Theoretical assessment of avian influenza vaccine
Folashade B. Agusto Abba B. Gumel
This study presents a deterministic model for theoretically assessing the potential impact of an imperfect avian influenza vaccine (for domestic birds) in two avian populations on the transmission dynamics of avian influenza in the domestic and wild birds population. The model is analyzed to gain insights into the qualitative features of its associated equilibria. This allows the determination of important epidemiological thresholds such as the basic reproduction number and a measure for vaccine impact. A sub-model without vaccination is first considered, where it is shown that it has a globally-asymptotically stable disease-free equilibrium whenever a certain reproduction threshold is less than unity. Unlike the sub-model without vaccination, the model with vaccination undergoes backward bifurcation, a phenomenon associated with the co-existence of multiple stable equilibria. In other words, for the model with vaccination, the classical epidemiological requirement of having the associated reproduction number less than unity does not guarantee disease elimination in the model. It is shown that the possibility of backward bifurcation occurring decreases with increasing vaccination rate (for susceptible domestic birds). Further, the study shows that the vaccine impact (in reducing disease burden) is dependent on the sign of a certain threshold quantity (denoted by $\nabla_{\mathcal P}$). The vaccine will have positive or no impact if $\nabla_{\mathcal P}$ is less than or equal to unity. Numerical simulations suggest that the prospect of effectively controlling the disease in the avian population increases with increasing vaccine efficacy and coverage.
keywords: backward bifurcation vaccine impact. avian influenza; equilibria; stability vaccine efficacy reproduction number
Mathematical analysis of a model for HIV-malaria co-infection
Zindoga Mukandavire Abba B. Gumel Winston Garira Jean Michel Tchuenche
A deterministic model for the co-interaction of HIV and malaria in a community is presented and rigorously analyzed. Two sub-models, namely the HIV-only and malaria-only sub-models, are considered first of all. Unlike the HIV-only sub-model, which has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction number is less than unity, the malaria-only sub-model undergoes the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium, for a certain range of the associated reproduction number less than unity. Thus, for malaria, the classical requirement of having the associated reproduction number to be less than unity, although necessary, is not sufficient for its elimination. It is also shown, using centre manifold theory, that the full HIV-malaria co-infection model undergoes backward bifurcation. Simulations of the full HIV-malaria model show that the two diseases co-exist whenever their reproduction numbers exceed unity (with no competitive exclusion occurring). Further, the reduction in sexual activity of individuals with malaria symptoms decreases the number of new cases of HIV and the mixed HIV-malaria infection while increasing the number of malaria cases. Finally, these simulations show that the HIV-induced increase in susceptibility to malaria infection has marginal effect on the new cases of HIV and malaria but increases the number of new cases of the dual HIV-malaria infection.
keywords: stability. equilibrium basic reproduction number HIV-malaria model co-infection
Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs
Robert G. McLeod John F. Brewster Abba B. Gumel Dean A. Slonowsky
This paper presents a statistical study of a deterministic model for the transmission dynamics and control of severe acute respiratory syndrome (SARS). The effect of the model parameters on the dynamics of the disease is analyzed using sensitivity and uncertainty analyses. The response (or output) of interest is the control reproduction number, which is an epidemiological threshold governing the persistence or elimination of SARS in a given population. The compartmental model includes parameters associated with control measures such as quarantine and isolation of asymptomatic and symptomatic individuals. One feature of our analysis is the incorporation of time-dependent functions into the model to reflect the progressive refinement of these SARS control measures over time. Consequently, the model contains continuous time-varying inputs and outputs. In this setting, sensitivity and uncertainty analytical techniques are used in order to analyze the impact of the uncertainty in the parameter estimates on the results obtained and to determine which parameters have the largest impact on driving the disease dynamics.
keywords: epidemiological model functional out-put control reproduction number Latin hypercube sampling partial rank correlation coefficients.
Mathematical analysis of the transmission dynamics of HIV/TB coinfection in the presence of treatment
Oluwaseun Sharomi Chandra N. Podder Abba B. Gumel Baojun Song
This paper addresses the synergistic interaction between HIV and mycobacterium tuberculosis using a deterministic model, which incorporates many of the essential biological and epidemiological features of the two dis- eases. In the absence of TB infection, the model (HIV-only model) is shown to have a globally asymptotically stable, disease-free equilibrium whenever the associated reproduction number is less than unity and has a unique endemic equilibrium whenever this number exceeds unity. On the other hand, the model with TB alone (TB-only model) undergoes the phenomenon of back- ward bifurcation, where the stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction threshold is less than unity. The analysis of the respective reproduction thresholds shows that the use of a targeted HIV treatment (using anti-retroviral drugs) strategy can lead to effective control of HIV provided it reduces the relative infectiousness of individuals treated (in comparison to untreated HIV-infected individuals) below a certain threshold. The full model, with both HIV and TB, is simu- lated to evaluate the impact of the various treatment strategies. It is shown that the HIV-only treatment strategy saves more cases of the mixed infection than the TB-only strategy. Further, for low treatment rates, the mixed-only strategy saves the least number of cases (of HIV, TB, and the mixed infection) in comparison to the other strategies. Thus, this study shows that if resources are limited, then targeting such resources to treating one of the diseases is more beneficial in reducing new cases of the mixed infection than targeting the mixed infection only diseases. Finally, the universal strategy saves more cases of the mixed infection than any of the other strategies.
keywords: equilibria HIV/TB treatment. stability bifurcation
Global asymptotic dynamics of a model for quarantine and isolation
Mohammad A. Safi Abba B. Gumel
The paper presents an SEIQHRS model for evaluating the combined impact of quarantine (of asymptomatic cases) and isolation (of individuals with clinical symptoms) on the spread of a communicable disease. Rigorous analysis of the model, which takes the form of a deterministic system of nonlinear differential equations with standard incidence, reveal that it has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. Further, the model has a unique endemic equilibrium when the threshold quantity exceeds unity. Using a Krasnoselskii sub-linearity trick, it is shown that the unique endemic equilibrium is locally-asymptotically stable for a special case. A nonlinear Lyapunov function of Volterra type is used, in conjunction with LaSalle Invariance Principle, to show that the endemic equilibrium is globally-asymptotically stable for a special case. Numerical simulations, using a reasonable set of parameter values (consistent with the SARS outbreaks of 2003), show that the level of transmission by individuals isolated in hospitals play an important role in determining the impact of the two control measures (the use of quarantine and isolation could offer a detrimental population-level impact if isolation-related transmission is high enough).
keywords: reproduction number Isolation stability. quarantine equilibria
Existence of multiple-stable equilibria for a multi-drug-resistant model of mycobacterium tuberculosis
Abba B. Gumel Baojun Song
The resurgence of multi-drug-resistant tuberculosis in some parts of Europe and North America calls for a mathematical study to assess the impact of the emergence and spread of such strain on the global effort to effectively control the burden of tuberculosis. This paper presents a deterministic compartmental model for the transmission dynamics of two strains of tubercu- losis, a drug-sensitive (wild) one and a multi-drug-resistant strain. The model allows for the assessment of the treatment of people infected with the wild strain. The qualitative analysis of the model reveals the following. The model has a disease-free equilibrium, which is locally asymptotically stable if a cer- tain threshold, known as the effective reproduction number, is less than unity. Further, the model undergoes a backward bifurcation, where the disease-free equilibrium coexists with a stable endemic equilibrium. One of the main nov- elties of this study is the numerical illustration of tri-stable equilibria, where the disease-free equilibrium coexists with two stable endemic equilibrium when the aforementioned threshold is less than unity, and a bi-stable setup, involving two stable endemic equilibria, when the effective reproduction number is greater than one. This, to our knowledge, is the first time such dynamical features have been observed in TB dynamics. Finally, it is shown that the backward bifurcation phenomenon in this model arises due to the exogenous re-infection property of tuberculosis.
keywords: Key words and phrases. multi-drug-resistant TB dynamical system bifurcation analysis epidemic model multi-stable steady state.
Sex-biased prevalence in infections with heterosexual, direct, and vector-mediated transmission: a theoretical analysis
Andrea Pugliese Abba B. Gumel Fabio A. Milner Jorge X. Velasco-Hernandez

