Farai Nyabadza | Stellenbosch University (original) (raw)

Papers by Farai Nyabadza

Journal of Theoretical Biology, 2010

A deterministic model for assessing the dynamics of mixed species malaria infections in a human p... more A deterministic model for assessing the dynamics of mixed species malaria infections in a human population is presented to investigate the effects of dual infection with Plasmodium malariae and Plasmodium falciparum. Qualitative analysis of the model including positivity and boundedness is performed. In addition to the disease free equilibrium, we show that there exists a boundary equilibrium corresponding to each species. The isolation reproductive number of each species is computed as well as the reproductive number of the full model. Conditions for global stability of the disease free equilibrium as well as local stability of the boundary equilibria are derived. The model has an interior equilibrium which exists if at least one of the isolation reproductive numbers is greater than unity. Among the interesting dynamical behaviours of the model, the phenomenon of backward bifurcation where a stable boundary equilibrium coexists with a stable interior equilibrium, for a certain range of the associated invasion reproductive number less than unity is observed. Results from analysis of the model show that, when cross-immunity between the two species is weak, there is a high probability of coexistence of the two species and when cross-immunity is strong, competitive exclusion is high. Further, an increase in the reproductive number of species i increases the stability of its boundary equilibrium and its ability to invade an equilibrium of species j. Numerical simulations support our analytical conclusions and illustrate possible behaviour scenarios of the model.

Mathematical and Computer Modelling, 2011

Physica A: Statistical …, Jan 1, 2011

A drug use epidemic can be represented by a finite number of states and transition rules that gov... more A drug use epidemic can be represented by a finite number of states and transition rules that govern the dynamics of drug use in each discrete time step. This paper investigates the spread of drug use in a community where some users are in treatment and others are not in treatment, citing South Africa as an example. In our analysis, we consider the neighbourhood prevalence of each individual, ie, the proportion of the individual's drug user contacts who are not in treatment amongst all of his or her contacts. We introduce ...

Contemporary Mathematics, 2006

Physica A: Statistical Mechanics and its Applications, 2009

Epidemics such as tuberculosis (TB), can be represented by a finite number of states and transiti... more Epidemics such as tuberculosis (TB), can be represented by a finite number of states and transition rules that govern the spread of the disease in each discrete time step. This paper uses a graph theoretic approach to investigate TB interactions in a community where infectives are categorized. A threshold value, α= 1− 1n, for 'reasonable'infectives is proposed. The results show that an epidemic will not ensue as long as the threshold is surpassed. Simulations presented show that unreasonable infectives can amplify the ...

Journal of Interdisciplinary Mathematics, 2008

ABSTRACT Churches have played an important role in the development and shaping of societies in th... more ABSTRACT Churches have played an important role in the development and shaping of societies in the world. An extension of a previously proposed theory for church growth is presented and analyzed using a mathematical modelling approach. The extension presented incorporates exposure before conversion and internal revival that results in the restoration of enthusiasm. The results are presented in terms of R 0 , the church reproduction potential. The incorporation of a class of the exposed and internal revival allowed the existence of a subcritical (backward) bifurcation at the critical value R 0 = 1 and hence the existence of multiple equilibria for R 0 < 1. The results suggest that having R 0 < 1 does not necessarily lead to a decline in church growth leading to extinction. Numerical results show that when internal revival is taken into account, R 0 does not accurately describe the invasion strength of the church.

Computational and Mathematical Methods in Medicine, 2014

Mathematical Modelling and Analysis, 2008

Mathematical Biosciences and Engineering, 2013

Substance abuse is a global menace with immeasurable consequences to the health of users, the qua... more Substance abuse is a global menace with immeasurable consequences to the health of users, the quality of life and the economy of countries affected. Although the prominently known routes of initiation into drug use are; by contact between potential users and individuals already using the drugs and self initiation, the role played by a special class of individuals referred to as drug lords can not be ignored. We consider a simple but useful compartmental model of drug use that accounts for the contribution of contagion and drug lords to initiation into drug use and drug epidemics. We show that the model has a drug free equilibrium when the threshold parameter R0 is less that unity and a drug persistent equilibrium when R0 is greater than one. In our effort to ascertain the effect of policing in the control of drug epidemics, we include a term accounting for law enforcement. Our results indicate that increased law enforcement greatly reduces the prevalence of substance abuse. In addition, initiation resulting from presence of drugs in circulation can be as high as seven times higher that initiation due to contagion alone.

