Theses and Dissertations

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A Multi-level Model for a Vector-Borne Organ to Tissue life Cycle Dynamics
(2025-09-05) Mahada, Awelani Sydney; Netshikweta, R.; Garira, W.
Introduction: Malaria is among the World’s most lethal infectious disease. It is caused by a parasitic pathogen transmitted by the Anopheles mosquito, which inoculates sporozoites into the human host during a blood meal. The population dynamics of malaria are well-known for their complexity, stemming not only from the parasite’s lifecycle, which involves two hosts (humans and mosquitoes)but also from the intricate replication and transmission cycles across different levels of the infectious disease system organization. Like other infectious disease systems, malaria infections inherit multilevel and multiscale systems, which pose significant challenges to efforts aimed at eliminating and ultimately eradicating the infection in a malaria-endemic population. Methodology Mathematical modeling in the study of complex system has proven to be an invaluable tool for understanding and predicting the behaviour and dynamics of a complex system within the domain of complexity science. Thus, in this study, we propose a multiscale modelling framework that captures the dynamics of malaria across three organizational levels within infectious disease systems implicated in the spread of malaria in a community. We begin by formulating a mathematical model to describe the development and progression of malaria parasites within the liver and tissue(blood) stages of an infected human host. This is followed by the formulation of a multiscale model that integrates both the inside(i.e.,the organ-tissue level)host and the outside (i.e., the host level) host malaria dynamics. Results Mathematical analysis for both the malaria models presented in this study was carried out and proved that all the models are mathematically and epidemiologically well-posed. We also compute the basic reproduction number R0 for both models and use the R0 to determine the local and global stability of the disease-free equilibriumas well as the local stability of endemic equilibrium of both models, respectively. We demonstrate that if R0 < 1, then the diseasefree equilibrium pointy of both models is locally and globally asymptotically stable, respevctively. However, if R0 > 1 the endemic equilibrium point of both models is locally asymptotically stable. The numerical results for both the models have demonstrated that the goal of intervention during malaria infection should be to reduce the rates at which merozoites and gametocytes invade healthy liver tissue as well as the blood cells. Hence it is recommended that interventions during malaria infection be directed on reducing the pace at which merozoites infect healthy blood cells and the density of merozoites in circulation. Conclusion The study presents a method that incoporates the complexity of malaria pathogens which is significant not only for malaria treatment but also for other vector-borne disease system control treatment strategies.
Open Access
Analysis of a boundary value problem for a system on non-homogeneous ordinary differential equations (ODE), with variable coefficients
(2015-01-16) Makhabane, Paul Suunyboy; Hlomuka, V. J.; Garira, W.
In this study we present a condition for the existence and uniqueness of the solution y(x) for a system of nonhomogeneous linear first order Ordinary Differential Equations (ODE). The existence and uniqueness of the solution of y(x) was confirmed through the Picard Lindelof Theorem. We then study the stability of matrix A(x) using its spectrum, moreover, A(x) is symmetric. This is a pre-condition for the application of Lefschetz direct stability method. We then modify the given Lefschetz system (Meyer, 1964) to suit the problem at hand. The direct method requires the construction of a suitable Lyapunov function; not easy for a time-independent (non-dynamic) problem. For a time-dependent problem the energy thereof becomes a suitable candidate for a Lyapunov function. For a non-dynamic problem it is harder to construct a Lyapunov function as there are no rules for that purpose. In our study we modified the Lefschetz system for the direct stability method and applied it to confirm the Lefschetz stability criterion using the modified systems of linear first order ODEs with variable coefficients. The Lefschetz method afforded us the construction of a credible Lyapunov function which enabled us to confirm the stability of the null solution to our problem. From our modified Lefschetz direct stability system, we solved the Makhabane / Hlomuka equation (5) for B(x) (7) which we later confirmed as both symmetric and positive definite.
Open Access
The Development and Application of Coupled Multiscale Models of Malaria Disease System
(2022-11-10) Maregere, Bothwell; Garira, W.; Mathebula, D.
