Reproduction Number Research Articles

Insights into the community structure and environmental functions of water hyacinth rhizobiome in urban river ecosystem.

Water hyacinth (WH) is a widespread floating invasive aquatic plant with a prolific reproductive and dispersion rate. With the aid of its root-associated microbes, WH significantly modulates the ecosystem's functioning. Despite their irrevocable importance, the WH microbiome remains unexplored in detail. Here, we present a shotgun sequencing analysis of WH rhizobiome predominant in urban rivers and their surrounding water to unveil the diversity drivers and functional relationship. The core microbiome of the WH mainly consisted of the methane-metabolizing archaebacteria and sulfate-reducing bacteria, which are probably driving the methane and sulfur metabolic flux along the vegetative zone in the water. The beta diversity analysis revealed temporal variations (River WH_2020 vs. WH_2022) (R of 0.8 to 1 and R2 of 0.17 to 0.41), which probably could be attributed to the transient taxa as there was a higher sharing of core bacteria (48%). Also, the WH microbiome significantly differed (R = 0.46 to 1.0 and R2 of 0.18 to 0.52) from its surrounding water. Further, the functional analysis predicted 140 pollutant-degrading enzymes (PDEs) well-implicated in various xenobiotic pollutant degradation, including hydrocarbons, plastics, and aromatic dyes. These PDEs were mapped to bacterial genera such as Hydrogenophaga, Ideonella, Rubrivivax, Dechloromonas, and Thauera, which are well-reported for facilitating the metabolism of xenobiotic compounds. The higher prevalence of metal and biocide resistance genes further highlighted the persistence of resistant microbes assisting WH in bioremediation applications.

Evolution of COVID-19 dynamics in Guangdong Province, China: an endemic-epidemic modeling study

BackgroundFrom January 2020 to June 2022, strict interventions against COVID-19 were implemented in Guangdong Province, China. However, the evolution of COVID-19 dynamics remained unclear in this period.ObjectivesThis study aims to investigate the evolution of within- and between-city COVID-19 dynamics in Guangdong, specifically during the implementation of rigorous prevention and control measures. The intent is to glean valuable lessons that can be applied to refine and optimize targeted interventions for future crises.MethodsData of COVID-19 cases and synchronous interventions from January 2020 to June 2022 in Guangdong Province were collected. The epidemiological characteristics were described, and the effective reproduction number (Rt) was estimated using a sequential Bayesian method. Endemic-epidemic multivariate time-series model was employed to quantitatively analyze the spatiotemporal component values and variations, to identify the evolution of within- and between-city COVID-19 dynamics.ResultsThe incidence of COVID-19 in Guangdong Province was 12.6/100,000 population (15,989 cases) from January 2020 to June 2022. The Rt predominantly remained below 1 and increased to a peak of 1.39 in Stage 5. As for the evolution of variations during the study period, there were more spatiotemporal components in stage 1 and 5. All components were fewer from Stage 2 to Stage 4. Results from the endemic-epidemic multivariate time-series model revealed a strong follow-up impact from previous infections in Dongguan, Guangzhou and Zhanjiang, with autoregressive components of 0.48, 0.45 and 0.36, respectively. Local risk was relatively high in Yunfu, Shanwei and Shenzhen, with endemic components of 1.17, 1.04 and 0.71, respectively. The impact of the epidemic on the neighboring regions was significant in Zhanjiang, Shenzhen and Zhuhai, with epidemic components of 2.14, 1.92, and 1.89, respectively.ConclusionThe findings indicate the presence of spatiotemporal variation of COVID-19 in Guangdong Province, even with the implementation of strict interventions. It’s significant to prevent transmissions within cities with dense population. Preventing spatial transmissions between cities is necessary when the epidemic is severe. To better cope with future crises, interventions including vaccination, medical resource allocation and coordinated non-pharmaceutical interventions were suggested.