South African Journal of Science, 2012

ABSTRACT The production and use of addictive stimulants has been a major problem in South Africa.... more ABSTRACT The production and use of addictive stimulants has been a major problem in South Africa. Although research has shown increased demand for drug abuse treatment, the actual size of the drug-abusing population remains unknown. Thus the prevalence of drug abuse requires estimation through available tools. Many questions remain unanswered with regard to interventions, new cases of substance abuse and relapse in recovering persons. A six-state compartmental model including a core and non-core group, with fast and slow progression to addiction, was formulated with the aim of qualitatively investigating the dynamics of substance abuse and predicting drug abuse trends. The analysis of the model was presented in terms of the substance abuse epidemic threshold R0. Numerical simulations were performed to fit the model to available data for methamphetamine use in the Western Cape and to determine the role played by some key parameters. The model was also fitted to data on methamphetamine users who enter rehabilitation using the least squares curve fitting method. It was shown that the model exhibits a backward bifurcation where a stable drug-free equilibrium coexists with a stable drug-persistent equilibrium for a certain defined range of values of R0. The stabilities of the model equilibria were ascertained and persistence conditions established. It was found that it is not sufficient to reduce R0 below unit to control the substance abuse epidemic. The reproduction number should be brought below a determined threshold, R0c. The results also suggested that the substance abuse epidemic can be reduced by intervention programmes targeted at light drug users and by increasing the uptake rate into treatment for those addicted. Projected trends showed a steady decline in the prevalence of methamphetamine abuse until 2015.

South African Journal of Science, 2013

Differential Equations and Dynamical Systems, 2011

Mathematical Medicine and Biology, 2015

The role of antibodies in HIV-1 infection is investigated using a discrete-time mathematical mode... more The role of antibodies in HIV-1 infection is investigated using a discrete-time mathematical model that considers cell-free and cell-associated transmission of the virus. Model analysis shows that the effect of each type of antibody is dependent on the stage of the infection. Neutralizing antibodies are efficient in controlling the viral levels in the early days after seroconversion and antibodies that coat HIV-1-infected cells and recruit effector cells to either kill the HIV-1-infected cells or inhibit viral replication are efficient when the infection becomes established. Model simulations show that antibodies that inhibit viral replication are more effective in controlling the infection than those that recruit Natural Killer T cells after infection establishment. The model was fitted to subjects of the Tsedimoso study conducted in Botswana and conclusions similar to elasticity analysis results were obtained. Model fitting results predicted that neutralizing antibodies are more efficient in controlling the viral levels than antibodies that coat HIV-1-infected cells and recruit effector cells to either kill the HIV-1-infected cells or inhibit viral replication in the early days after seroconversion.

Applied Mathematics and Computation, 2014

ABSTRACT A metapopulation model is developed to describe the spread of cholera between two commun... more ABSTRACT A metapopulation model is developed to describe the spread of cholera between two communities connected by migratory movement. Disease threshold ratios specific to the communities are given, considering a case when the communities are isolated and when the communities are connected. The connection of the threshold ratios to disease spread and stability is discussed. The disease free equilibrium is globally stable whenever, the corresponding community specific disease threshold ratios are less than one and unstable otherwise. Community specific endemic equilibrium points are unique, locally asymptotically stable, and only exist when the corresponding disease thresholds are greater than unit. Disease spread is explosive in nature at the beginning of the outbreak but more severe in a community with poor facilities relative to a community with more improved facilities. In isolated communities, in the case of endemic cholera, the infection is characterised by a big outbreak, followed by a small episode of the infection. Only one typically big outbreak is observed in the community with improved facilities with no recurrence of the epidemic. In connected communities, movement of individuals across communities not only influences persistence of the infection but also results in more pronounced outbreak in a relatively well facilitated community in the long term. Synchronous fluctuation of the population is observed when there is unrestricted movement of both immunologically naive and infected individuals across the communities. Our results suggest that during times of cholera, movement to and fro cholera endemic areas should be avoided if the outbreak is to be contained. Otherwise, with continued migration, the infection may potentially worsen even in communities with relatively good facilities.

