1227 1214 1. Introduction Visceral Leishmaniasis (VL), also known as Al Kala-azar, has been a longstanding issue in Sudan, with its initial documented instance dating back to 1908 [7]. The disease has been observed in various parts of the country, including the Darfur region in western Sudan, the Nuba Mountains in Kordofan, and along the banks of the White Nile, among others [19]. However, it is the eastern region that has experienced the highest incidence of VL, making it a focal point for public health efforts due to the significant concern it poses in those areas [4]. Despite considerable intervention programs aimed at controlling and eliminating VL, particularly in the eastern region. The utilization of mathematical modeling has become indispensable in comprehend- ing the intricate dynamics and transmission patterns of infectious diseases, as exemplified in references [1, 25, 26]. By employing equations that delineate the interactions among populations and infections, as well as the routes of transmission, researchers can gain profound insights into disease dynamics. However, the complexity of these models and the presence of unforeseen variables often hinder the attainment of analytical solutions. Hence, crucial parameters and approximate solutions are frequently derived through nu- merical approaches [5, 9, 14]. Numerical models enable scientists to predict treatment outcomes, evaluate medication efficacy, and refine strategies for managing clinical cases. These methodologies play a pivotal role in enhancing public health measures, optimizing vaccination programs, and advancing the understanding of epidemiological factors con- tributing to disease transmission [12, 13, 28]. Various mathematical models have been proposed to analyze the dynamics of VL transmission, typically based on SEIR-type mod- els and focusing on three key groups: humans, animal reservoirs, and vectors [27, 30, 31]. Extensive research has generated theoretical and simulation-based insights into VL dy- namics from diverse perspectives. Studies have shown that under certain circumstances, integrating human therapy with vector control can effectively eradicate the disease [11, 17]. Additionally, research has explored the interaction between malaria and VL co-infection [23]. Optimal control theory has been employed to analyze dynamical models for VL transmission, determining the most effective approaches for disease management [2, 24]. In a recent study [29], we developed a comprehensive SEIR-type model to analyze the transmission dynamics of VL, incorporating the effectiveness of interventions, and pub- lished a Caputo fractional-order derivative version of it along with rigorous mathematical analysis. In the current study we focus on the deterministic version of that model, using it to estimate the efficacy of interventions made to combat VL in eastern Sudan from 2000 to 2021. We analyse the impact of interventions using percentages and consider that the number of reported human VL cases is only a portion of the total community infections. The article is organized as follows. In Section 2, we revisit the model published in [29]. Section 4 provides an overview of the parameter estimation procedure, while Section 6 deals with a comprehensive overview and analysis of the findings. R. Saadeh et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1213-1227 1215 2. The Model As mentioned above, the model is described in detail in [29]. However, for the sake of readability, we rewrite it here. Briefly, the model considers the dynamics of VL in three populations: humans, reservoirs, and vectors. Each population is divided into a number of distinct compartments, described in Table 1, and the transmission channels between these compartments are captured by system (1) of ordinary differential equations, governed by parameters described in Table 2. Table 1: Description of the model’s compartments Compartment Description Nh Total human population Sh Susceptible human population Eh Exposed human population Iu Undetected VL-infected human population Id Detected VL-infected human population Ph PKDL VL-infected human population Rh Human population who are immune to VL disease Nr Total reservoir population Ir Susceptible reservoir population Ir VL-infected reservoir population Nv Total vector population Iv Susceptible vector population Iv VL-infected