Bifurcation analysis and nonstandard finite difference schemes for Kermack and McKendrick type epidemiological models

The classical SIR and SIS epidemiological models are extended by considering the number of adequate contacts per infective in unit time as a function of the total population in such a way that this number grows less rapidly as the total population increases. A diffusion term is added to the SIS model and this leads to a reaction–diffusion equation, which governs the spatial spread of the disease. With the parameter R0 representing the basic reproduction number, it is shown that R0 = 1 is a forward bifurcation for the SIR and SIS models, with the disease–free equilibrium being globally asymptotic stable when R0 is less than 1. In the case when R0 is greater than 1, for both models, the endemic equilibrium is locally asymptotically stable and traveling wave solutions are found for the SIS diffusion model. Nonstandard finite difference (NSFD) schemes that replicate the dynamics of the continuous SIR and SIS models are presented. In particular, for the SIS model, a nonstandard version of the Runge-Kutta method having high order of convergence is investigated. Numerical experiments that support the theory are provided. On the other hand the SIS model is extended to a Volterra integral equation, for which the existence of multiple endemic equilibria is proved. This fact is confirmed by numerical simulations.Dissertation (MSc)--University of Pretoria, 2012.Mathematics and Applied Mathematicsunrestricte

A nonstandard Volterra difference equation for the SIS epidemiological model

By considering the contact rate as a function of infective individuals and by using a general distribution of the infective period, the SIS-model extends to a Volterra integral equation that exhibits complex behaviour such as the backward bifurcation phenomenon.We design a nonstandard finite difference (NSFD) scheme, which is reliable in replicating this complex dynamics. It is shown that the NSFD scheme has no spurious fixed-points compared to the equilibria of the continuous model. Furthermore, there exist two threshold parameters Rc 0 andRm0 , Rc 0 ≤ 1 ≤ Rm0 , such that the disease-free fixed-point is globally asymptotically stable (GAS) for R0, the basic reproduction number, less than Rc 0 and unstable for R0 > 1, while it is locally asymptotically stable (LAS) and coexists with a LAS endemic fixed-point forRc 0 Rm0 andRm0 < ∞. Numerical experiments that support the theory are provided.DST/NRF SARChI Chair in Mathematical Models and Methods in Bioengineering and Biosciences.http://www.thelancet.com/2016-09-30hb201

Analysis and dynamically consistent nonstandard discretization for a rabies model in humans and dogs

Rabies is a fatal disease in dogs as well as in humans. A possible model to represent rabies transmission dynamics in human and dog populations is presented. The next generation matrix operator is used to determine the threshold parameter R0, that is the average number of new infective individuals produced by one infective individual intro- duced into a completely susceptible population. If R0 < 1, the disease-free equilibrium is globally asymptotically stable, while it is unstable and there exists a locally asymptot- ically stable endemic equilibrium when R0 > 1. A nonstandard nite di erence scheme that replicates the dynamics of the continuous model is proposed. Numerical tests to support the theoretical analysis are provided.DST/NRF SARChI Chair in Mathematics Models and Methods in Bioengineering and Biosciences.http://link.springer.com/journal/133982017-09-30hb2016Mathematics and Applied Mathematic

Constructive treatment of reaction-diffusion and Volterra integral equations for the SIS epidemiological model

We design and investigate the reliability of various nonstandard nite di erence (NSFD) schemes for the SIS epidemiological model in three di erent settings. For the classical SIS model, we construct two new NSFD schemes which faithfully replicate the property of the continuous model of having the parameter R0, the basic reproduction number, as a threshold to determine the stability properties of equilibrium points: the disease-free equilibrium (DFE) is globally asymptotically stable (GAS) when R0 1; it is unstable when R0 > 1 and there appears a unique GAS endemic equilibrium (EE) in this case. These schemes also preserve the positivity and boundedness properties of solutions of the classical SIS model. The schemes are further used to derive NSFD schemes for the SIS-di usion model which constitutes the second setting of the study. The designed NSFD schemes are dynamically consistent with the global asymptotic stability of the disease-free equilibrium for R0 1 and the instability of the disease-free equilibrium for R0 > 1. In the latter case, the schemes replicate the global asymptotic stability of the endemic equilibrium. Positivity and boundedness properties of solutions of the SIS-di usion model are also preserved by the NSFD schemes. In a third step, the classical SIS model is extended into a SIS-Volterra integral equation model in which the contact rate is a function of fraction of infective individuals and allows a distributed period of infectivity. The qualitative analysis is now based on two threshold parameters Rc 0 1 Rm0 . The system can undergo the backward bifurcation phenomenon as follows. The DFE is the only equilibrium and it is GAS when R0 < Rc 0; there exists only one EE, which is GAS when R0 > Rm0 with the DFE being unstable when R0 > 1; for Rc 0 < R0 < 1, the DFE is locally asymptotically stable (LAS) and coexists with at least one LAS endemic equilibrium. We design a NSFD scheme and prove theoretically and computationally that it preserves the above-stated stability properties of equilibria as well as positivity and boundedness of the solutions of the continuous model.Thesis (PhD)--University of Pretoria, 2015.tm2015Mathematics and Applied MathematicsPhDUnrestricte

