Quantifying early COVID-19 outbreak transmission in South Africa and exploring vaccine efficacy scenarios

The emergence and fast global spread of COVID-19 has presented one of the greatest public health challenges in modern times with no proven cure or vaccine. Africa is still early in this epidemic, therefore the extent of disease severity is not yet clear. We used a mathematical model to fit to the observed cases of COVID-19 in South Africa to estimate the basic reproductive number and critical vaccination coverage to control the disease for different hypothetical vaccine efficacy scenarios. We also estimated the percentage reduction in effective contacts due to the social distancing measures implemented. Early model estimates show that COVID-19 outbreak in South Africa had a basic reproductive number of 2.95 (95% credible interval [CrI] 2.83-3.33). A vaccine with 70% efficacy had the capacity to contain COVID-19 outbreak but at very higher vaccination coverage 94.44% (95% Crl 92.44-99.92%) with a vaccine of 100% efficacy requiring 66.10% (95% Crl 64.72-69.95%) coverage. Social distancing measures put in place have so far reduced the number of social contacts by 80.31% (95% Crl 79.76-80.85%). These findings suggest that a highly efficacious vaccine would have been required to contain COVID-19 in South Africa. Therefore, the current social distancing measures to reduce contacts will remain key in controlling the infection in the absence of vaccines and other therapeutics.

Modelling the dynamics of crystal meth ('tik') abuse in the presence of drug-supply chains in South Africa

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 in the Western Cape province of South Africa. We consider a model for 'tik' use that tracks drug-supply chains, and accounts for rehabilitation and amelioration for the addicted. We analyse the model and show that it has a unique drug-free equilibrium. We prove that the drug-free equilibrium is globally stable when the reproduction number is less than one. We also consider both slow and fast dynamics, and show that there is a unique drug-persistent equilibrium when the reproduction number exceeds one. The model is fitted to data on 'tik' users in rehabilitation in the Western Cape province. A sensitivity analysis reveals that the parameters with the most control over the epidemic are the quitting rate of light-drug users and the person-to-person contact rate between susceptible individuals and 'tik' users. This suggests that programs aimed at light-drug users that encourage them to quit will be significantly more effective than targeting hard-drug users, either in quitting or in rehabilitation. Similarly, the person-to-person contact rate may be reduced by social programs that raise awareness of the dangers of 'tik' use and discourage light users from recruiting others.

Mathematical Modelling of Bacterial Meningitis Transmission Dynamics with Control Measures

Vaccination and treatment are the most effective ways of controlling the transmission of most infectious diseases. While vaccination helps susceptible individuals to build either a long-term immunity or short-term immunity, treatment reduces the number of disease-induced deaths and the number of infectious individuals in a community/nation. In this paper, a nonlinear deterministic model with time-dependent controls has been proposed to describe the dynamics of bacterial meningitis in a population. The model is shown to exhibit a unique globally asymptotically stable disease-free equilibrium ℰ0, when the effective reproduction number ℛVT ≤ 1, and a globally asymptotically stable endemic equilibrium ℰ1, when ℛVT > 1; and it exhibits a transcritical bifurcation at ℛVT = 1. Carriers have been shown (by Tornado plot) to have a higher chance of spreading the infection than those with clinical symptoms who will sometimes be bound to bed during the acute phase of the infection. In order to find the best strategy for minimizing the number of carriers and ill individuals and the cost of control implementation, an optimal control problem is set up by defining a Lagrangian function L to be minimized subject to the proposed model. Numerical simulation of the optimal problem demonstrates that the best strategy to control bacterial meningitis is to combine vaccination with other interventions (such as treatment and public health education). Additionally, this research suggests that stakeholders should press hard for the production of existing/new vaccines and antibiotics and their disbursement to areas that are most affected by bacterial meningitis, especially Sub-Saharan Africa; furthermore, individuals who live in communities where the environment is relatively warm (hot/moisture) are advised to go for vaccination against bacterial meningitis.

