Articles: Department of Mathematics and Statistics

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Forecasting Kenya's public debt using time series analysis
(University of Embu, 2024-08-03) Obwoge Frankline Keraro; Zakayo Ndiku Morris; Kitavi Dominic Makaa; Maurice Wanyony
Accurately forecasting public debt is essential for developing countries like Kenya to maintain fiscal sustainability and economic stability. This study aimed to identify the best time series forecasting model for predicting Kenya's future public debt to help policymakers create effective fiscal reforms. The Autoregressive Integrated Moving Average (ARIMA) and Holt-Winters exponential smoothing models were tested due to their ability to handle complex patterns and seasonality in time series data. Public debt data from Kenya from 2001 to 2021 were analyzed, and both models were applied to the processed data. The ARIMA (0,2,1) model, which uses second-order differencing and a moving average component, was found to be the best model based on information criteria. The Holt-Winters additive method also showed good performance, adapting well to recent data and seasonal trends with optimized smoothing parameters. Both models produced forecasts that closely matched the actual debt figures for 2022 and 2023, with an error margin of only 0.73. Measures of accuracy, such as Mean Absolute Percentage Error (MAPE) and Mean Absolute Scaled Error (MASE), confirmed the reliability of the models, with ARIMA performing slightly better than Holt-Winters. While previous studies have looked at debt forecasting for Kenya, this research offers a thorough evaluation and comparison of two strong time series models. Unlike existing literature, this study provides a rigorous out-of-sample forecasting assessment, identifying the best approach for reliably predicting Kenya's debt. However, the study is limited by its focus on univariate time series models, which could be improved by including relevant external economic variables. The findings show that the ARIMA and Holt-Winters models are accurate tools for forecasting Kenya's public debt, helping policymakers to develop sustainable debt management strategies and fiscal reforms based on reliable future projections.
Perspectives on Judicial Efficiency: Case Study of Predictive Modeling in Kenyan High Courts
(University of Embu, 2025-10-08) KANDIE ALEX KANGOGO; ATITWA EDWINE BENSON; WESLEY NYALUKE
Persistent case backlogs and delays in providing timely justice undermine judicial efficiency and impose significant societal costs through legal expenditure, incentives for vexatious litigation, weakened rule of law, and erosion of public trust. In Kenya, quantitative analysis examining factors influencing resolution timelines within the High Court system has been limited. This study addresses this critical knowledge gap by applying survival analysis techniques to a comprehensive dataset spanning over 90,000 cases from 2012–2022. Specifically, this is aimed to perform explanatory data analysis of factors influencing the survival time of cases in Kenyan High Courts; to predict the time for a case to be determined in Kenyan High Courts using survival models; and to compare the model precision between Cox regression and other parametric survival models. This study applied exploratory visualization and regression modeling, where key predictors of case duration were identified, and the performance of parametric and semi-parametric survival models were fitted to establish comparison. Exploratory investigation indicated that family cases exhibited the longest durations, whereas criminal cases and anti-corruption cases were resolved the quickest. Appealed cases generally had a longer duration as well. According to the Akaike Information Criterion, the parametric Weibull distribution provided a better fit for the complicated event history data than the semi-parametric Cox Proportional Hazards model. Weibull regression showed civil, constitutional, family, and appeal cases had considerably lower resolution risks than anti-corruption proceedings. While certain case types may have counterintuitive estimates, this analysis provides a robust quantitative framework for resource allocation, legislation, and initiatives to reduce disproportionate backlogs across litigation areas and uphold constitutional rights to timely justice, which is critical for economic growth and societal prosperity.