Journal of Theoretical Biology, 2011

ISRN Biomathematics, 2014

Journal of Biological Systems, 2010

Journal of Biological Systems, 2013

ABSTRACT Substance abuse remains a global menace in spite of recurrent warnings, seizures, social... more ABSTRACT Substance abuse remains a global menace in spite of recurrent warnings, seizures, social and pharmacological effects associated with addiction to drugs. In this paper, we use a mathematical model which is a combination of the classical SIS and SIR models to investigate the dynamics of substance abuse. Initiation into drug use is based on contact of those at risk (the susceptible population) with drug users at different levels of drug use. We evaluate the threshold number and use it to analyze the model. We show that when this threshold number is less than unity, the drug-free steady state is globally asymptotically stable and when this threshold number is greater than unity the drug-persistent steady state is also globally stable. The impact of amelioration, rehabilitation and re-initiation on drug epidemics is investigated. Amelioration in presence of quitting for light users is observed to reduce the prevalence of substance abuse and this is supported by numerical simulations. The results show that both prevention and treatment/rehabilitation are necessary strategies for reduction of drug epidemics. Our recommendation is that preventive strategies should be directed toward reducing the contact rate and treatment should be combined with psychotherapy to accelerate quitting and reduce re-initiation.

Bulletin of Mathematical Biology, 2013

Substance abuse remains a global problem, with immense health and social consequences. Crystal me... more Substance abuse remains a global problem, with immense health and social consequences. Crystal meth, known as 'tik' in South Africa, is a growing problem, and its supply chains have equally grown due to increased numbers of 'tik' users, especially…

Journal of Biological Systems, 2006

... Title: A MATHEMATICAL MODEL FOR COMBATING HIV/AIDS IN SOUTHERN AFRICA: WILL MULTIPLE STRATEGI... more ... Title: A MATHEMATICAL MODEL FOR COMBATING HIV/AIDS IN SOUTHERN AFRICA: WILL MULTIPLE STRATEGIES WORK? Author(s): FARAI NYABADZA Department of Mathematics, University of Botswana, P. Bag UB 00704, Gaborone, Botswana. ...

Journal of Theoretical Biology, 2010

A deterministic model for assessing the dynamics of mixed species malaria infections in a human p... more A deterministic model for assessing the dynamics of mixed species malaria infections in a human population is presented to investigate the effects of dual infection with Plasmodium malariae and Plasmodium falciparum. Qualitative analysis of the model including positivity and boundedness is performed. In addition to the disease free equilibrium, we show that there exists a boundary equilibrium corresponding to each species. The isolation reproductive number of each species is computed as well as the reproductive number of the full model. Conditions for global stability of the disease free equilibrium as well as local stability of the boundary equilibria are derived. The model has an interior equilibrium which exists if at least one of the isolation reproductive numbers is greater than unity. Among the interesting dynamical behaviours of the model, the phenomenon of backward bifurcation where a stable boundary equilibrium coexists with a stable interior equilibrium, for a certain range of the associated invasion reproductive number less than unity is observed. Results from analysis of the model show that, when cross-immunity between the two species is weak, there is a high probability of coexistence of the two species and when cross-immunity is strong, competitive exclusion is high. Further, an increase in the reproductive number of species i increases the stability of its boundary equilibrium and its ability to invade an equilibrium of species j. Numerical simulations support our analytical conclusions and illustrate possible behaviour scenarios of the model.

Mathematical and Computer Modelling, 2011

Physica A: Statistical …, Jan 1, 2011

A drug use epidemic can be represented by a finite number of states and transition rules that gov... more A drug use epidemic can be represented by a finite number of states and transition rules that govern the dynamics of drug use in each discrete time step. This paper investigates the spread of drug use in a community where some users are in treatment and others are not in treatment, citing South Africa as an example. In our analysis, we consider the neighbourhood prevalence of each individual, ie, the proportion of the individual's drug user contacts who are not in treatment amongst all of his or her contacts. We introduce ...