vector population R. Saadeh et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1213-1227 1216 Table 2: Description of the model’s parameters Par. Description ϵ Average detecting rate αh Effectiveness of therapies for human population αr Effectiveness of the interventions for reservoir population αv Effectiveness of interventions for the control of population β1 Likelihood of a virus being transmitted from an infected vector to a host that is susceptible to the virus β2 Likelihood of transmission of VL from an infected human that has been identified to a vector that is susceptible to the virus β3 Likelihood of transmission of from undetected VL-infected person to a susceptible vector β4 Likelihood of transmission from a PKDL individual to susceptible vectors β5 Likelihood of transmitted from an infected reservoir to a susceptible vector 1 ω Average incubation period for VL in human δ1 Average mortality rate in detected VL-infected human population δ2 Average mortality rate in undetected VL-infected human population δ3 VL-induced death rate for individuals in Ph class σ1 Treatment rate for individuals in Id class σ2 Treatment rate for individuals in Ph class γ1 Natural recovery rate for individuals in Iu class γ2 Natural recovery for individuals in Ph class ρ1 Rate of progression from Id class to Ph class ρ2 Rate of progression from Iu class to Ph class η Rate of progression from Rh class to Sh class λh Constant growth rate of human population λr Constant growth rate of reservoir population λv Constant growth rate of vector population µh Constant death rate of human population µr Constant death rate of human reservoir µv Death rate of the vector population that remains constant S′ h(t) = λh − β1Iv(t) Sh(t)(1−αh(t)) Nh(t) + ηRh(t)− µhSh(t), (1) E′ h(t) = β1Iv(t) Sh(t)(1−αh(t)) Nh(t) − (ω + µh)Eh(t), I ′d(t) = ϵ(t)ωEh(t)− (σ1 + ρ1 + δ1 + µh) Id(t), I ′u(t) = [1− ϵ(t)]ωEh(t)− (γ1 + ρ2 + δ2 + µh) Iu(t), P ′ h(t) = ρ1Id(t) + ρ2Iu(t)− (σ2 + γ2 + µh)Ph(t), R′ h(t) = σ1Id(t) + γ1Iu(t) + (σ2 + γ2)Ph(t)− (η + µh)Rh(t), S′ r(t) = λr − β1Iv(t) Sr(t) Nr(t) − (αr(t) + µr)Sr(t), I ′r(t) = β1Iv(t) Sr(t) Nr(t) − (αr(t) + µr)Ir(t), S′ v(t) = − (( β2 Id(t) Nh(t) + β3 Iu(t) Nh(t) + β4 Ph(t) Nh(t) ) (1− αh(t)) + β5 Ir(t) Nr(t) ) Sv(t) +λv − (αv(t) + µv)Sv(t), I ′v(t) = (( β2 Id(t) Nh(t) + β3 Iu(t) Nh(t) + β4 Ph(t) Nh(t) ) (1− αh(t)) + β5 Ir(t) Nr(t) ) Sv(t) −(αv(t) + µv)Iv(t). R. Saadeh et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1213-1227 1217 In addition, the following three equations are also implicit in model (1) construction: N ′ h(t) = λh − µhNh(t)− δ1Id(t)− δ2Iu(t), N ′ r(t) = λr − (αr(t) + µr)Nr(t), (2) N ′ v(t) = λv − (αv(t) + µv)Nv(t). All parameters of the model are non-negative, and since the model tracks the number of living things, all compartments within the model are also non-negative. A detailed mathematical treatment of model (1) was conducted in [29], including positivity, invariant region, stability and sensitivity analysis, and a formula for calculating the basic reproduc- tion number. 3. Numerical Simulations The data from the regional Ministry of Health’s VL control campaign is used to repre- sent the annual VL confirmed cases in Eastern Sudan from 2000 to 2021 in Figure 1. Each point on the line chart, which represents a different year, shows the annual variations in VL cases in the plot. To further show the average yearly VL cases over the whole time, a blue horizontal line is superimposed. This persistent presence emphasizes how difficult it is to eradicate VL in Sudan. Figure 1: (A) Annual VL confirmed cases in Eastern Sudan from 2000 to 2021 and (B) cumulative VL cases. R. Saadeh et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1213-1227 1218 To evaluate the consistency of model (1), we performed numerical simulations within a hypothetical ecosystem incorporating human, reservoir, and vector populations. Through systematic manipulation of control parameters, namely the average case detection rate ϵ, interventions to protect human population αh, measures against the vector population, and measures to control reservoir population αv and αr. We investigated the impact of different values of the parameters on the cumulative number of human VL cases, as elaborated in Section 4. For our simulations, we initialized the human population at 1 million individuals, denoted as Nh = 106 