arXiv subjects
Publications and source records attributed to Michael T. Meehan.
We investigate the global dynamics of a renewal-type epidemic model with variable susceptibility. We show that in this extended model there exists a unique endemic equilibrium and prove that it is globally asymptotically stable when $R_0 > 1$, i.e. when it exists. We also show that the infection-free equilibrium, which exists always, is globally asymptotically stable for $R_0 \leq 1$.
We investigate a two-strain disease model with amplification to simulate the prevalence of drug-susceptible (s) and drug-resistant (m) disease strains. We model the emergence of drug resistance as a consequence of inadequate treatment, i.e. amplification. We perform a dynamical analysis of the resulting system and find that the model contains three equilibrium points: a disease-free equilibrium; a mono-existent disease-endemic equilibrium with respect to the drug-resistant strain; and a co-existent disease-endemic equilibrium where both the drug-susceptible and drug-resistant strains persist. We found two basic reproduction numbers: one associated with the drug-susceptible strain $R_{0s}$; the other with the drug-resistant strain $R_{0m}$,and showed that at least one of the strains can spread in a population if ($R_{0s}$,$R_{0m}$) > 1 (epidemic).Furthermore, we also showed that if $R_{0m}$ > max($R_{0s}$,1), the drug-susceptible strain dies out but the drug-resistant strain persists in the population; however if $R_{0s}$ > max($R_{0m}$,1), then both the drug-susceptible and drug-resistant strains persist in the population. We conducted a local stability analysis of the system equilibrium points using the Routh-Hurwitz conditions and a global stability analysis using appropriate Lyapunov functions. Sensitivity analysis was used to identify the most important model parameters through the partial rank correlation coefficient (PRCC) method. We found that the contact rate of both strains had the largest influence on prevalence. We also investigated the impact of amplification and treatment rates of both strains on the equilibrium prevalence of infection; results suggest that poor quality treatment make coexistence more likely but increase the relative abundance of resistant infections.
The rise of antimicrobial drug resistance is an imminent threat to global health that has warranted, and duly received, considerable attention within the medical, microbiological and modelling communities. Outbreaks of drug-resistant pathogens are ignited by the emergence and transmission of mutant variants descended from wild-type strains circulating in the community. In this work we investigate the stochastic dynamics of the emergence of a novel disease strain, introduced into a population in which it must compete with an existing endemic strain. In analogy with past work on single-strain epidemic outbreaks, we apply a branching process approximation to calculate the probability that the new strain becomes established. As expected, a critical determinant of the survival prospects of any invading strain is the magnitude of its reproduction number relative to that of the background endemic strain. Whilst in most circumstances this ratio must exceed unity in order for invasion to be viable, we show that differential control scenarios can lead to less-fit novel strains invading populations hosting a fitter endemic one. This analysis and the accompanying findings will inform our understanding of the mechanisms that have led to past instances of successful strain invasion, and provide valuable lessons for thwarting future drug-resistant strain incursions.
We extend a recent investigation by Meehan et al. (2019) regarding the global stability properties of the general Kermack-McKendrick (renewal) model to the multi-strain case. We demonstrate that the basic reproduction number of each strain $R_{0j}$ represents a sharp threshold parameter such that when $R_{0j} \leq 1$ for all $j$ each strain dies out and the infection-free equilibrium is globally asymptotically stable; whereas for $R_{01} \equiv \mathrm{max}_j\, R_{0j} > 1$ the endemic equilibrium point $\bar{P}^1$, at which only the fittest strain (i.e. strain 1) remains in circulation, becomes globally asymptotically stable.
Arboviral infections, especially dengue, continue to cause significant health burden in their endemic regions. One of the strategies to tackle these infections is to replace the main vector agent, Ae. aegypti, with the ones incapable of transmitting the virus. Wolbachia, an intracellular bacterium, has shown promise in achieving this goal. However, key factors such as imperfect maternal transmission, loss of Wolbachia infection, reduced reproductive capacity and shortened life-span affect the dynamics of Wolbachia in different forms in the Ae. aegypti population. In this study, we developed a Wolbachia transmission dynamic model adjusting for imperfect maternal transmission and loss of Wolbachia infection. The invasive reproductive number that determines the likelihood of replacement of the Wolbachia-uninfected (WU) population is derived and with it, we established the local and global stability of the equilibrium points. This analysis clearly shows that cytoplasmic incompatibility (CI) does not guarantee establishment of the Wolbachia-infected (WI) mosquitoes as imperfect maternal transmission and loss of Wolbachia infection could outweigh the gains from CI. Optimal release programs depending on the level of imperfect maternal transmission and loss of Wolbachia infection are shown. Hence, it is left to decision makers to either aim for replacement or co-existence of both populations.