Three deterministic Kermack-McKendrick-type models for studying the transmission dynamics of an infection in a two-sex closed population are analyzed here. In each model it is assumed that infection can be transmitted through heterosexual contacts, and that there is a higher probability of transmission from one sex to the other than vice versa. The study is focused on understanding whether and how this bias in transmission reflects in sex differences in final attack ratios (i.e. the fraction of individuals of each sex that eventually gets infected). In the first model, where the other two transmission modes are not considered, the attack ratios (fractions of the population of each sex that will eventually be infected) can be obtained as solutions of a system of two nonlinear equations, that has a unique solution if the net reproduction number exceeds unity. It is also shown that the ratio of attack ratios depends solely on the ratio of gender-specific susceptibilities and on the basic reproductive number of the epidemic $ \mathcal{R}_0 $, and that the gender-specific final attack-ratio is biased in the same direction as the gender-specific susceptibilities. The second model allows also for infection transmission through direct, non-sexual, contacts. In this case too, an analytical expression is derived from which the attack ratios can be obtained. The qualitative results are similar to those obtained for the previous model, but another important parameter for determining the value of the ratio between the attack ratios in the two sexes is obtained, the relative weight of direct vs. heterosexual transmission (namely, ρ). Quantitatively, the ratio of final attack ratios generally will not exceed 1.5, if non-sexual transmission accounts for most transmission events (ρ ≥ 0.6) and the ratio of gender-specific susceptibilities is not too large (say, 5 at most).