to closely resemble the population of eastern Sudan [22, 33], with the reservoir population set at half of the human population, Nr = 0.5 × Nh, and the vector population at double the human population, Nv = 2 × Nh. Additionally, we assumed that 1% of each population was infected at the onset of the simulations. The impact of various values of ϵ on the annual cumulative VL cases is illustrated in Figure 2. The figure highlights how enhancing the average case detection rate, attainable through heightened surveillance and diagnostic capabilities, health education, capacity building, and ensuring affordable and accessible healthcare, correlates with a rise in cu- mulative VL cases. However, this increase is accompanied by a positive outcome: detected cases can be effectively managed and their spread contained, thereby contributing to the overall control of the disease within the community. Figure 2: Simulations of model (1) for different values of ϵ and their impact on cumulative VL cases. The primary intervention to protect the human population against VL often involves the widespread use of insecticide-treated nets (ITNs). These nets act as a physical bar- R. Saadeh et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1213-1227 1219 rier against the sandflies that transmit the parasite causing VL, thereby reducing the risk of infection among individuals sleeping under them. Additionally, ITNs can also be impregnated with insecticides, which further enhance their effectiveness by repelling or killing the sandflies upon contact. Figure 3 illustrates the impact of different values of αh, representing the cumulative efforts aimed at enhancing the efficacy of interventions to protect populations, on the cumulative VL cases. It suggests that higher values of αh correspond to lower cumulative VL cases, which aligns with intuitive understanding. Implementing mass-scale distribution of ITNs and promoting their consistent and correct use can significantly reduce the incidence of the disease. Figure 3: Simulations of model (1) for different values of αh and their impact on cumu- lative VL cases. The main intervention for controlling vector populations is the application of insecticide spray. Figure 4 illustrates the simulation results depicting the impact of varying intensities of αv. It indicates that an increase in the efficacy of these interventions is correlated with a decrease in the cumulative cases of VL. R. Saadeh et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1213-1227 1220 Figure 4: Simulations of model (1) for different values of αv and their impact on cumu- lative VL cases. The primary intervention for controlling VL in the reservoir population has tradition- ally been culling. However, this approach may have negative consequences, as detailed in Elmojtaba et al. [17], because eliminating animals creates a vacant niche for vectors, potentially increasing human exposure. Our simulation results align with this conclusion (see Figure 5), showing that different values of αr have similar effects on cumulative VL cases Figure 5: Simulations of model (1) for different values of αr and their impact on cumu- lative VL cases. R. Saadeh et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1213-1227 1221 4. Parameters Estimation Many parameters related to VL disease features in model (1) have approximate values that can be gathered from relevant literature (see Table 4). However, parameters repre- senting interaction rates (β1, β2, β3, β4, and β5), intervention effectiveness (ϵ, αh, αr, and αv), and the initial values of Nh, Eh, Iu, Ph, Rh, Nr, Nv, Ir, and Iv are unknown. We denote these unknown parameters collectively by θ: θ = {ϵ, αh, αr, αv, β1, β2, β3, β4, β5, Nh(0), Eh(0), Iu(0), Ph(0), Nr(0), Ir(0), Nv(0), Iv(0)}. (3) Furthermore, as in [11], the initial number of the human population can be modeled as Nh(0) = Kh × 10m, for m > 0 and Kh ∈ (a, b) for a and b non-negative numbers. Similarly, we set Sh(0) = Ksh × Nh(0), Eh(0) = Keh × Sh(0), Nr(0) = Kr × Nh(0), and Nv(0) = Kv ×Nh(0). Given annual cases of VL, we seek to estimate the values of these known parameters using a Bayesian approach. In particular, we are interested in the cumulative number of reported human VL cases at time t, denoted by y, which can be calculated from system (1) as follows: dy(t) dt = ϵωE(t). (4) Cumulative number of annual VL cases are count data, and thus can be modelled using negative binomial (NB) distribution as a random number of occurrences. Therefore, the joint posterior distribution of the parameters vector θ: J(θ, ϕ|y1, y2, ....yL) = L∏ t=1 NB(y(t)|θ, ϕ)P (θ)P (ϕ), (5) where L is the number of years, y(t) is derived is given by (4),P (θ) is the joint prior distribution of all parameters in vector θ, and P (ϕ) is the prior distribution for dispersion parameter ϕ. We assumed flat uninformative prior distributions u(0, 1) or N(0, 1) (see Table 4) for the parameters in θ, and ϕ−1 ∼ Gamma(0.1, 0.1). 