Modelling the effects of psychological fear and vaccination on the dynamics of typhoid fever
(University of Embu, 2024-12-05) WANGUI JACKLINE WANJIKU; MARILYN RONOH; KANYIRI CAROLINE W.; MANDA EDNA CHILENJE
Despite the great advancements in healthcare systems and sanitary improvements globally, sub-Saharan Africa including Kenya bears a significant burden of infectious diseases, among which typhoid fever continues to exert a notable toll. In this study, we developed a deterministic mathematical model to examine the interplay between human responses driven by the psychological factor of fear of infection, vaccination efforts, and the dynamics of human-to-human and environmental transmission of typhoid fever. The mathematical model was analyzed using theories of first order ordinary differential equations to establish the existence of equilibrium points and their conditions for local and global stability. The reproduction number, R0, was established and distinct pathways for the transmission of infection were identified, shedding light on the crucial interactions among key population groups fueling the spread of typhoid fever disease. The model results suggests that, typhoid fever infection is heightened by both direct and indirect contact with infected individuals and contaminated environments. Additionally, lack or limited awareness contributes to decreased fear of infection and reluctance towards vaccination, further exacerbating the situation. Moreover, an increase in environmental transmission is observed due to elevated discharge rates from infected individuals. This study contributes valuable insights into the design of effective mitigation strategies aimed at combating typhoid fever in resource-limited settings.
Modeling the socioeconomic determinants of health insurance coverage among boda-boda riders in Kenya
(University of Embu, 2025) Kihara, Perpetual Wambui; Morris, Zakayo Ndiku; Mwilu, Nicholas Mutothya
Achieving universal health coverage remains a major challenge in low- and middle-income countries such as Kenya, especially for vulnerable groups like informal workers. This study focuses on boda-boda riders, an important part of Kenya’s transport sector, who often do not have access to formal health insurance. The aim of the study was to examine how socioeconomic factors influence health insurance enrollment among boda-boda riders in Kenya, with the goal of supporting fair access to healthcare. A descriptive cross-sectional design was used, and data were collected from 370 boda-boda riders in Embu County, Kenya, using a structured questionnaire. The questionnaire gathered information on socioeconomic characteristics, insurance enrollment status, perceived affordability of health insurance, and demographic details. Logistic regression was used to assess the effect of these factors on the likelihood of enrolling in health insurance. Results showed that 62.2% of participants had health insurance. The analysis found that riders with higher income (OR = 1.000, p < 0.001), more working hours per week (OR = 1.655, p < 0.001), and older age (OR = 1.270, p < 0.001) were more likely to be enrolled. In contrast, having more dependents (OR = 0.385, p < 0.001) and more years of experience in the boda-boda business (OR = 0.118, p < 0.001) were linked to a lower chance of enrollment. Additionally, those who viewed health insurance as affordable were significantly more likely to enroll (OR = 4.529, p < 0.001) compared to those who saw it as expensive. These findings highlight that both socioeconomic status and the perceived cost of insurance are key factors affecting enrollment in health insurance among boda-boda riders in Kenya.
SEMI-MARKOVIAN ANALYSIS OF THE PROGNOSIS OF BREAST CANCER BETWEEN DIAGNOSIS AND TREATMENT INITIATION IN KENYA: A CASE STUDY OF TWO COUNTIES
(University of Embu, 2025) WAWERU, PETER KAMAU; ATITWA, EDWINE BENSON; MORRIS, ZAKAYO NDIKU; KAILEMIA, PETER NTOITI
Breast cancer is a major health burden not only globally. It is the most commonly diagnosed type of cancer globally and in Kenya. In 2022, 7,243 new cases of breast cancer were reported accounting for approximately 16.2% of all cancer cases diagnosed with a mortality rate of 11.6% which translates to 3,398 deaths. This study aimed to determine the prevalence and analyze female breast cancer (FBC) prognosis between diagnosis and treatment, taking a case study of two counties in Kenya. Data for this study was obtained from two cancer registries in two county hospitals with a sample of 300 health records. After data cleaning, 150 records were eligible for analysis. Key variables of interest in the study were staging information of FBC at diagnosis and treatment, time taken between diagnosis and treatment, as well as the waiting time before transiting to the subsequent stage. One of the approaches that can be used to gain insight into how breast cancer progresses over time is the application of semi-Markov analysis which was used to analyze the prognosis of breast cancer in two counties in Kenya. This was obtained by determining the prevalence of FBC at diagnosis and at treatment and finding the transitional probabilities between different cancer states. The results of the analysis showed that FBC stage III was the most prevalent at diagnosis and treatment initiation with a prevalence of 36% and 34.7% respectively. The probability of remaining at stage II or stage III after diagnosis was found to decrease with the increase in the waiting time before treatment initiation. The results outline the necessity of timely diagnosis and initiation of interventions, which may help in clinical decision-making, resource allocation and inform public health policies.