The purpose of this thesis is to develop coupled multi-scale dynamics of infectious disease systems. An infectious disease system consists of three subsystems interacting, which are the host, the pathogen, and the environment. Each level has two different interaction scales (micro-scale and macro-scale) and is organized into hierarchical levels of an organization, from the cellular level to the macro-ecosystem level, and is arranged into hierarchical levels of an organization. There are two main theories of infectious diseases: (i) the transmission mechanism theory, (ii) the replication-transmission relativity theory. A significant difference exists between these theories in that (i) the transmission mechanism theory considers transmission to be the primary cause of infectious disease spread at the macro-scale, while (ii) replicationtransmission relativity theory is an extension of the first theory. It is important to consider the interaction between two scales when pathogen replication occurs within the host and transmission occurs between hosts (macro-scale). Our research primarily focuses on the replication-transmission relativity theory of pathogens. The main purpose of this study is to develop coupled multi-scale models of direct vectorborne diseases using malaria as a paradigm. We have developed a basic coupled multi-scale model with a combination of two other categories of multi-scale models, which are a nested multi-scale model in the human host and an embedded multi-scale model in the mosquito host. The developed multi-scale model consists of approaches of nonlinear differential equations that are employed to provide the mathematical results to the underlying issues of the multi-scale cycle of pathogen replication and transmission of malaria disease. Stability analyses of the models were evolved to substantiate that the infection-free equilibrium is locally and globally asymptotically stable whenever R0 < 1, and the endemic equilibrium exists and is globally asymptotically stable whenever R0 > 1. We applied the vaccination process as a governing measure on the multi-scale model of malaria with mosquito life cycle by comprising the three stages of vaccination, namely pre-erythrocyte stage vaccines, blood stage vaccines and transmission stage vaccines. The impact of vaccination on malaria disease has been proven. Through numerical simulation, it was found that when the comparative of vaccination efficacy is high, the community pathogen load (GH and PV ) decreases and the reproductive number can be reduced by 89.09%, that is, the transmission of malaria can be reduced on the dynamics of individual level and population-level.We also evolved the multi-scale model with the human immune response on a within-human sub-model which is stimulated by the malaria parasite. We investigated the effect of immune cells on reducing malaria infection at both the betweenhost scale and within-host scale. We incorporate the environmental factor, such as temperature in the multi-scale model of the malaria disease system with a mosquito life cycle. We discovered that as the temperature enhances the mosquito population also increases which has the impact of increasing malaria infection at the individual level and at the community-scale. We also investigated the influence of the mosquito life cycle on the multi-scale model of the malaria disease system. The increase in eggs, larval and pupal stages of mosquitoes result in the increase of mosquito density and malaria transmission at the individual level and community-scale. Therefore, the suggestion is that immature and mature mosquitoes be controlled to lessen malaria transmission. The results indicated that the combination of malaria health interventions with the highest efficacy has the influence of reducing malaria infection at the populationlevel. Models developed and analyzed in this study can play a significant role in preventing malaria outbreaks. Using the coupled multi-scale models that were developed in this study, we made conclusions about the malaria disease system based on the results obtained. It is possible to apply the multi-scale framework in this study to other vector-borne diseases as well.
Open Access
Exploring the Multi-scale character of infectious disease dynamics
(2023-05-19) Mufoya, Blessings; Garira, W.; Mathebula, D.
This research study characterised multiscale models of infectious disease dynamics. This was achieved by establishing when it is appropriate to implement particular mathematical methods for different multiscale models. The study of infectious disease systems has been elucidated ever since the discovery of mathematical modelling. Due to the vast complexities in the dynamics of infectious disease systems, modellers are increasingly gravitating towards multiscale modelling approach as a favourable alternative. Among the diseases that have persistently plagued most developing countries are vector-borne diseases like Malaria and directly transmitted diseases like Foot-and-Mouth disease (FMD). Globally, FMD has caused major losses in the economic sector (particularly agriculture) as well as tourism. On the other hand, Malaria remains amongst the most severe public health problems worldwide with millions of people estimated to live in permanent risk of contracting the disease. We developed multiscale models that can describe both local transmission and global transmission of infectious disease systems at any hierarchical level of organization using FMD and Malaria disease as paradigms. The first stage in formulating the multiscale models in this study was to integrate two submodels namely: (i) the between-host submodel and (ii) within-host submodel of an infectious disease system using the nested approach. The outcome was a system of nonlinear ordinary differential equations which described the local transmission mechanism of the infectious disease system. The next step was to incorporate graph theoretic methods to the system of differential equations. This approach enabled modelling the migration of humans/animals between communities (also called patches or geographical distant locations) thereby describing the global transmission mechanism of infectious disease systems. At whole organism-level we considered the organs in a host as patches and the transmission within-organ scale as direct transmission represented by ordinary differential equations. However, at between-organ scale there was movement of pathogen between the organs through the blood. This transmission mechanism called global transmission was represented by graph-theoretic methods. At macrocommunity-level we considered communities as patches and established that at withincommunity scale there was direct transmission of pathogen represented by ordinary differental equations and at between-community scale there was movement of infected individuals. Furthermore, the systems of differential equations were extended to stochastic differential equations in order to incorporate randomness in the infectious disease dynamics. By adopting a cocktail of computational and analytical tools we sufficiently analyzed the impact of the transmission mechanisms in the different multiscale models. We established that once we used a graph-theoretic method at host level it would be difficult to extend this to community level. However, when we used different methods then it was easy to extend to community level. This was the main aspect of the characterization of multiscale models that we investigated in this thesis which has not been done before. We also established distinctions between local transmission and global transmission mechanisms which enable us to implement intervention strategies targeted torwards both local transmission such as vaccination and global transmission such as travel restrictions. In spite of the fact that the results collected in this study are restricted to FMD and Malaria, the multiscale modelling frameworks established are suitable for other directly transmitted diseases and vector-borne diseases.