A mathematical model for Ebola epidemic with self-protection measures

A mathematical model presented in Berge T, Lubuma JM-S, Moremedi GM, Morris N Shava RK, A simple mathematical model for Ebola in Africa, J Biol Dyn 11(1): 42–74 (2016) for the transmission dynamics of Ebola virus is extended to incorporate vaccination and change of behavior for self-protection of susceptible individuals. In the new setting, it is shown that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number R0 is less than or equal to unity and unstable when R0>1. In the latter case, the model system admits at least one endemic equilibrium point, which is locally asymptotically stable. Using the parameters relevant to the transmission dynamics of the Ebola virus disease, we give sensitivity analysis of the model. We show that the number of infectious individuals is much smaller than that obtained in the absence of any intervention. In the case of the mass action formulation with vaccination and education, we establish that the number of infectious individuals decreases as the intervention efforts increase. In the new formulation, apart from supporting the theory, numerical simulations of a nonstandard finite difference scheme that we have constructed suggests that the results on the decrease of the number of infectious individuals is valid.The South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation: SARChI Chair in Mathematical Models and Methods in Bioengineering and Biosciences. TB and YAT acknowledge the support, in part, of DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS).https://www.worldscientific.com/worldscinet/jbs2019-03-01hj2018Mathematics and Applied Mathematic

Mathematics of a sex-structured model for syphilis transmission dynamics

Syphilis, a major sexually transmitted disease, continues to pose major public health burden in both underdeveloped and developed nations of the world. This study presents a new 2‐group sex‐structured model for assessing the community‐level impact of treatment and condom use on the transmission dynamics and control of syphilis. Rigorous analysis of the model shows that it undergoes the phenomenon of backward bifurcation. In the absence of this phenomenon (which is shown to arise because of the reinfection of recovered individuals), the disease‐free equilibrium of the model is shown to be globally asymptotically stable when the associated reproduction number is less than unity. Furthermore, the model can have multiple endemic equilibria when the reproduction threshold exceeds unity. Numerical simulations of the model, using data relevant to the transmission dynamics of the disease in Nigeria, show that, with the assumed 80% condom efficacy, the disease will continue to persist (ie, remain endemic) in the population regardless of the level of compliance in condom usage by men. Furthermore, detailed optimal control analysis (using Pontraygin's maximum principle) reveals that, for situations where the cost of implementing the controls (treatment and condom‐use) considered in this study is low, channelling resources to a treatment‐only strategy is more effective than channelling them to a condom‐use only strategy. Furthermore, as expected, the combined condom‐treatment strategy provides a higher population‐level impact than the treatment‐only strategy or the condom‐use only strategy. When the cost of implementing the controls is high, the 3 strategies are essentially equally as ineffective.The DST/NRF SARChI Chair in Mathematical Models and Methods in Bioengineering and Biosciences.http://wileyonlinelibrary.com/journal/mma2019-12-01hj2018Mathematics and Applied Mathematic

Prevalence, years lived with disability, and trends in anaemia burden by severity and cause, 1990–2021: findings from the Global Burden of Disease Study 2021

Background: Anaemia is a major health problem worldwide. Global estimates of anaemia burden are crucial for developing appropriate interventions to meet current international targets for disease mitigation. We describe the prevalence, years lived with disability, and trends of anaemia and its underlying causes in 204 countries and territories. Methods: We estimated population-level distributions of haemoglobin concentration by age and sex for each location from 1990 to 2021. We then calculated anaemia burden by severity and associated years lived with disability (YLDs). With data on prevalence of the causes of anaemia and associated cause-specific shifts in haemoglobin concentrations, we modelled the proportion of anaemia attributed to 37 underlying causes for all locations, years, and demographics in the Global Burden of Disease Study 2021. Findings: In 2021, the global prevalence of anaemia across all ages was 24·3% (95% uncertainty interval [UI] 23·9–24·7), corresponding to 1·92 billion (1·89–1·95) prevalent cases, compared with a prevalence of 28·2% (27·8–28·5) and 1·50 billion (1·48–1·52) prevalent cases in 1990. Large variations were observed in anaemia burden by age, sex, and geography, with children younger than 5 years, women, and countries in sub-Saharan Africa and south Asia being particularly affected. Anaemia caused 52·0 million (35·1–75·1) YLDs in 2021, and the YLD rate due to anaemia declined with increasing Socio-demographic Index. The most common causes of anaemia YLDs in 2021 were dietary iron deficiency (cause-specific anaemia YLD rate per 100 000 population: 422·4 [95% UI 286·1–612·9]), haemoglobinopathies and haemolytic anaemias (89·0 [58·2–123·7]), and other neglected tropical diseases (36·3 [24·4–52·8]), collectively accounting for 84·7% (84·1–85·2) of anaemia YLDs. Interpretation: Anaemia remains a substantial global health challenge, with persistent disparities according to age, sex, and geography. Estimates of cause-specific anaemia burden can be used to design locally relevant health interventions aimed at improving anaemia management and prevention. Funding: Bill & Melinda Gates Foundation