Optimal control analysis on the spread of COVID-19: Impact of contact transmission and environmental contamination

The study investigates the intricate dynamics of SARS-CoV-2 transmission, with a particular focus on both close-contact interactions and environmental factors. Using advanced mathematical modeling and epidemiological analysis, explored the effects of these transmission pathways on the spread of COVID-19. The equilibrium points for both the disease-free and endemic states calculated and evaluated their global stability. Additionally, the basic reproduction number (R0) is derived to quantify the transmission potential of the virus.To ensure model accuracy, numerical simulations are performed using MATLAB, utilizing daily COVID-19 case data from India. Parameter values are sourced from existing literature, with certain parameters estimated through fitting the model to observed data. Crucially, the model incorporates environmental transmission factors, such as surface contamination and airborne spread. The inclusion of these factors provides a more comprehensive understanding of the virus’s spread, demonstrating the importance of interventions like use of face masks, environmental sanitization, vaccine efficacy, availability of treatment resources underappreciated when focusing solely on direct human contact. A sensitivity analysis is conducted to assess the impact of different parameters on R0, with results visualized through heat maps to identify the most influential factors.Furthermore, Pontryagin’s maximum principle is employed to develop an optimal control model, enabling the formulation of effective intervention strategies. By analysing both interpersonal and environmental transmission mechanisms, this study offers a more holistic framework for understanding SARS-CoV-2 transmission. The insights gained are critical for informing public health strategies, emphasizing the necessity of addressing both direct contact and environmental sources of infection to more effectively manage current and future outbreaks.

Using the basic reproduction ratio to quantify transmission and identify data gaps for epizootic haemorrhagic disease virus.

Epizootic haemorrhagic disease virus (EHDV) is an arbovirus transmitted by Culicoides biting midges that has recently emerged in Europe. Here, the basic reproduction ratio (R 0) was used to quantify the transmission of EHDV and its dependence on temperature for cattle and deer. Using data from the published literature the parameters needed to calculate R 0 were estimated with Bayesian methods to incorporate uncertainty in the calculations. The Sobol method of sensitivity analysis was used to determine the parameters having the greatest influence on R 0 and, hence, to identify important data gaps. Depending on the strain, the maximum R 0 for EHDV varied from 0.7 to 2.5 in cattle and 1.3 to 4.3 in deer. The maximum R 0 occurred at temperatures between 22 and 25°C, while the lowest temperature at which R 0 exceeded one was between 16 and 20°C. The sensitivity analysis identified the threshold temperature for virus replication, the probability of transmission from host to vector and the vector- to- host ratio as the most important parameters influencing R 0. Furthermore, there are only limited data on EHDV in European deer species and on transmission in wildlife and at the livestock/wildlife interface. These data gaps should be the focus of future research.

Epidemiologic Quantities for Monkeypox Virus Clade I from Historical Data with Implications for Current Outbreaks, Democratic Republic of the Congo.

We used published data from outbreak investigations of monkeypox virus clade I in the Democratic Republic of the Congo to estimate the distributions of critical epidemiological parameters. We estimated a mean incubation period of 9.9 days (95% credible interval [CrI] 8.5-11.5 days) and a mean generation time of 17.2 days (95% CrI 14.1-20.9 days) or 11.3 days (95% CrI 9.4-14.0 days), depending on the considered dataset. Presymptomatic transmission was limited. Those estimates suggest generally slower transmission dynamics in clade I than in clade IIb. The time-varying reproduction number for clade I in the Democratic Republic of the Congo was estimated to be below the epidemic threshold in the first half of 2024. However, in the South Kivu Province, where the newly identified subclade Ib has been associated with sustained human-to-human transmission, we estimated an effective reproduction number above the epidemic threshold (95% CrI 0.96-1.27).

Digital technologies in dairy cattle breeding to improve the reproductive function of cows and heifers: A case study in Northern Kazakhstan

Background and Aim: In some countries, the application of digital technologies in dairy cattle breeding is still under development. This study aimed to investigate the use of digital technologies in dairy cattle breeding to improve the reproductive function of cows and heifers in three northern regions of Kazakhstan. Materials and Methods: This study explores the application of Austrian smaXtec bolus sensors, which enable the daily monitoring of the reproductive functions of cows and heifers in livestock. To control indicators of reproductive function in Simmental and Holstein-Friesian cattle breeds, a series of experiments were conducted before and after the introduction of boluses in the rumen. Results: It was established that the application of smaXtec boluses increases milk yield in 305 days, the percentage of conception in the first insemination and in cows with up to three inseminations, the duration of dry secretion, and the percentage of calve output per 100 heads. Moreover, the use of smaXtec boluses reduced the insemination index, duration of the calving-to-conception interval (open days), reproductive rate, and percentage of abortions and culls due to gynecological problems. Conclusion: The use of smaXtec boluses allows farmers and veterinarians to determine indicators, such as the period of sexual heat in livestock and diseases, in a timely manner and to increase the efficiency of feeding and controlling drinking cycles. Moreover, the application of smaXtec boluses minimizes labor costs associated with collecting data on indicators of reproductive function in cows and heifers and increases accuracy. Keywords: bolus, cows, digital technology, reproductive function, smaXtec.