Open Access
Hybrid multi-scale mathematical modelling of malaria infection transmission
(2017-09-18) Vele, Khathutshelo; Garira, W.; Moyo, S.
See the attached abstract below
Open Access
A mathematical modelling frame-work for immuno-epidemiology of Guinea worm infection
(2016-02-12) Netshikweta, Rendani; Garira, W.; Moyo, S.
Open Access
Mathematical modelling of Cholera Immunology
(2016-05) Maphiri, Azwindini Delinah; Garira, W.; Musie, E.
See the attached abstract below
Open Access
Mathematical modelling of fungal contamination of citrus produce along the pre-harvest supply chain
(2016-05) Muleya, Nqobile; Garira, W.; Mchau, G. R. A.
See the attached abstract below
Open Access
Multi-scale modelling of soil-transmitted Helminths infections in humans
(2019-05-18) Makhuvha, Mulalo; Mathebula, D.; Garira, W.
In this study, we develop a multiscale model of soil transmitted helminths in humans with a special reference to hookworm infection. Firstly, we develop a single scale model that comprises of five between host scale populations namely; susceptible humans, infected humans, eggs in the physical environment, noninfective worms in the physical environment and infective worms in the physical environment. Secondly, we extend the single scale model to incorporate within-host scales namely; infective larvae within-host, immature worms in small intestine, mature worm population and within-host egg population which resulted to a multiscale model. The models are analysed both numerically and analytically. The models are epidemiologically and mathematically well posed. Numerical simulation results show that there is a bidirectional relationship between the between-host and within-host scales. This is in agreement with the sensitivity analysis results, we noted that the same parameters that reduce reproductive number R0 are the same parameters that reduce the infective worms endemic equilibrium point. From the comparative effectiveness of hookworm interventions analysis results, we notice that any intervention combination that include wearing shoes controls and reduces the spread of the infection. The modelling framework developed in this study is vigorous to be applicable to other soil transmitted helminths infections
Open Access
Multi-Scale Modelling of Vector-Borne Diseases
(2018-09-21) Mathebula, Dephney; Garira, W.; Moyo, S.
In this study, we developed multiscale models of vector-borne diseases. In general, the transmission of vector-borne diseases can be considered as falling into two categories, i.e. direct transmission and environmental transmission. Two representative vector-borne diseases, namely; malaria which represents all directly transmitted vector-borne diseases and schistosomiasis which represents all environmentally transmitted vector-borne diseases were studied. Based on existing mathematical modelling science base, we established a new multiscale modelling framework that can be used to evaluate the effectiveness of vector-borne diseases treatment and preventive interventions. The multiscale models consisted of systems of nonlinear ordinary differential equations which were studied for the provision of solutions to the underlying problem of the disease transmission dynamics. Relying on the fact that there is still serious lack of knowledge pertaining to mathematical techniques for the representation and construction of multiscale models of vector-bone diseases, we have developed some grand ideas to placate this gap. The central idea in multiscale modelling is to divide a modelling problem such as a vector-bone disease system into a family of sub-models that exist at different scales and then attempt to study the problem at these scales while simultaneously linking the sub-models across these scales. For malaria, we formulated the multiscale models by integrating four submodels which are: (i) a sub-model for the mosquito-to-human transmission of malaria parasite, (ii) a sub-model for the human-to-mosquito transmission of malaria parasite, (iii) a within-mosquito malaria parasite population dynamics sub-model and (iv) a within-human malaria parasite population dynamics sub-model. For schistosomiasis, we integrated the two subsystems (within-host and between-host sub-models) by identifying the within-host and between-host variables and parameters associated with the environmental dynamics of the pathogen and then designed a feedback of the variables and parameters across the within-host and between-host sub-models. Using a combination of analytical and computational tools we adequately accounted for the influence of the sub-models in the different multiscale models. The multiscale models were then used to evaluate the effectiveness of the control and prevention interventions that operate at different scales of a vector-bone disease system. Although the results obtained in this study are specific to malaria and schistosomiasis, the multiscale modelling frameworks developed are robust enough to be applicable to other vector-borne diseases.