The Dynamics of an HIV/AIDS Model with Screened Disease Carriers

The presence of carriers usually complicates the dynamics and prevention of a disease. They are not recognized as disease cases themselves unless they are screened and they usually spread the infection without them being aware. We argue that this has been one of the major causes of the spread of human immunodeficiency virus (HIV). We propose, in this paper, a model for the heterogeneous transmission of HIV/acquired immunodeficiency syndrome in the presence of disease carriers. The model allows us to assess the role of screening, as an intervention program that can slow the epidemic. A threshold value ψ*, for the screening rate is obtained. It is shown numerically that if 80% or more of the carrier population is screened, the epidemic can be contained. The qualitative analysis is done in terms of the model reproduction numberR. The model has two equilibria, the disease free equilibrium and a unique endemic equilibrium. The disease free equilibrium is globally stable ofR < 1 and the endemic equilibrium is is locally stable forR > 1. A detailed discussion of the model reproduction number is given and numerical simulations are done to show the role of some of the important model parameters.

Modelling the Potential Impact of Social Distancing on the COVID-19 Epidemic in South Africa

The novel coronavirus (COVID-19) pandemic continues to be a global health problem whose impact has been significantly felt in South Africa. With the global spread increasing and infecting millions, containment efforts by countries have largely focused on lockdowns and social distancing to minimise contact between persons. Social distancing has been touted as the best form of response in managing a rapid increase in the number of infected cases. In this paper, we present a deterministic model to describe the impact of social distancing on the transmission dynamics of COVID-19 in South Africa. The model is fitted to data from March 5 to April 13, 2020, on the cumulative number of infected cases, and a scenario analysis on different levels of social distancing is presented. The model shows that with the levels of social distancing under the initial lockdown level between March 26 and April 13, 2020, there would be a projected continued rise in the number of infected cases. The model also looks at the impact of relaxing the social distancing measures after the initial announcement of the lockdown. It is shown that relaxation of social distancing by 2% can result in a 23% rise in the number of cumulative cases whilst an increase in the level of social distancing by 2% would reduce the number of cumulative cases by about 18%. The model results accurately predicted the number of cases after the initial lockdown level was relaxed towards the end of April 2020. These results have implications on the management and policy direction in the early phase of the epidemic.

Modelling the trends of inpatient and outpatient rehabilitation for methamphetamine in the Western Cape province of South Africa

BACKGROUND

Dependence on methamphetamine remains one of the major health and social problem in the Western Cape province of South Africa. We consider a mathematical model that takes into account two forms of rehabilitation, namely; inpatient and outpatient. We examine the trends of these two types of rehabilitation. We also seek to investigate the global dynamics of the developed methamphetamine epidemic model.

METHODS

The model is designed by likening the initiation process to an infection that spreads in a community through interactions between methamphetamine users and non-users. We make use of Lyapunov functions obtained from a suitable combination of common quadratic and Volterra-type functions to establish the global stability of the methamphetamine-persistent steady state. The least squares curve fit routine (lsqcurvefit) in Matlab with optimization is used to estimate the parameter values.

RESULTS

The model analysis shows that the model has two equilibria, the methamphetamine free equilibrium and the methamphetamine persistent equilibrium, that are both globally stable when the threshold R(a) < 1 and R(a) > 1, respectively. Upon fitting the model to data on drug users under rehabilitation, parameter values that give the best fit were obtained. The projections carried out the long term trends of these forms of rehabilitation.

CONCLUSION

The results suggest that inpatient rehabilitation programs have an increased potential of enhancing the chances of recovery for methamphetamine addicts.

Modeling the role of public health education in Ebola virus disease outbreaks in Sudan

Public involvement in Ebola Virus Disease (EVD) prevention efforts is key to reducing disease outbreaks. Targeted education through practical health information to particular groups and sub-populations is crucial to controlling the disease. In this paper, we study the dynamics of Ebola virus disease in the presence of public health education with the aim of assessing the role of behavior change induced by health education to the dynamics of an outbreak. The power of behavior change is evident in two outbreaks of EVD that took place in Sudan only 3 years apart. The first occurrence was the first documented outbreak of EVD and produced a significant number of infections. The second outbreak produced far fewer cases, presumably because the population in the region learned from the first outbreak. We derive a system of ordinary differential equations to model these two contrasting behaviors. Since the population in Sudan learned from the first outbreak of EVD and changed their behavior prior to the second outbreak, we use data from these two instances of EVD to estimate parameters relevant to two contrasting behaviors. We then simulate a future outbreak of EVD in Sudan using our model that contains two susceptible populations, one being more informed about EVD. Our finding show how a more educated population results in fewer cases of EVD and highlights the importance of ongoing public health education.