Modelling and Simulation of Competition for Students’ Population with Holling Type II Response
(UoEm, 2023) Odhiambo, Brian; Ngari, Cyrus; Kimani, Patrick; Njori, Peter
Te increase in the country’s population attracted the establishment of more schools, both public and private schools, to cater for the increasing number of students. However, there have been dynamics of students’ population both in public and private schools through transfer from one category of school to the other, through completion of the learning period, and through dropout due to unknown reasons which have subjected both the public and private schools to compete in order to maintain a good number of students. In this work, a modifed Lotka–Volterra model of schools and nonenrolled entities population in the education system is studied. Private schools and nonenrolled entities play the role of a predator in public schools. Again, public schools and nonenrolled entities play the role of predators in private schools. Holling type II functional responses have been integrated in the analysis of the Lotka–Volterra model. Te equilibrium points are established and their stability are determined using the Routh–Hurwitz criterion and eigenvalue method. Global stability has been done for the positive equilibrium point. Bifurcation is also done around the positive equilibrium point. Finally, a graphical illustration of various parameter is derived to show their efect on schools when they are varied. It is revealed that the increase in parameters θ2 , θ3 , and η3 greatly afects the schools population as they are the ones leading to predation in school. Terefore, proper strategies should be developed to focus on reducing the mentioned parameters to avoid leading schools’ population to extinct
Transitivity Action of An on (n=4,5,6,7) on Unordered and Ordered Quadrupples
(UoEm, 2015) kimani, Gachago j; N, Kinyanjui J.; j, Rimberia; kimani, Patrick; muchemi, Jacob kiboi
In this paper, we study some transitivity action properties of the alternating group An(n=4,5,6,7 ,) acting on unordered and ordered pairs from the set X = {1,2,...,n} through determination of the number of disjoint equivalence classes called orbits.when n≤ 7 ,the alternating group acts transitively on both X (4) and X [4] . key words : Orbits ,alternating group An , An on unordered and ordered quadruples from the set X.
Rank and Subdegrees of P GL(2, q) Acting Cosets of P GL(2, e) for q an Even Power of e
(UoEm, 2019) Kimani, Patrick; Kamuti, Ireri; Rimberia, Jane
The action of projective general group on the cosets of its maximal subgroups has been studied. For instance, [9] studied the action of G on the cosets of P GL(2, e) when q is an odd prime power of e. In this paper, we determine the rank and subdegrees of the action of P GL(2, q) on the cosets of its subgroup P GL(2, e) for odd q and an even power of e. We apply the table of marks to achieve this.
Predicting the Number of Tourists in-Flow to Kenya Using Seasonal Autoregressive Integrated Moving Average Model
(UoEm, 2022-12) Gechore, Dennis; Atitwa, Edwin; Kimani, Patrick; Wanyonyi, Maurice
Tourism is the leading source of revenue to the Kenyan Government, contributing about 8.8% to the Kenya’s Gross Domestic Product. Based on the 2019 report released by the ministry of tourism and wildlife, tourism industry contributed approximately $7.9 billion to the Kenya’s budget. This study was therefore developed to predict the future numbers of tourists that will visit Kenya between 2023 and 2025. The Seasonal Autoregressive Integrated Moving Average time series model was applied for the prediction. The study used secondary data collected from the Ministry of Tourism and Wildlife. The data covered a period of 11 years from 2011 to 2022. The model was fitted to the real tourists’ data using the time series algorithm implemented in R statistical software. Based on the Akaike Information Criterion, the ARIMA(2,1,1)(0,1,0)12 was identified as the perfect model with minimum errors. The model passed the diagnostic test performed. Importantly, 95% confidence level prediction done for 3 years (2023-2025) using the model showed that the number of tourists expected to visit Kenya will increase significantly. Therefore, the study recommended that recreational facilities and accommodations should be maintained to cater for the high projected numbers of tourists. The study also recommended that the government of Kenya should strategize on how to beef up security to curb terrorism attacks and tribal conflicts which might discourage tourists.