Open Access
Multiscale Modelling of Environmentally Transmitted Infectious Diseases
(2021-11-19) Netshikweta, Rendani; Garira, W.; Mathebula
In the field of mathematical biology, researchers are beginning to witness an overwhelming appreciation of multiscale modelling as an essential and suitable technique as opposed to a traditional single-scale modelling approach in predicting the dynamics of infectious disease systems. Yet, there is still a lack of evidence that generally indicates which among the different categories of multiscale models of infectious disease systems is more appropriate to use in multiscale modelling of infectious disease systems at different levels of their organization. This research study is the first of its kind to compare the suitability of the two fundamental categories of multiscale models of infectious disease systems which are nested multiscale models and embedded multiscale models in predicting disease dynamics with specific reference to environmentallytransmitted diseases. Two environmentally transmitted diseases are used as case studies, namely ruminant paratuberculosis and human ascariasis, to compare the two fundamental categories of multiscale models in predicting disease dynamics. The two environmentally-transmitted diseases considered in this study represent infectious disease systems with replication-cycle at microscale (i.e. ruminant paratuberculosis) and infectious disease systems without replication cycle at the microscale (i.e. human ascariasis). Firstly, the author develop a single-scale model at the host-level that we progressively extend to different categories of multiscale models that we later compare. The findings of this study (through both mathematical and numerical analysis of the multiscale models) are that for ruminant paratuberculosis which has a pathogen replication-cycle at the within-host scale both nested and embedded multiscale models can be used because both the models provide the same prediction of disease dynamics. However, for human ascariasis the findings are such that nested multiscale model is not appropriate in characterizing the disease dynamics, only the embedded is appropriate. Although the comparison of different categories of multiscale models in disease prediction carried out in this study are specific to paratuberculosis in ruminants and human ascariasis, the results obtained in this study are robust enough to be applicable to other infectious disease systems. Our results can be generalized to imply that for any level of organization of an infectious disease systems, if the disease has a replication cycle at the microscale, the nested multiscale and the embedded multiscle model provide the same accuracy in predicting disease dynamics. However, when the disease has no replication cycle at the microscale, only the embedded multiscle model is appropriate for predicting disease dynamics. In such a case, a nested multiscale model is inappropriate. We anticipate that this study will enable modelers to choose appropriate multiscale model category in the study of infectious diseases.
Open Access
Multiscale Modelling of Foodborne Diseases
(2024-09-06) Maphiri, Azwindini Delinah; Muzhinyi, K.; Garira, W.; Mathebula, D.
Infectious disease systems are essentially multiscale complex system wherein pathogens multiply within hosts, spread across people, and infect entire populations of hosts. The description of most biological processes involves multiple, interconnected phenomena occurring on different spatial and temporal scales in the human body. Traditional approaches for modelling infectious disease systems rely on the principles and concepts of the transmission mechanism theory that considers transmission to be the primary cause of infectious disease spread at the macroscale. Modellers of infectious diseases are increasingly using multiscale modelling approach in response to this challenge. Multiscale models of infectious disease systems encompass intricate structures that revolve around the interplay of three distinct sub-systems: the host, the pathogen, and the environmental subsystems. The replication-transmission relativity theory is a novel theory designed for the purpose of multiscale modeling of infectious disease systems, accounting for variations in time and space by incorporating pathogen replication that leads to transmission. Replicationtransmission relativity theory consists of seven distinct levels of organization within an infectious disease system, each level including the within-host scale (microscale) and between-host scale (macroscale). Five separate classifications of multiscale models can be formulated that integrate the microscale and macroscale. A research gap has been created in an attempt to establish a multiscale framework in order to understand the mechanisms on how foodborne pathogens cause infections on human beings and animals, as very little has been done in modelling of foodborne disease. The primary goal of this study is to create multiscale models for foodborne diseases to examine whether a mutual influence exists between the microscale and macroscale, guided by the principles of replication-relativity theory. The multiscale models are developed by considering three environmental transmitted diseases at host level caused by pathogens: norovirus, E. coli O157:H7 and taenia solium. We start by developing a single-scale model of foodborne diseases caused by viruses in general, which is then extended to create a multiscale model for norovirus. We formulate a non-standard finite difference scheme for the single-scale model, norovirus, and E. coli O157:H7. For taenia solium, we use ODE solvers in Python, specifically, ODE int function in the sci.integrate. The numerical findings from the study confirm the applicability of the replication-transmission relativity theory in cases where the reciprocal impact between the within-host scale and the between-host scale involves both infection/super-infection (for the effect of the between-host scale on the within-host scale) and pathogen excretion/shedding (for the effect of the within-host scale on the between-host scale). We expect that our study will help modellers integrate microscale and macroscale dynamics across various levels of organization within infectious disease systems.