5. Results The cumulative number of reported cases of VL in eastern Sudan from 2000 to 2021 (L = 22) was modeled using (5), employing an MCMC algorithm. Specifically, we em- ployed the No-U-Turn sampler, a sophisticated MCMC method integrated into the Turing framework for Bayesian computation in Julia [10]. This method offers rapid convergence to the target distribution, avoids random walks, and automatically adjusts the learning rate. The program was executed for 15,000 iterations, with the initial 5,000 samples of θ discarded as burn-in. Subsequently, point estimates and 95% credible intervals for each parameter were computed from the remaining samples. It’s worth noting that the R. Saadeh et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1213-1227 1222 human population in Gadaref state in 2000 is estimated to be around one million [22, 33], Therefore, in our implementation, we set m = 6, a = 0.8, and b = 1.4, i.e., Kh ∼ U(0.8, 0.14). Figure 6: Comparison between actual and predicted annual Cumulative VL Cases in eastern Sudan from (2000 - 2021) and projection for (2022- 2026). Figure 6a compares the simulation results of model (1) with the actual cumulative VL cases in Eastern Sudan. Despite missing a few data points, the model’s findings closely match the real data. Furthermore, we utilized the estimated parameter values to project cumulative VL cases in the region for five year period, from 2022 to 2027. If existing measures remain unchanged, VL is expected to persist in the region for the foreseeable future, as projected in Figure 6b. Point estimates and 95% credible intervals for the initial values of system (1) compartments and parameters are provided in Tables 3 and 4, respectively. Table 3: Estimates for the initial value of the compartment. Compartment Value Nh(0) 10859000 Sh(0) 0.27×Nh(0) Eh(0) 2.1× 10−1 × Sh(0) Iu(0) 3.659× 10−1 × Eh(0) Ph(0) 3.659× 10−1 × [Id(0) + Iu(0)] Rh(0) Nr(0) 18× 10−2 ×Nh(0) Ir(0) 7× 10−2 ×Nr(0) Nv(0) 6.15×Nh(0) Iv(0) 0.2×Nv(0) R. Saadeh et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 1213-1227 1223 Table 4: Description of the model’s parameters. Parameter Value Range Source ϵ 0.18 [0.12,0.57] Fitted αh 0.7 [0.3, 0.9] Fitted αr 0.015 [0.01, 0.44 ] Fitted αv 0.5 [0.13 , 1] Fitted β1 0.6 [0.22 , 0.94] Fitted β2 0.34 [0.004 , 0.92 ] Fitted β3 0.46 [0.04, 0.93] Fitted β4 0.37 [0.02.0, 0.94] Fitted β5 0.74 [0.3, 1] Fitted 1 ω 60 days [30, 180] [31, 34] δ1 4.015 – [31, 32] δ2 4.015 – [31, 32] δ3 4.015 – [31, 32] σ1 0.03 day−1 - [32] σ2 0.033 day−1 - [18] γ1 0.84 [0.8, 0.9] [32] γ2 0.005 day−1 [0.001, 0.005] [32] ρ1 0.36 - [18] ρ2 0.0001 [0.0001,0.0002] [32] η 0.002 day−1 [0.001, 0.006] [32] λh 0.027 - [8] λr µh ×Nr(0) - [11, 32] λv µh ×Nv(0) - [11, 32] µh 0.0067 - [8] µr 0.0017 - [15] µv 0.067 - [32] 6. Discussion and Conclusion VL remains a severe health concern in eastern Sudan despite numerous control ef- forts. In this study, we employ a Bayesian inference method to fit a previously developed mathematical model for VL transmission dynamics, incorporating the effects of control measures. We analyzed 22 years of cumulative VL cases and evaluated intervention ef- fectiveness over this period. The best-fit results (see Figure 6a and Table 4) suggest that VL is prevalent, with reported cases representing less than 20% of community infections, highlighting a significant gap in disease surveillance and diagnosis. Our analysis indicates that the efficacy of interventions targeting vector population control is limited, with α estimated around 50% (see Table 4). Notably, there are no documented interventions aimed at controlling the disease reservoir in the region. Our model predicts a negligible effect for αr > 2%. Furthermore, the model projects a potential increase in cumulative VL