On the Number of Cyclotomic Cosets and Cyclic Codes over Z13
(UoEm, 2018-06) Hussein, Lao; Kivunge, Benard; Kimani, Patrick; Muthoka, Geoffrey
Let Zq be a finite field with q element and x n − 1 be a given cyclotomic polynomial. The number of cyclotomic cosets and cyclic codes has not been done in general. Although for different values of q the polynomial x n − 1 has been characterised. This paper will determine the number of irreducible monic polynomials and cyclotomic cosets of x n − 1 over Z13 .The factorization of x n − 1 over Z13 into irreducible polynomials using cyclotomic cosets of 13 modulo n will be established. The number of irreducible polynomials factors of x n − 1 over Zq is equal to the number of cyclotomic cosets of q modulo n. Each monic divisor of x n − 1 is a generator polynomial of cyclic code in Fq n . This paper will further show that the number of cyclic codes of length n over a finite field F is equal to the number of polynomials that divide x n − 1. Finally, the number of cyclic codes of length n, when n = 13k, n = 13k , n = 13k − 1, k, 13 = 1 are determine.
Enumeration of cyclic codes over GF(17)
(UoEm, 2015-05) Hussein, Lao; Kivunge, Benard; Muthoka, Geoffrey; Mwangi, Patrick
In this paper we seek the number of irreducible polynomials of xn− 1 over GF(17). We factorize Xn− 1 over GF(17)into irreducible polynomials using cyclotomic cosets of 17 modulo n . The number of irreducible polynomials factors of Xn− 1 over fq is equal to the number of q cyclotomic cosets of modulo n. Each monic divisor of Xn− 1 is a generator polynomial of cyclic code in Fqn. We show that the number of cyclic codes of length n over a finite field f is equal to Xn− 1. Lastly, the number of cyclic codes of length n , when n= 17 , = the number of polynomials that divide 17k ,n = 17k,n=17k − 1, ( , 17) = 1 are enumerated.
Cycle Index Formulas for Dn Acting on Ordered pairs
(UoEm, 2016-04) Muthoka, Geoffrey; Kamuti, Ireri; Kimani, Patrick; Hussein, Lao
The cycle index of dihedral group Dn acting on the set of the vertices of a regular n-gon was studied by Harary and Palmer in 1973 [1]. Since then a number of researchers have studied the cycle indices dihedral group acting on different sets X={1,2,...,n} and the resulting formulas Dn have found applications in enumeration of a number of items. Muthoka (2015) [2] studied the cycle index formula of the –the n vertices of a regular -gon. In this paper we study the dihedral group acting on unordered pairs from the set X={1,2,..,n} cycle index formulas of acting on ordered pairs from the set . In each case the actions of the cyclic part and the reflection part are studied separately for both an even value of and an odd value of n .
Cycle Index Formulas for Dn Acting on Unordered Pairs
(UoEm, 2015) Muthoka, Geoffrey; Kamuti, Ireri; Lao, Hussein; kimani, patrick
The cycle index of dihedral group Dn acting on the set X of the vertices of a regular -gon was studied (See [1]). In this paper we study the cycle index formulas of Dn acting on unordered pairs from the set x . In each case the actions of the cyclic part and the reflection part are studied separately for both an even value of n and an odd value of n .
Action ofPGL (2 ,q ) on the Cosets of the Centralizer of an Eliptic Element
(UoEm, 2021) Kimani, patrick; Adicka, Daniel
Most researchers consider the action of projective general group on the cosets of its maximal subgroups leaving out non-maximal subgroupsCq+1. In this paper, we consider the action ofPGL (2, ) centralizer of an elliptic element which is a non maximal subgroup . In particular, we determine the subdegrees, rank and properties of the suborbital graphs of the action. We achieve this through the application of the action of a group by conjugation. We have proved that the rank is and the subdegrees are [1][ ] and [ + 1][ ] .