Contemporary Mathematics, 2006

Physica A: Statistical Mechanics and its Applications, 2009

Epidemics such as tuberculosis (TB), can be represented by a finite number of states and transiti... more Epidemics such as tuberculosis (TB), can be represented by a finite number of states and transition rules that govern the spread of the disease in each discrete time step. This paper uses a graph theoretic approach to investigate TB interactions in a community where infectives are categorized. A threshold value, α= 1− 1n, for 'reasonable'infectives is proposed. The results show that an epidemic will not ensue as long as the threshold is surpassed. Simulations presented show that unreasonable infectives can amplify the ...

Journal of Interdisciplinary Mathematics, 2008

ABSTRACT Churches have played an important role in the development and shaping of societies in th... more ABSTRACT Churches have played an important role in the development and shaping of societies in the world. An extension of a previously proposed theory for church growth is presented and analyzed using a mathematical modelling approach. The extension presented incorporates exposure before conversion and internal revival that results in the restoration of enthusiasm. The results are presented in terms of R 0 , the church reproduction potential. The incorporation of a class of the exposed and internal revival allowed the existence of a subcritical (backward) bifurcation at the critical value R 0 = 1 and hence the existence of multiple equilibria for R 0 < 1. The results suggest that having R 0 < 1 does not necessarily lead to a decline in church growth leading to extinction. Numerical results show that when internal revival is taken into account, R 0 does not accurately describe the invasion strength of the church.

Computational and Mathematical Methods in Medicine, 2014

Mathematical Modelling and Analysis, 2008

Mathematical Biosciences and Engineering, 2013

Substance abuse is a global menace with immeasurable consequences to the health of users, the qua... more Substance abuse is a global menace with immeasurable consequences to the health of users, the quality of life and the economy of countries affected. Although the prominently known routes of initiation into drug use are; by contact between potential users and individuals already using the drugs and self initiation, the role played by a special class of individuals referred to as drug lords can not be ignored. We consider a simple but useful compartmental model of drug use that accounts for the contribution of contagion and drug lords to initiation into drug use and drug epidemics. We show that the model has a drug free equilibrium when the threshold parameter R0 is less that unity and a drug persistent equilibrium when R0 is greater than one. In our effort to ascertain the effect of policing in the control of drug epidemics, we include a term accounting for law enforcement. Our results indicate that increased law enforcement greatly reduces the prevalence of substance abuse. In addition, initiation resulting from presence of drugs in circulation can be as high as seven times higher that initiation due to contagion alone.

South African Journal of Science, 2012

ABSTRACT The production and use of addictive stimulants has been a major problem in South Africa.... more ABSTRACT The production and use of addictive stimulants has been a major problem in South Africa. Although research has shown increased demand for drug abuse treatment, the actual size of the drug-abusing population remains unknown. Thus the prevalence of drug abuse requires estimation through available tools. Many questions remain unanswered with regard to interventions, new cases of substance abuse and relapse in recovering persons. A six-state compartmental model including a core and non-core group, with fast and slow progression to addiction, was formulated with the aim of qualitatively investigating the dynamics of substance abuse and predicting drug abuse trends. The analysis of the model was presented in terms of the substance abuse epidemic threshold R0. Numerical simulations were performed to fit the model to available data for methamphetamine use in the Western Cape and to determine the role played by some key parameters. The model was also fitted to data on methamphetamine users who enter rehabilitation using the least squares curve fitting method. It was shown that the model exhibits a backward bifurcation where a stable drug-free equilibrium coexists with a stable drug-persistent equilibrium for a certain defined range of values of R0. The stabilities of the model equilibria were ascertained and persistence conditions established. It was found that it is not sufficient to reduce R0 below unit to control the substance abuse epidemic. The reproduction number should be brought below a determined threshold, R0c. The results also suggested that the substance abuse epidemic can be reduced by intervention programmes targeted at light drug users and by increasing the uptake rate into treatment for those addicted. Projected trends showed a steady decline in the prevalence of methamphetamine abuse until 2015.