Minimal wave speed for a two-group epidemic model with nonlocal dispersal and spatial-temporal delay

In this paper, a two-group SIR reaction-diffusion epidemic model with nonlocal dispersal and spatial-temporal delay based on within-group and inter-group transmission mechanisms is investigated. The basic reproduction number R0 is calculated using the method of next-generation matrix. The critical wave speed cm* is determined by analyzing the distribution of roots of the characteristic equation. When R0>1 and wave speed c⩾cm*, the existence of traveling waves connecting disease-free and endemic steady states is obtained by constructing sub- and super-solutions and a Lyapunov functional, and applying Schauder’s fixed-point theorem and a limit argument. When R0>1 and 0<c<cm*, the nonexistence of traveling waves connecting disease-free and endemic steady states is proven by contradiction and two-sided Laplace transform. This indicates that the critical wave speed cm* is exactly the minimum wave speed. Numerical simulations are carried out to illustrate theoretical results. The dependence of the minimal speed cm* on time delay, diffusion rates and contact rates is discussed, showing that the longer the latent period and the lower the diffusion rates of infected individuals and the inter-group transmission rates between groups, the slower the spread of disease.

Analysis and optimal control for SEIR mathematical modeling of COVID-19

In this paper a mathematical model of SEIR type is formulated. represented by modeling the coronavirus epidemic. In this present study, we consider a mathematical model that incorporates the whole population and variability in transmission between reported and unreported populations. The global stability of the disease free equilibrium (DFE) point is established. The basic reproduction number R0 is calculated. We introduce into our model two controls which are vaccination ofsusceptible humans denoted by u and treatment of infected humans designed by v. In addition, this model takes into consideration the control of contact between infectious individuals and susceptible persons A numerical simulation of the model is made.

A Regional-Scale Assessment-Based SARS-CoV-2 Variants Control Modeling with Implications for Infection Risk Characterization.

The emergence and progression of highly divergent SARS-CoV-2 variants have posed increased risks to global public health, triggering the significant impacts on countermeasures since 2020. However, in addition to vaccination, the effectiveness of non-pharmaceutical interventions, such as social distancing, masking, or hand washing, on different variants of concern (VOC) remains largely unknown. This study provides a mechanistic approach by implementing a control measure model and a risk assessment framework to quantify the impacts of control measure combinations on the transmissions of five VOC (Alpha, Beta, Delta, Gamma, and Omicron), along with a different perspective of risk assessment application. We applied uncontrollable ratios as an indicator by adopting basic reproduction number (R 0) data collected from a regional-scale survey. A risk assessment strategy was established by constructing VOC-specific dose-response profiles to implicate practical uses in risk characterization when exposure data are available. We found that social distancing alone was ineffective without vaccination in almost all countries and VOC when the median R 0 was greater than two. Our results indicated that Omicron could not be contained, even when all control measure combinations were applied, due to its low threshold of infectivity (~3×10-4 plague-forming unit (PFU) mL-1). To facilitate better decision-making in future interventions, we provide a comprehensive evaluation of how combined control measures impact on different countries and various VOC. Our findings indicate the potential application of threshold estimates of infectivity in the context of risk communication and policymaking for controlling future emerging SARS-CoV-2 variant infections.

Stochastic differential equations model for the interaction of HCV with liver cells

The main objective of this paper is to study the dynamic behavior of the hepatitis C virus to control the spread of this disease. The study was conducted by creating a new stochastic mathematical model. Three equilibriums specific to this system are discussed, namely, disease-free equilibrium, equilibrium in the absence of an antibody response, and equilibrium when immune response CTLs are zero. We also showed conditions that must be met for the injured person to recover by searching for the basic reproduction number. If it means that the liver will get rid of the virus and heal the infected person. while in this case, the infection grows and the disease can invade all liver cells. These results have also been demonstrated by computer simulations.