cases in the future (see Figure 6b), underscoring the necessity for adaptable multifaceted disease control initiatives. To achieve its stated goal of eradicating VL as a public health concern by 2030, the Sudanese National Leishmaniasis Control Program must implement concerted efforts. Em- bracing innovative research methodologies and strengthening public health infrastructure are essential steps toward mitigating the impact of VL. We advocate for continuous mon- REFERENCES 1224 itoring and evaluation of VL control programs, emphasizing the development of dynamic and responsive interventions tailored to the complex landscape of VL transmission. This study not only provides insights into the current state of VL management in eastern Sudan but also emphasizes the imperative for innovation in surveillance, treatment methodolo- gies and disease control strategies.In future research, we aim to extend our focus beyond visceral leishmaniasis control measures to the exploration of new fractional models using alternative techniques [3, 6, 9, 16, 20, 21, 29] . References [1] Mohammad A Abdoon, Rania Saadeh, Mohammad Berir, and Guma F E. Analysis, modeling and simulation of a fractional-order influenza model. Alexandria Engineer- ing Journal, 74:231–240, 2023. [2] Folashade B Agusto and Ibrahim M ELmojtaba. Optimal control and cost-effective analysis of malaria/visceral leishmaniasis co-infection. PLoS One, 12(2):e0171102, 2017. [3] Mawada Ali, Salem Mubarak Alzahrani, Rania Saadeh, Mohamed A Abdoon, Ahmad Qazza, Naseam Al-kuleab, and Fathelrhman EL Guma. Modeling covid-19 spread and non-pharmaceutical interventions in south africa: A stochastic approach. Scientific African, page e02155, 2024. [4] Dalal Khalid Almutairi, Mohamed A Abdoon, Salih Yousuf Mohamed Salih, Shahi- naz A Elsamani, Fathelrhman EL Guma, and Mohammed Berir. Modeling and anal- ysis of a fractional visceral leishmaniosis with caputo and caputo–fabrizio derivatives. Journal of the Nigerian Society of Physical Sciences, pages 1453–1453, 2023. [5] Abdulrahman BMAlzahrani, Rania Saadeh, Mohamed A Abdoon, Mohamed Elbadri, Mohammed Berir, and Ahmad Qazza. Effective methods for numerical analysis of the simplest chaotic circuit model with atangana–baleanu caputo fractional derivative. Journal of Engineering Mathematics, 144(1):9, 2024. [6] Salem Mubarak Alzahrani, Rania Saadeh, Mohamed A Abdoon, Ahmad Qazza, Fa- thelrhman EL Guma, and Mohammed Berir. Numerical simulation of an influenza epidemic: Prediction with fractional seir and the arima model. Appl. Math, 18(1):1– 12, 2024. [7] RWAshford, J Seaman, J Schorscher, and F Pratlong. Epidemic visceral leishmaniasis in southern sudan: identity and systematic position of the parasites from patients and vectors. Transactions of the Royal Society of Tropical Medicine and Hygiene, 86(4):379–380, 1992. [8] The World Bank. Sudan data, 2023. data retrieved from World Development Indi- cators, https://data.worldbank.org/country/sudan. REFERENCES 1225 [9] Mohammed Berir, Rania Saadeh, Mohamed A Abdoon, Ahmad Qazza, and Dalal Almutairi. A fractional study for solving the sir model and chaotic system. IAENG International Journal of Applied Mathematics, 54(2), 2024. [10] Jeff Bezanson, Alan Edelman, Stefan Karpinski, and Viral B Shah. Julia: A fresh approach to numerical computing. SIAM Review, 59(1):65–98, 2017. [11] Santanu Biswas, Abhishek Subramanian, Ibrahim M ELMojtaba, Joydev Chat- topadhyay, and Ram Rup Sarkar. Optimal combinations of control strategies and cost-effective analysis for visceral leishmaniasis disease transmission. PLoS One, 12(2):e0172465, 2017. [12] Youness Chatibi, El Hassan El Kinani, and Abdelaziz Ouhadan. Lie symmetry analy- sis and conservation laws for the time fractional black–scholes equation. International Journal of Geometric Methods in Modern Physics, 17(01):2050010, 2020. [13] Youness Chatibi, El Hassan El Kinani, and Abdelaziz Ouhadan. On the discrete sym- metry analysis of some classical and fractional differential equations. Mathematical Methods in the Applied Sciences, 44(4):2868–2878, 2021. [14] Youness Chatibi, Abdelaziz Ouhadan, et al. Continuous and discrete symmetry meth- ods for fractional differential equations. In Fractional-Order Modeling of Dynamic Systems with Applications in Optimization, Signal Processing and Control, pages 1– 35. Elsevier, 2022. [15] Christopher