We investigate the global dynamics of a general Kermack-McKendrick-type epidemic model formulated in terms of a system of renewal equations. Specifically, we consider a renewal model for which both the force of infection and the infected removal rates are arbitrary functions of the infection age, $τ$, and use the direct Lyapunov method to establish the global asymptotic stability of the equilibrium solutions. In particular, we show that the basic reproduction number, $R_0$, represents a sharp threshold parameter such that for $R_0\leq 1$, the infection-free equilibrium is globally asymptotically stable; whereas the endemic equilibrium becomes globally asymptotically stable when $R_0 > 1$, i.e. when it exists.
We introduce and analyze coupled, multi-strain epidemic models designed to simulate the emergence and dissemination of mutant (e.g. drug-resistant) pathogen strains. In particular, we investigate the mathematical and biological properties of a general class of multi-strain epidemic models in which the infectious compartments of each strain are coupled together in a general manner. We derive explicit expressions for the basic reproduction number of each strain and highlight their importance in regulating the system dynamics (e.g. the potential for an epidemic outbreak) and the existence of nonnegative endemic solutions. Importantly, we find that the basic reproduction number of each strain is independent of the mutation rates between the strains --- even under quite general assumptions for the form of the infectious compartment coupling. Moreover, we verify that the coupling term promotes strain coexistence (as an extension of the competitive exclusion principle) and demonstrate that the strain with the greatest reproductive capacity is not necessarily the most prevalent. Finally, we briefly discuss the implications of our results for public health policy and planning.
We revisit the calculation of dark matter relic abundances in scalar-tensor gravity using a generic form $A(φ_*) = e^{βφ_*^2/2}$ for the coupling between the scalar field $φ_*$ and the metric, for which detailed Big Bang Nucleosynthesis constraints are available. We find that BBN constraints restrict the modified expansion rate in these models to be almost degenerate with the standard expansion history at the time of dark matter decoupling. In this case the maximum level of enhancement of the dark matter relic density was found to be a factor of $\sim 3$, several orders of magnitude below that found in previous investigations.
The relic density of symmetric and asymmetric dark matter in a Gauss-Bonnet (GB) modified Randall-Sundrum (RS) type II braneworld cosmology is investigated. The existing study of symmetric dark matter in a GB braneworld (Okada and Okada, 2009) found that the expansion rate was reduced compared to that in standard General Relativity (GR), thereby delaying particle freeze-out and resulting in relic abundances which are suppressed by up to $\mathcal{O}(10^{-2})$. This is in direct contrast to the behaviour observed in RS braneworlds where the expansion rate is enhanced and the final relic abundance boosted. However, this finding that relic abundances are suppressed in a GB braneworld is based upon a highly contrived situation in which the GB era evolves directly into a standard GR era, rather than passing through a RS era as is the general situation. This collapse of the RS era requires equating the mass scale $m_α$ of the GB modification and the mass scale $m_σ$ of the brane tension. However, if the GB contribution is to be considered as the lowest order correction from string theory to the RS action, we would expect $m_α > m_σ$. We investigate the effect upon the relic abundance of choosing more realistic values for the ratio $\mathcal{R}_{m} \equiv m_α/m_σ$ and find that the relic abundance can be either enhanced or suppressed by more than two orders of magnitude. However, suppression only occurs for a small range of parameter choices and, overwhelmingly, the predominant situation is that of enhancement as we recover the usual Randall-Sundrum type behaviour in the limit $\mathcal{R}_m \gg 1$. We use the latest observational bound $Ω_{DM}h^2 = 0.1187 \pm 0.0017$ to constrain the various model parameters and briefly discuss the implications for direct/indirect dark matter detection experiments as well as dark matter particle models.
We investigate the effect of a braneworld expansion era on the relic density of asymmetric dark matter. We find that the enhanced expansion rate in the early universe predicted by the Randall-Sundrum II (RSII) model leads to earlier particle freeze-out and an enhanced relic density. This effect has been observed previously by Okada and Seto (2004) for symmetric dark matter models and here we extend their results to the case of asymmetric dark matter. We also discuss the enhanced asymmetric annihilation rate in the braneworld scenario and its implications for indirect detection experiments.