Open Access
Multiscale Modelling of HIV/AIDS Transmission Dynamics
(2018-09-21) Mafunda, Martin Canaan; Garira, W.; Moyo, S.
Infectious diseases remain a major public health concern. Well-known for causing sickness and death, enormous pain and suffering, increased time spent on patient care and huge economic losses due to lost production. Infectious diseases continue to be a scourge without equal. In this work, we address the following research question: Can we use a multiscale model of HIV/AIDS transmission dynamics to assess the comparative effectiveness of health interventions that are implemented at different scale domains? To achieve the set objectives of the study, we use multiscale modelling approach, a new and emerging computational high-throughput technique for mathematically studying problems that have many characteristics across several scales. To be more specific, we perform three tasks in addressing the research question. First, we develop a within-host submodel and use it to show it’s associated limitations which only a multiscale model can resolve. Second, we develop a between-host submodel and use it to motivate the need for multiscale modelling of the HIV/AIDS disease system. Finally, we link the two submodels to produce a nested HIV/AIDS multiscale model that affords us the opportunity to compare effectiveness of five preventive and treatment HIV/AIDS health interventions. Analysis of the multiscale model shows that it is possible to jointly study two key aspects (immunology and epidemiology) of infectious diseases. The multiscale model provides the means for making meaningful comparative effectiveness on available preventive and treatment health interventions. Consequently, we employ the multiscale model to show that impact of HIV/AIDS packages increases as more interventions are integrated into the packages. Specifically, the study shows that combined HAART and male circumcision is more effective than an intervention involving HAART alone. Overall, our study successfully illustrates the utility of multiscale modelling methodology as a tool for assessing the comparative effectiveness of HIV/AIDS preventive and treatment interventions. For purposes of informing public health policy, we use the study results to infer that condom use, male circumcision and pre-exposure prophylaxis are more effective in controlling the transmission dynamics of HIV/AIDS at the start of the epidemic as compared to when the disease is endemic in the community while the converse is also true for HAART.
Open Access
Share Price Prediction for Increasing Market Efficiency using Random Forest
(2022-11-10) Mbedzi, Tshinanne Angel; Chagwiza, W.; Garira, W.
The price of a single share of a collection of sell-able shares, options, or other financial assets, shall be the price of a share price. The share price is unpredictable since it primarily depends on buyers’ and sellers’ expectations. Share is a primary and secondary market equity security. In this study we will use machine learning techniques to predict the share price for increasing market efficiency. In addition, it is important for us to build a models to create appropriate features to improve the performance of the models. The random forest and the recurrent neural network will be used to achieve this. To fix class imbalance, we analyse preprocessing of the data set, like the selection of the features using filter and wrapper methods and selected oversampling techniques. The model’s performance will be evaluated using Mean absolute error (MAE), Mean square error (MSE), Root mean square error (RMSE), Relative MAE (rMAE), and Relative RMSE (rRMSE). The performance of the RNN and Rf algorithms was compared for the prediction of the closing price. The Rf model was found to be the best model for predicting the stock price (closing price). This research project together with its findings will have an impact in increasing market efficiency. This will also promote potential economic growth.
Open Access
A stochastic programming framework for financial intermediaries liquidity in South Africa
(2015-05) Chagwiza, Wilbert; Garira, W.; Moyo, S.