Dye. The logic of visceral leishmaniasis control. The American journal of tropical medicine and hygiene, 55(2):125–130, 1996. [16] Mohamed Elbadri, Mohamed A Abdoon, Mohammed Berir, and Dalal Khalid Almu- tairi. A symmetry chaotic model with fractional derivative order via two different methods. Symmetry, 15(6):1151, 2023. [17] Ibrahim M ELmojtaba. Mathematical model for the dynamics of visceral leishmaniasis–malaria co-infection. Mathematical Methods in the Applied Sciences, 39(15):4334–4353, 2016. [18] S Gasim, AM Elhassan, A Kharazmi, EAG Khalil, A Ismail, and TG Theander. The development of post-kala-azar dermal leishmaniasis (pkdl) is associated with acquisition of leishmania reactivity by peripheral blood mononuclear cells (pbmc). Clinical & Experimental Immunology, 119(3):523–529, 2000. [19] Fathelrhman EL Guma. Comparative analysis of time series prediction models for visceral leishmaniasis: based on sarima and lstm. Appl. Math, 18(1):125–132, 2024. [20] Fathelrhman EL Guma, Ossama M Badawy, Mohammed Berir, and Mohamed A Abdoon. Numerical analysis of fractional-order dynamic dengue disease epidemic in sudan. Journal of the Nigerian Society of Physical Sciences, pages 1464–1464, 2023. REFERENCES 1226 [21] Faeza Lafta Hasan, Mohamed A. Abdoon, Rania Saadeh, Ahmad Qazza, and Dalal Khalid Almutairi. Exploring analytical results for (2+1) dimensional breaking soliton equation and stochastic fractional broer-kaup system. AIMS Mathematics, 9(5):11622–11643, 2024. [22] John F Hermance. Historical variability of rainfall in the African East Sahel of Sudan: implications for development. Springer Science & Business Media, 2013. [23] C Malaria Consortium et al. Leishmaniasis control in eastern africa: Past and present efforts and future needs. Situation and gap analysis, 86, 2010. [24] Buddhi Pantha, Folashade B Agusto, and Ibrahim M Elmojtaba. Optimal control applied to a visceral leishmaniasis model. Electronic Journal of Differential Equations, 2020(80):1–24, 2020. [25] A Qazza, M Abdoon, R Saadeh, and M Berir. A new scheme for solving a fractional differential equation and a chaotic system. European Journal of Pure and Applied Mathematics, 16(2):1128–1139, 2023. [26] R Qazza, A Saadeh. On the analytical solution of fractional sir epidemic model. Applied Computational Intelligence and Soft Computing, 2023, 2023. [27] KS Rock, RJ Quinnell, GF Medley, and O Courtenay. Progress in the mathematical modelling of visceral leishmaniasis. Advances in parasitology, 94:49–131, 2016. [28] R Saadeh, M Abdoon, A Qazza, and M Berir. A numerical solution of generalized caputo fractional initial value problems. Fractal And Fractional, 7(4):332, 2023. [29] Rania Saadeh, Mohamed A Abdoon, Ahmad Qazza, Mohammed Berir, Fathel- rhman EL Guma, Naseam Al-Kuleab, and Abdoelnaser M Degoot. Mathematical modeling and stability analysis of the novel fractional model in the caputo derivative operator: a case study. Heliyon, 2024. [30] Tridip Sardar, Sourav Rana, Sabyasachi Bhattacharya, Kamel Al-Khaled, and Joydev Chattopadhyay. A generic model for a single strain mosquito-transmitted disease with memory on the host and the vector. Mathematical biosciences, 263:18–36, 2015. [31] Yateng Song, Tailei Zhang, Hui Li, Kai Wang, and Xiaobo Lu. Mathematical model analysis and simulation of visceral leishmaniasis, kashgar, xinjiang, 2004–2016. Com- plexity, 2020:1–14, 2020. [32] Anette Stauch, Ram Rup Sarkar, Albert Picado, Bart Ostyn, Shyam Sundar, Suman Rijal, Marleen Boelaert, Jean-Claude Dujardin, and Hans-Peter Duerr. Visceral leish- maniasis in the indian subcontinent: modelling epidemiology and control. PLoS ne- glected tropical diseases, 5(11):e1405, 2011. REFERENCES 1227 [33] Temmy Sunyoto, Gamal K Adam, Atia M Atia, Yassin Hamid, Rabie Ali Babiker, Nugdalla Abdelrahman, Catiane Vander Kelen, Koert Ritmeijer, Gabriel Alcoba, Margriet den Boer, et al. “kala-azar is a dishonest disease”: community perspectives on access barriers to visceral leishmaniasis (kala-azar) diagnosis and care in southern gadarif, sudan. The American journal of tropical medicine and hygiene, 98(4):1091, 2018. [34] Songnian Zhao, Yan Kuang, Chih-Hang Wu, David Ben-Arieh, Marcelo Ramalho- Ortigao, and Kaiming Bi. Zoonotic visceral leishmaniasis transmission: modeling, backward bifurcation, and optimal control. Journal of mathematical biology, 73:1525– 1560, 2016.