South African Journal of Science, 2013

Differential Equations and Dynamical Systems, 2011

Mathematical Medicine and Biology, 2015

The role of antibodies in HIV-1 infection is investigated using a discrete-time mathematical mode... more The role of antibodies in HIV-1 infection is investigated using a discrete-time mathematical model that considers cell-free and cell-associated transmission of the virus. Model analysis shows that the effect of each type of antibody is dependent on the stage of the infection. Neutralizing antibodies are efficient in controlling the viral levels in the early days after seroconversion and antibodies that coat HIV-1-infected cells and recruit effector cells to either kill the HIV-1-infected cells or inhibit viral replication are efficient when the infection becomes established. Model simulations show that antibodies that inhibit viral replication are more effective in controlling the infection than those that recruit Natural Killer T cells after infection establishment. The model was fitted to subjects of the Tsedimoso study conducted in Botswana and conclusions similar to elasticity analysis results were obtained. Model fitting results predicted that neutralizing antibodies are more efficient in controlling the viral levels than antibodies that coat HIV-1-infected cells and recruit effector cells to either kill the HIV-1-infected cells or inhibit viral replication in the early days after seroconversion.

Applied Mathematics and Computation, 2014

ABSTRACT A metapopulation model is developed to describe the spread of cholera between two commun... more ABSTRACT A metapopulation model is developed to describe the spread of cholera between two communities connected by migratory movement. Disease threshold ratios specific to the communities are given, considering a case when the communities are isolated and when the communities are connected. The connection of the threshold ratios to disease spread and stability is discussed. The disease free equilibrium is globally stable whenever, the corresponding community specific disease threshold ratios are less than one and unstable otherwise. Community specific endemic equilibrium points are unique, locally asymptotically stable, and only exist when the corresponding disease thresholds are greater than unit. Disease spread is explosive in nature at the beginning of the outbreak but more severe in a community with poor facilities relative to a community with more improved facilities. In isolated communities, in the case of endemic cholera, the infection is characterised by a big outbreak, followed by a small episode of the infection. Only one typically big outbreak is observed in the community with improved facilities with no recurrence of the epidemic. In connected communities, movement of individuals across communities not only influences persistence of the infection but also results in more pronounced outbreak in a relatively well facilitated community in the long term. Synchronous fluctuation of the population is observed when there is unrestricted movement of both immunologically naive and infected individuals across the communities. Our results suggest that during times of cholera, movement to and fro cholera endemic areas should be avoided if the outbreak is to be contained. Otherwise, with continued migration, the infection may potentially worsen even in communities with relatively good facilities.

Journal of Theoretical Biology, 2011

ISRN Biomathematics, 2014

Journal of Biological Systems, 2010

Journal of Biological Systems, 2013

ABSTRACT Substance abuse remains a global menace in spite of recurrent warnings, seizures, social... more ABSTRACT Substance abuse remains a global menace in spite of recurrent warnings, seizures, social and pharmacological effects associated with addiction to drugs. In this paper, we use a mathematical model which is a combination of the classical SIS and SIR models to investigate the dynamics of substance abuse. Initiation into drug use is based on contact of those at risk (the susceptible population) with drug users at different levels of drug use. We evaluate the threshold number and use it to analyze the model. We show that when this threshold number is less than unity, the drug-free steady state is globally asymptotically stable and when this threshold number is greater than unity the drug-persistent steady state is also globally stable. The impact of amelioration, rehabilitation and re-initiation on drug epidemics is investigated. Amelioration in presence of quitting for light users is observed to reduce the prevalence of substance abuse and this is supported by numerical simulations. The results show that both prevention and treatment/rehabilitation are necessary strategies for reduction of drug epidemics. Our recommendation is that preventive strategies should be directed toward reducing the contact rate and treatment should be combined with psychotherapy to accelerate quitting and reduce re-initiation.

Bulletin of Mathematical Biology, 2013

Substance abuse remains a global problem, with immense health and social consequences. Crystal me... more Substance abuse remains a global problem, with immense health and social consequences. Crystal meth, known as 'tik' in South Africa, is a growing problem, and its supply chains have equally grown due to increased numbers of 'tik' users, especially…

Journal of Biological Systems, 2006

... Title: A MATHEMATICAL MODEL FOR COMBATING HIV/AIDS IN SOUTHERN AFRICA: WILL MULTIPLE STRATEGI... more ... Title: A MATHEMATICAL MODEL FOR COMBATING HIV/AIDS IN SOUTHERN AFRICA: WILL MULTIPLE STRATEGIES WORK? Author(s): FARAI NYABADZA Department of Mathematics, University of Botswana, P. Bag UB 00704, Gaborone, Botswana. ...