We provide a thorough overview of the current liquidity problem, dis cussing what are considered to be state-of-the-art approaches in both in dustry and in academia, and clearly establish our motivation to depart from current liquidity management standards. We noted that during 2007 to 2009 financial crisis different countries throughout the whole world faced serious challenges that ranged from chronic liquidity problems, deep-rooted risk management deficiencies and poor corporate governance practices. The liq uidity crisis, especially in the USA, was caused by low real interest rates stimulating an asset price bubble fuelled by new financial products that were not stress-tested and that failed in the downturn. Large depository institu tions face the question of how to optimally utilise funds and manage liquid ity problems. The existing liquidity models are not standardised, and do not take into consideration the complexity and the nature of the bank. As evi denced by the late 2000 financial crisis, it is evident that nowadays there is not a metric that seems to be completely adequate to prevent liquidity crises. Our contribution is fourfold. First we investigate appropriate scenario gener ation methods and perform a rigorous investigation on how to generate mul tistage scenario trees. Further, we investigate the inclusion of all qualifying liquid asset instruments into the portfolio optimisation. We propose a novel multistage stochastic programming methodology for liquid asset control. Thus we define how to construct and solve stochastic programming models for liquidity needs-driven sub-portfolios. Our approach is based on scenario trees and makes no assumption on the distributions of random variables. Fi nally, we investigate the inclusion of liquidity needs-driven strategies which are core liquidity, cash cushion, operational cash and discretionary liquidity into the overall liquidity portfolio. Stochastic programming is a multi-faceted problem, and even the most fo cused treatment necessarily incorporates techniques from a wide range of disciplines. Through the development of fully coherent models and a suffi ciently robust solution methodology, we provide a thorough overview of the problem at hand, discussing approaches in both industry and in academia. We introduce practical and theoretical advances that are to our best knowl edge unexplored in the current literature, and document the usefulness of these avenues through a systematic series of increasingly complex applica tions and experiments. The data used is from the South African Reserve Bank (SARB) and International Monetary Fund (IMF) from January 1988 to May 2014. Most corporations such as commercial banks generally hold a unitary liquidity portfolio and not necessarily segmented to focus on bank specific cash and liquidity needs. Banks do have different cash needs at different times. Financial intermediaries to manage liquidity efficiently and effectively, segmenting the optimal liquid asset portfolio is the best manage ment strategy. We construct different stochastic programming models based on decision making under risk and provision of powerful paradigm for de cision making under uncertainty. The stochastic programming models with recourse are clear, easy to implement, very efficient and provide the optimal solution according to future possible set of scenarios. In defining and con structing stochastic programming models with recourse, sensitivity analysis should be carried out to increase the decision maker's understanding of the problem and to show the effect of different assumptions. Constructing an SP model with more stages provides better approximations compared to a single-period model. We found that the use of the proposed models through segmenting the unitary liquidity portfolio improves the management of liq uidity compared to current techniques which are based on simulation, ex perience and trial-and-error. Thus high quality liquidity buffers insure the bank against any adverse liquidity risk. We found that liquidity buffer is best modelled as a stochastic process than deterministic. High net cash flow enhance effective and efficient liquidity management. Finally, excess re serves improve country monetary policy and broaden the scope of central bank lending programmes to address conditions in credit markets but on the other hand, may lead banks to loosen standards by weakening lending crite ria in an attempt to increase returns. In designing the sub-portfolios, liquidity determinants such as liquid asset returns, loan returns, payment flow, interest rates and gross income are un certain, and to avoid liquidity problems caused by these variables, we need to include randomness on them. To effectively and efficiently manage liq uidity, a bank may be required to follow certain key liquidity principles. The first is that the bank needs to understand and categorise its cash needs into at least four liquidity sub-portfolios as explained in the research. In this study, we strongly recommend that the banks' management should design the following sub-portfolios; liquidity buffer, cash cushion, operational cash and discretionary liquidity. The bank should clearly maintain the holdings of high quality liquid assets that can provide reliable reserves under all con ditions. In addition, strict and relatively comprehensive disclosure practices in relation to liquidity risk management objectives should be submitted to central banks. Finally, there should be improvement on funding markets and public confidence by broadening the scope of bank guarantees to ensure future financial stability. We need to emphasise the point that bank liquid ity is restricted by capital adequacy, required reserves, liability and deposit insurance defined by the central bank. Research can therefore be done on models that can be used to efficiently and accurately forecast future cash outflow taking into consideration the behavioural cash flows and estimating the discretionary and excess reserves. Further research can also be done on comparing the benefits of the calculated strategies in the stochastic programming context to more traditional methods.

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