On the Role of Imitation on Adolescence Methamphetamine Abuse Dynamics

Adolescence methamphetamine use is an issue of considerable concern due to its correlation with later delinquency, divorce, unemployment and health problems. Understanding how adolescents initiate methamphetamine abuse is important in developing effective prevention programs. We formulate a mathematical model for the spread of methamphetamine abuse using nonlinear ordinary differential equations. It is assumed that susceptibles are recruited into methamphetamine use through imitation. An epidemic threshold value, [Formula: see text], termed the abuse reproduction number, is proposed and defined herein in the drug-using context. The model is shown to exhibit the phenomenon of backward bifurcation. This means that methamphetamine problems may persist in the population even if [Formula: see text] is less than one. Sensitivity analysis of [Formula: see text] was performed to determine the relative importance of different parameters in methamphetamine abuse initiation. The model is then fitted to data on methamphetamine users less than 20 years old reporting methamphetamine as their primary substance of abuse in the treatment centres of Cape Town and parameter values that give the best fit are chosen. Results show that the proportion of methamphetamine users less than 20 years old reporting methamphetamine as their primary substance of abuse will continue to decrease in Cape Town of South Africa. The results suggest that intervention programs targeted at reducing adolescence methamphetamine abuse, are positively impacting methamphetamine abuse.

Modelling Drug Abuse Epidemics in the Presence of Limited Rehabilitation Capacity

The abuse of drugs is now an epidemic globally whose control has been mainly through rehabilitation. The demand for drug abuse rehabilitation has not been matched with the available capacity resulting in limited placement of addicts into rehabilitation. In this paper, we model limited rehabilitation through the Hill function incorporated into a system of nonlinear ordinary differential equations. Not every member of the community is equally likely to embark on drug use, risk structure is included to help differentiate those more likely (high risk) to abuse drugs and those less likely (low risk) to abuse drugs. It is shown that the model has multiple equilibria, and using the centre manifold theory, the model exhibits the phenomenon of backward bifurcation whose implications to rehabilitation are discussed. Sensitivity analysis and numerical simulations are performed. The results show that saturation in rehabilitation will in the long run lead to the escalation of drug abuse. This means that limited access to rehabilitation has negative implications in the fight against drug abuse where rehabilitation is the main form of control. This suggests that increased access to rehabilitation is likely to lower the drug abuse epidemic.

Modelling cholera transmission dynamics in the presence of limited resources

Objectives

We study the transmission dynamics of cholera in the presence of limited resources, a common feature of the developing world. The model is used to gain insight into the impact of available resources of the health care system on the spread and control of the disease. A deterministic model that includes a nonlinear recovery rate is formulated and rigorously analyzed. Limited treatment is described by inclusion of a special treatment function. Center manifold theory is used to show that the model exhibits the phenomenon of backward bifurcation. Matlab has been used to carry out numerical simulations to support theoretical findings.

Results

The model analysis shows that the disease free steady state is locally stable when the threshold \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}_{0} < 1$$\end{document}R0<1. It is also shown that the model has multiple equilibria and the model exhibits the phenomenon of backward bifurcation whose implications to cholera infection are discussed. The results are useful for the public health planning in resource allocation for the control of cholera transmission.

On the transmission dynamics of Buruli ulcer in Ghana: Insights through a mathematical model

Background

Mycobacterium ulcerans is know to cause the Buruli ulcer. The association between the ulcer and environmental exposure has been documented. However, the epidemiology of the ulcer is not well understood. A hypothesised transmission involves humans being bitten by the water bugs that prey on mollusks, snails and young fishes.

Methods

In this paper, a model for the transmission of Mycobacterium ulcerans to humans in the presence of a preventive strategy is proposed and analysed. The model equilibria are determined and conditions for the existence of the equilibria established. The model analysis is carried out in terms of the reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R}_0$$\end{document}R0. The disease free equilibrium is found to be locally asymptotically stable for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R}_0<1.$$\end{document}R0<1. The model is fitted to data from Ghana.

Results

The model is found to exhibit a backward bifurcation and the endemic equilibrium point is globally stable when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{R}_0>1.$$\end{document}R0>1. Sensitivity analysis showed that the Buruli ulcer epidemic is highly influenced by the shedding and clearance rates of Mycobacterium ulcerans in the environment. The model is found to fit reasonably well to data from Ghana and projections on the future of the Buruli ulcer epidemic are also made.