A clustered-LPSEIRS malware propagation model in complex networks

In this paper, we present a new malware propagation model that integrates epidemic spread, clustering, and link prediction techniques, tailored for complex network networks. Our model is based on the clustered-link prediction-susceptible-exposed-infected-recovered (clustered-LPSEIRS) epidemic model, which simulates malware dissemination within the network. Our findings reveal a significant decrease in the rate of malware spread compared to the traditional SEIR model, with this enhancement in containment attributed to the integration of clustering and link prediction methods. We also compute the basic reproduction ratio (R0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R_{0}$$\\end{document}) for our model, providing insights into the potential ramifications of malware within the network. By examining parameter variations, we enhance our understanding of the model's behavior under diverse scenarios. Additionally, we assess the influence of clustering and link prediction on mitigating malware spread, emphasizing its effectiveness in diminishing the overall impact.

Serologic Evidence for Early SARS-CoV-2 Circulation in Lima, Peru, 2020.

During early 2021, Peru had the highest COVID-19-associated per-capita mortality rate. Socioeconomic inequality, insufficiently prepared healthcare, and surveillance systems are factors explaining the mortality rate, which can be severely worsened by early undetected SARS-CoV-2 circulation. We tested 1,441 individuals with fever sampled during August 2019-May 2021, several months before the first SARS-CoV-2 seroprevalence study available so far in Lima, Peru, for SARS-CoV-2-specific antibodies. The testing algorithm included a chemiluminescence immunoassay and surrogate virus neutralization test. Early positive samples (N = 24) from January-March 2020 were further tested using a plaque-reduction neutralization test (PRNT) and avidity test against the SARS-CoV-2 spike and nucleoprotein. None of the early samples were PRNT-confirmed, in contrast to 81.8% (18/22) of a subsample from April 2020 onward (Fisher exact test; P <0.0001). Therefore, we excluded non-PRNT-confirmed samples from subsequent analyses. The SARS-CoV-2 antibody detection rate was 0.9% in mid-April 2020 (1/104; 95% CI: 0.1-5.8%), suggesting viral circulation in early-middle March 2020, consistent with the first molecular detection of SARS-CoV-2 in Peru on March 2020. Mean avidity increase of 62-77% to 81-94% from all PRNT-confirmed SARS-CoV-2-positive samples during early 2020 were consistent with onset of SARS-CoV-2 circulation during late February/March 2020. Early circulation was also confirmed in a susceptible, exposed, infected, and recovered mathematical model that calculated an effective reproduction number >1 during February-March 2020. Early introduction of SARS-CoV-2 thus contributed to the high COVID-19 mortality rate in Peru. Emphasizing the role of diagnostic confirmation in understanding the pandemic's trajectory, this study highlights the importance of early detection and accurate testing in managing infectious disease outbreaks.

Regional demography of Icelandic rock ptarmigan and its implications for harvest management

Abstract The rock ptarmigan (Lagopus muta) is a popular game bird in Iceland, but management of hunting has tended to be controversial. We were interested in whether regional differences in ptarmigan demography exist, and if so, their implications for harvest management. We fit integrated population models (IPMs) to monitoring data from six hunting regions of Iceland during 2005–2023, and then examined equilibrium and non‐equilibrium harvest strategies based on an objective to maximize sustainable harvest. Survival and reproductive rates tended to be similar amongst regions, except in the demographically important Northeast, where survival was lower and productivity was higher. There was a negative relationship between post‐breeding age ratio (juveniles/adults) and spring abundance in all regions, although the strength of density dependence varied. Spring abundance was temporally stable in all regions, although harvest rates tended to decline, and adult survival rates tended to increase. Evidence for cyclical patterns in abundance was weak except in the Northeast, but this may be an artifact of the relatively short time series. Sustainable harvest potential varied amongst regions and was greatest in the East and Northeast. It appears that harvests during 2005–2022 were somewhat less than the sustainable maximum. Non‐equilibrium harvest strategies, in which the allowable harvests depend on spring population size and anticipated productivity, are so‐called ‘bang‐bang’ strategies, meaning that no or very little harvest is optimal until the population is at or above its maximum level of net production. Practical implications. Regional differences in ptarmigan demography warrant different harvest strategies. Yet, the hunting season in Iceland historically has been regulated on a country‐wide basis, with population abundance and dynamics in the Northeast playing a key role. To implement regional harvest management, decision makers must first agree on harvest management objectives that satisfy most stakeholders, who tend to hold very diverse values. Moreover, the historical relationship between season length and harvest rate is tenuous, yet season length is the primary mechanism traditionally used in Iceland to control harvest. The uncertainty in this relationship could be addressed in an adaptive‐management framework.