The third model considers vector-borne, instead of direct transmission. In this case, we were not able to find an analytical expression for the final attack ratios, but used instead numerical simulations. The results on final attack ratios are actually quite similar to those obtained with the second model. It is interesting to note that transient patterns can differ from final attack ratios, as new cases will tend to occur more often in the more susceptible sex, while later depletion of susceptibles may bias the ratio in the opposite direction.

The analysis of these simple models, despite their lack of realism, can help in providing insight into, and assessment of, the potential role of gender-specific transmission in infections with multiple modes of transmission, such as Zika virus (ZIKV), by gauging what can be expected to be seen from epidemiological reports of new cases, disease incidence and seroprevalence surveys.

keywords: Epidemic model heterosexual transmission vector transmission sex-biased prevalence Zika epidemic model
An sveir model for assessing potential impact of an imperfect anti-SARS vaccine
Abba B. Gumel C. Connell McCluskey James Watmough
The control of severe acute respiratory syndrome (SARS), a fatal contagious viral disease that spread to over 32 countries in 2003, was based on quarantine of latently infected individuals and isolation of individuals with clinical symptoms of SARS. Owing to the recent ongoing clinical trials of some candidate anti-SARS vaccines, this study aims to assess, via mathematical modelling, the potential impact of a SARS vaccine, assumed to be imperfect, in curtailing future outbreaks. A relatively simple deterministic model is designed for this purpose. It is shown, using Lyapunov function theory and the theory of compound matrices, that the dynamics of the model are determined by a certain threshold quantity known as the control reproduction number ($\R_{v}$). If $\R_{v}\le 1$, the disease will be eliminated from the community; whereas an epidemic occurs if $\R_{v}>1$. This study further shows that an imperfect SARS vaccine with infection-blocking efficacy is always beneficial in reducing disease spread within the community, although its overall impact increases with increasing efficacy and coverage. In particular, it is shown that the fraction of individuals vaccinated at steady-state and vaccine efficacy play equal roles in reducing disease burden, and the vaccine must have efficacy of at least 75% to lead to effective control of SARS (assuming $\R=4$). Numerical simulations are used to explore the severity of outbreaks when $\R_{v}>1$.
keywords: severe acute respiratory syndrome (SARS) disease transmission model control reproduction number. vaccination epidemiology
Mathematical Analysis of a Weather-driven Model for the Population Ecology of Mosquitoes
Kamaldeen Okuneye Ahmed Abdelrazec Abba B. Gumel

A new deterministic model for the population biology of immature and mature mosquitoes is designed and used to assess the impact of temperature and rainfall on the abundance of mosquitoes in a community. The trivial equilibrium of the model is globally-asymptotically stable when the associated vectorial reproduction number $({\mathcal R}_0)$ is less than unity. In the absence of density-dependence mortality in the larval stage, the autonomous version of the model has a unique and globally-asymptotically stable non-trivial equilibrium whenever $1 < {\mathcal R}_0 < {\mathcal R}_0^C$ (this equilibrium bifurcates into a limit cycle, via a Hopf bifurcation at ${\mathcal R}_0={\mathcal R}_0^C$). Numerical simulations of the weather-driven model, using temperature and rainfall data from three cities in Sub-Saharan Africa (Kwazulu Natal, South Africa; Lagos, Nigeria; and Nairobi, Kenya), show peak mosquito abundance occurring in the cities when the mean monthly temperature and rainfall values lie in the ranges $[22 -25]^{0}$C, $[98 -121]$ mm; $[24 -27]^{0}$C, $[113 -255]$ mm and $[20.5 -21.5]^{0}$C, $[70 -120]$ mm, respectively (thus, mosquito control efforts should be intensified in these cities during the periods when the respective suitable weather ranges are recorded).

keywords: Mosquitoes stage-structure climate change autonomous and non-autonomous model stability Bézout matrix; reproduction number

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