Conclusions

The model reasonably fitted data from Ghana. The fitting process showed data that appeared to have reached a steady state and projections showed that the epidemic levels will remain the same for the projected time. The implications of the results to policy and future management of the disease are discussed.

On the efficiency of HIV transmission: Insights through discrete time HIV models

There are different views on which of the two forms of viral spread is more efficient in vivo between cell-free and cell-associated virus. In this study, discrete time human immunodeficiency virus models are formulated and analysed with the goal of determining the form of viral spread that is more efficient in vivo. It is shown that on its own, cell-free viral spread cannot sustain an infection owing to the low infectivity of cell-free virus and cell-associated virus can sustain an infection because of the high infectivity of cell-associated virus. When acting concurrently, cell-associated virus is more efficient in spreading the infection upon exposure to the virus. However, in the long term, the two forms of viral spread contribute almost equally. Both forms of viral spread are shown to be able to initiate an infection.

Stemming cholera tides in Zimbabwe through mass vaccination

BACKGROUND

In 2018, Zimbabwe declared another major cholera outbreak a decade after recording one of the worst cholera outbreaks in Africa.

METHODS

A mathematical model for cholera was used to estimate the magnitude of the cholera outbreak and vaccination coverage using cholera cases reported data. A Markov chain Monte Carlo method based on a Bayesian framework was used to fit the model in order to estimate the basic reproductive number and required vaccination coverage for cholera control.

RESULTS

The results showed that the outbreak had a basic reproductive number of 1.82 (95% credible interval [CrI] 1.53-2.11) and required vaccination coverage of at least 58% (95% Crl 45-68%) to be contained using an oral cholera vaccine of 78% efficacy. Sensitivity analysis demonstrated that a vaccine with at least 55% efficacy was sufficient to contain the outbreak but at higher coverage of 75% (95% Crl 58-88%). However, high-efficacy vaccines would greatly reduce the required coverage, with 100% efficacy vaccine reducing coverage to 45% (95% Crl 35-53%).

CONCLUSIONS

These findings reinforce the crucial need for oral cholera vaccines to control cholera in Zimbabwe, considering that the decay of water reticulation and sewerage infrastructure is unlikely to be effectively addressed in the coming years.

Modelling the dynamics of two political parties in the presence of switching

This paper generalizes the model proposed by Misra, by considering switching between political parties. In the model proposed, the movements of members from political party B to political party C and vice versa, are considered but the net movement is considered by assuming that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _1-\theta _2=\theta$$\end{document}θ1-θ2=θ (a constant), which implies that the movement of members is either from party B to party C or from party C to party B. In this paper we remodel these movements through switching functions to capture how individuals switch between parties. The results provide a more comprehensive synopsis of the dynamics between two political parties.

Modelling the role of correctional services on gangs: insights through a mathematical model

Research has shown that gang membership increases the chances of offending, antisocial behaviour and drug use. Gang membership should be acknowledged as part of crime prevention and policy designs, and when developing interventions and preventative programmes. Correctional services are designed to rehabilitate convicted offenders. We formulate a deterministic mathematical model using nonlinear ordinary differential equations to investigate the role of correctional services on the dynamics of gangs. The recruitment into gang membership is assumed to happen through an imitation process. An epidemic threshold value, [Formula: see text], termed the gang reproduction number, is proposed and defined herein in the gangs' context. The model is shown to exhibit the phenomenon of backward bifurcation. This means that gangs may persist in the population even if [Formula: see text] is less than one. Sensitivity analysis of [Formula: see text] was performed to determine the relative importance of different parameters in gang initiation. The critical efficacy ε* is evaluated and the implications of having functional correctional services are discussed.

A comparison of elasticities of viral levels to specific immune response mechanisms in human immunodeficiency virus infection

BACKGROUND

The presence of an asymptomatic phase in an HIV infection indicates that the immune system can partially control the infection. Determining the immune mechanisms that contribute significantly to the partial control of the infection enhance the HIV infection intervention strategies and is important in vaccine development. Towards this goal, a discrete time HIV model, which incorporates the life cycle aspects of the virus, the antibody (humoral) response and the cell-mediated immune response is formulated to determine immune system components that are most efficient in controlling viral levels. Ecological relationships are used to model the interplay between the immune system components and the HIV pathogen. Model simulations and transient elasticity analysis of the viral levels to immune response parameters are used to compare the different immune mechanisms.