Ecotoxicity of alkaline residue from the pulp and paper industry to soil faunal organisms

The increased production and expanding demand for cellulosic products have required companies to provide solutions for the proper disposal of generated residues without compromising soil quality. In this context, the current research aimed at evaluating the ecotoxicological impacts of dregs application in subtropical soils. The experiments were carried out with the earthworm Eisenia andrei, the springtail Folsomia candida, and the potworm Enchytraeus crypticus in two subtropical soils (Neosol and Cambisol) collected in the southern region of Brazil. The considerable differences in texture and organic matter content motivated the choosing of these soils. The reproductive rate of all organisms was influenced, with varying EC50 values depending on the soil type. However, more pronounced effects were observed for Neosol. The springtail F. candida proved to be the most sensitive to contamination (EC50 = 5.8 g kg−1 for Neosol and 48.5 g kg−1 for Cambisol), followed by the earthworm E. andrei (EC50 = 62.4 g kg−1 for Neosol and 129.5 g kg−1 for Cambisol) and the potworm E. crypticus (EC50 = 67.2 g kg−1 for Neosol and 230.4 g kg−1 for Cambisol). As a result, while dregs can be used to correct soil acidity, they have been shown to have negative effects on important organisms responsible for functions critical to maintaining ecosystem quality.

The antidepressant, sertraline, impacts growth and reproduction in the benthic deposit feeder, Tubifex tubifex

Among emerging contaminants, pharmaceuticals are considered one of the most pertinent substances that may threaten aquatic ecosystems. Pharmaceuticals are designed to be directed at specific metabolic- and molecular pathways. Thus, they are assumed to be still biologically active when entering the ecosystem and may result in unpremeditated impacts on non-target organisms. One of the most widely used selective serotonin reuptake inhibitors, sertraline (an antidepressant), is regularly found in aquatic environments. However, knowledge about the effects, and in particular, of sediment-associated sertraline in benthic invertebrates is limited. We examined the impacts of chronic exposure (28 d) to sediment-associated sertraline (3.3, 33, 330 μg/g dw sed.) on survival, growth and reproduction in the deposit-feeding oligochaete, Tubifex tubifex. Sertraline significantly decreased T. tubifex survival and growth. Worms exposed to high sertraline concentrations (330 μg/g) had a lower growth rate and reproduction, as indicated by a significantly lower number of cumulated cocoons. Worms exposed to an environmentally relevant concentration (3.3 μg/g) decreased growth but maintained a reproduction rate similar to that of the control. The implications are that adult worms exposed to high sertraline concentrations presumably required more energy for maintenance and detoxification, thereby reducing available energy for reproduction and growth. This represents a trade-off between survival, reproduction and growth. In contrast, T. tubifex exposed to environmentally relevant concentrations allocated more energy to reproduction by slightly increasing the number of cocoons produced and reducing growth. However, the quantity and quality of offspring may be impacted as we observed fewer juveniles in the environmentally relevant treatment than in the control. Overall, the results indicate that sediment-associated sertraline is bioavailable and negatively impacts T. tubifex survival, growth, and reproduction even at environmentally relevant concentrations.

Lethal and sublethal concentrations spirodiclofen stress may increase the adaptation of Panonychus citri (Acari: Tetranychidae).