Application of Marks to Computation of Ranks and Subdegrees of the Symmetric Group Acting on Ordered 4-Element and 5- Element Subsets
(UoEm, 2015) Kimani, Patrick; Rimberia, Jane; Muthoka, Geoffrey; Lao, Hussein; Kimani, Jacob
Ranks and subdegrees can be computed using combinatorial arguments, the Cauchy-Frobenius lemma and use of the concept of marks. However the concept of Marks has been given very little attention. In this paper we will apply the concept of marks to compute the ranks and subdegrees of the symmetric group Sn (n = 7,8,9) acting on ordered 4-element subsets and Sn (n = 8,9,10) acting on ordered 5-element subsets.
Application of Marks to Computation of Ranks and Subdegrees of the Symmetric Group Acting on Ordered Pairs and on Ordered Triples
(UoEm, 2014) Kimani, Patrick; Rimberia, Jane; Muthoka, Geoffrey; Lao, Hussein
Ranks and subdegrees can be computed using combinatorial arguments, the Cauchy-Frobenius lemma and use of the concept of marks. However the concept of Marks has been given very little attention. In this paper we will apply the concept of marks to compute the ranks and subdegrees of the symmetric group Sn (n = 5,6,7) and Sn (n = 6,7,8) acting on ordered pairs and triples respectively.
Ranks and Subdegrees of External Direct Product of Cn Dr Acting on X Y
(UoEm, 2020) Mutua, Felix Mutinda; Kamaku, Peter Waweru; Kimani, Patrick Mwangi
In this paper, transitivity, ranks and subdegrees of the action of ex- ternal direct product of Cyclic and Dihedral group on Cartesian Product of two sets are determined. The action is proved to be transitive. Also, it's established that the rank associated with the action is n( r+1 2 ) and subdegrees are [1][n] and [2][n( r􀀀1 2 )] when r is old. Additionally, the rank of the action for the case where r is even is proved to be n( r+2 2 ) and subdegrees are [1][2n] and [2][n( r􀀀2 2 )].
Rank and Subdegrees of PGL(2; q) Acting Cosets of PGL(2; e) for q an Even Power of e
(UoEm, 2019) Kimani, Patrick Mwangi; Kamuti, Ireri; Rimberia, Jane
The action of projective general group on the cosets of its maximal subgroups has been studied. For instance, [9] studied the action of G on the cosets of PGL(2; e) when q is an odd prime power of e. In this paper, we determine the rank and subdegrees of the action of PGL(2; q) on the cosets of its subgroup PGL(2; e) for odd q and an even power of e. We apply the table of marks to achieve this.
Ranks, Subdegrees and Suborbital graphs of the product action of Affine Groups
(UoEm, 2020) Agwanda, Siahi Maxwell; Kimani, Patrick; Kamuti, Ireri
The action of affine groups on Galois field has been studied. For instance, [3] studied the action of 𝐴𝑓𝑓 (𝑞 ) on Galois field 𝐺𝐹 (𝑞) for 𝑞 a power of prime 𝑝. In this paper, the rank and subdegree of the direct product of affine groups over Galois field acting on the cartesian product of Galois field is determined. The application of the definition of the product action is used to achieve this. The ranks and subdegrees are used in determination of suborbital graph, the non-trivial suborbital graphs that correspond to this action have been constructed using Sims procedure and were found to have a girth of 0, 3, 4 and
Application of Marks to Computation of Ranks and Subdegrees of the Symmetric Group Acting on Ordered 4-Element and 5- Element Subsets
(UoEm, 2015) Kimani, Patrick; Rimberia, Jane; Muthoka, Geoffrey; Lao, Hussein; Kimani, Jacob
Ranks and subdegrees can be computed using combinatorial arguments, the Cauchy-Frobenius lemma and use of the concept of marks. However the concept of Marks has been given very little attention. In this paper we will apply the concept of marks to compute the ranks and subdegrees of the symmetric group 𝑆𝑛(𝑛 = 7,8,9) acting on ordered 4-element subsets and 𝑆𝑛 (𝑛 = 8,9,10) acting on ordered 5-element subsets .

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