RESULTS

It is shown that cell-mediated immune response is more effective in controlling the viral levels than the antibody response. Killing of infected cells is shown to be crucial in controlling the viral levels. Our results show a negative correlation between the antibody response and the viral levels in the early stages of the infection, but we predicted this immune mechanism to be positively correlated with the viral levels in the late stage of the infection. A result that suggests lack of relevance of antibody response with infection progression. On the contrary, we predicted the cell-mediated immune response to be always negatively correlated with viral levels.

CONCLUSION

Neutralizing antibodies can only control the viral levels in the early days of the HIV infection whereas cell-mediated immune response is beneficial during all the stages of the infection. This study predicts that vaccine design efforts should also focus on stimulating killer T cells that target infected cells.

A mathematical model for the prediction of the prevalence of allergies in Zimbabwe

Background

The prevalence of allergies has been observed to be increasing in the past years in Zimbabwe. It is thus important to consider the long term prevalence of allergies. Our interest is in investigating the trends of allergies in the next 2 decades.

Method

We formulate a deterministic model with 6 compartments to predict the prevalence of allergies in Zimbabwe. The human population is divided into 4 distinct epidemiological, classes based on their exposure to 2 allergen groups (food and inhalants), represented by 2 compartments. The model is used to predict the prevalence of allergen sensitization. The number of human allergen groups in each compartment are tracked through a system of differential equations. Model parameters were obtained by fitting observed data to the model. Graphical solutions of the model were developed using Matlab and Excel.

Results

The rate of sensitisation to food allergen sources is found to be lower than the rate of sensitisation to inhalant allergens. The rate at which individuals develop tolerance to food allergen sources is found to be almost twice the rate of developing tolerance to inhalant allergies. The equilibrium solutions (the long-term states of the populations) of the model are found to be non-zero implying that there will never be an allergy-free population. Our results also show that the prevalence of food allergy is likely to increase in the next 2 decades while inhalant allergy prevalence is expected to decrease.

Conclusion

Our long-term solutions show endemicity in allergies in Zimbabwe. So, allergy will be endemic in the Zimbabwean population; hence there is a need for allergy care and management facilities to be increased. These results are critical in policy development and planning around allergies in the near future.

An Optimal Treatment Strategy for Diabetes

Consider a system on n variables involved in the regulation of glucose in the body, whose concentrations are given by stochastic differential equations driven by m-dimensional Brownian motion. We formulate a stochastic control problem and give sufficient conditions for the existence of an optimal treatment strategy. We study the following problem: what treatment strategy for the n variables, maximizes the expected benefit from treatment.

Effect of time-varying adherence to non-pharmaceutical interventions on the occurrence of multiple epidemic waves: A modeling study

Non-pharmaceutical interventions (NPIs) play a central role in infectious disease outbreak response and control. Their usefulness cannot be overstated, especially during the early phases of a new epidemic when vaccines and effective treatments are not available yet. These interventions can be very effective in curtailing the spread of infectious diseases when adequately implemented and sufficiently adopted by the public. However, NPIs can be very disruptive, and the socioeconomic and cultural hardships that come with their implementation interfere with both the ability and willingness of affected populations to adopt such interventions. This can lead to reduced and unsteady adherence to NPIs, making disease control more challenging to achieve. Deciphering this complex interaction between disease dynamics, NPI stringency, and NPI adoption would play a critical role in informing disease control strategies. In this work, we formulate a general-purpose model that integrates government-imposed control measures and public adherence into a deterministic compartmental epidemic model and study its properties. By combining imitation dynamics and the health belief model to encode the unsteady nature of NPI adherence, we investigate how temporal variations in NPI adherence levels affect the dynamics and control of infectious diseases. Among the results, we note the occurrence of multiple epidemic waves as a result of temporal variations in NPI adherence and a trade-off between the stringency of control measures and adherence. Additionally, our results suggest that interventions that aim at increasing public adherence to NPIs are more beneficial than implementing more stringent measures. Our findings highlight the necessity of taking the socioeconomic and cultural realities of affected populations into account when devising public health interventions.