Panonychus citri is one of the most destructive pests in citrus orchards, exhibiting varying degrees of tolerance to numerous insecticides, such as spirodiclofen. To effectively manage pests, this study explores the response of P. citri to spirodiclofen stress from the perspectives of life history, enzymatic parameters, and reproduction. The effects of two concentrations (LC30 and LC50) of spirodiclofen on the biological parameters of P. citri were evaluated by the life table method. The results showed that the development duration, fecundity, oviposition days, and lifespan were shortened, though the pre-oviposition period of two treatments was prolonged in comparison with the control. A significant decrease was recorded in the net reproductive rate (R0) and the mean generation time (T) for the two treatments. Nevertheless, the intrinsic rate of increase (r) and the rate of increase (λ) were not significantly affected in the LC30 treatment, whereas they declined in the LC50 treatment. The enzyme activity assay resulted in higher activities of catalase (CAT), superoxide dismutase (SOD), peroxidase (POD), and carboxylesterase (CarE), among the treatments than the control. In contrast, the treatments recorded lower cytochromeP450 (CYP450) and Glutathione S-transferase (GST) activities than the control. Furthermore, the study detected that relative mRNA expression of Vitellogenin (Vg) and Vitellogenin receptor (VgR) for two treatments were lower than the control. In summary, two concentrations of spirodiclofen inhibited progeny growth and fecundity of P. citri. Additionally, the results of this study may support further research on tolerance of P. citri in response to spirodiclofen stress.

Spotlight hunting of invasive deer (Axis axis) selects for individuals with greater reproductive value in a protected area of north-eastern Argentina

Spotlight hunting of invasive deer (Axis axis) selects for individuals with greater reproductive value in a protected area of north-eastern Argentina

Physiological effect of leptin and time-restricted feeding-in reproductive value of female rat

Physiological effect of leptin and time-restricted feeding-in reproductive value of female rat

Mathematical analysis of Ebola considering transmission at treatment centres and survivor relapse using fractal-fractional Caputo derivatives in Uganda

In this article, we seek to formulate a robust mathematical model to study the Ebola disease through fractal-fractional operators. The study thus incorporates the transmission rate in the treatment centers and the relapse rate, since the Ebola virus persists or mostly hides in the immunologically protected sites of survivors. The Ebola virus disease (EVD) is one of the infectious diseases that has recorded a high death rate in countries where it is endemic, and Uganda is not an exception. The world at large has suffered from this deadly disease since 1976 when it was declared epidemic by the World Health Organization. The study employed fractal-fractional operators to identify the epidemiological patterns of EVD, especially in treatment centers and relapse. Memory loss and relapse are mostly observed in EVD survivors and this justifies the use of fractional operators to capture the true dynamics of the disease. Through dynamical analysis, the model is proven to be positive and bounded in the region. The model is further explicitly shown to have a solution that is unique and stable. The reproduction number was duly computed by using the next-generation matrix approach. By taking EVD epidemic cases in Uganda, the study fitted all parameters to real data. It has been shown through sensitivity index analysis that the transmission rate outside treatment centers and relapse have a significant effect on the endemic state of the disease, as they lead to an increase in the basic reproduction ratio.

On a Symmetry-Based Structural Deterministic Fractal Fractional Order Mathematical Model to Investigate Conjunctivitis Adenovirus Disease

In the last few years, the conjunctivitis adenovirus disease has been investigated by using the concept of mathematical models. Hence, researchers have presented some mathematical models of the mentioned disease by using classical and fractional order derivatives. A complementary method involves analyzing the system of fractal fractional order equations by considering the set of symmetries of its solutions. By characterizing structures that relate to the fundamental dynamics of biological systems, symmetries offer a potent notion for the creation of mechanistic models. This study investigates a novel mathematical model for conjunctivitis adenovirus disease. Conjunctivitis is an infection in the eye that is caused by adenovirus, also known as pink eye disease. Adenovirus is a common virus that affects the eye’s mucosa. Infectious conjunctivitis is most common eye disease on the planet, impacting individuals across all age groups and demographics. We have formulated a model to investigate the transmission of the aforesaid disease and the impact of vaccination on its dynamics. Also, using mathematical analysis, the percentage of a population which needs vaccination to prevent the spreading of the mentioned disease can be investigated. Fractal fractional derivatives have been widely used in the last few years to study different infectious disease models. Hence, being inspired by the importance of fractal fractional theory to investigate the mentioned human eye-related disease, we derived some adequate results for the above model, including equilibrium points, reproductive number, and sensitivity analysis. Furthermore, by utilizing fixed point theory and numerical techniques, adequate requirements were established for the existence theory, Ulam–Hyers stability, and approximate solutions. We used nonlinear functional analysis and fixed point theory for the qualitative theory. We have graphically simulated the outcomes for several fractal fractional order levels using the numerical method.