Dynamics analysis of a multi-strain cholera model with an imperfect vaccine.

A new two-strain model, for assessing the impact of basic control measures, treatment and dose-structured mass vaccination on cholera transmission dynamics in a population, is designed. The model has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. The model has a unique, and locally-asymptotically stable, endemic equilibrium when the threshold quantity exceeds unity and another condition holds. Numerical simulations of the model show that, with the expected 50% minimum efficacy of the first vaccine dose, vaccinating 55% of the susceptible population with the first vaccine dose will be sufficient to effectively control the spread of cholera in the community. Such effective control can also be achieved if 50% of the first vaccine dose recipients take the second dose. It is shown that a control strategy that emphasizes the use of antibiotic treatment is more effective than one that emphasizes the use of basic (non-pharmaceutical) anti-cholera control measures only. Numerical simulations show that, while the universal strategy (involving all three control measures) gives the best outcome in minimizing cholera burden in the community, the combined basic anti-cholera control measures and treatment strategy also has very effective community-wide impact.

A mathematical model for SARS-CoV-2 in variable-order fractional derivative.

A new coronavirus mathematical with hospitalization is considered with the consideration of the real cases from March 06, 2021 till the end of April 30, 2021. The essential mathematical results for the model are presented. We show the model stability when R 0 < 1 in the absence of infection. We show that the system is stable locally asymptotically when R 0 < 1 at infection free state. We also show that the system is globally asymptotically stable in the disease absence when R 0 < 1 . Data have been used to fit accurately to the model and found the estimated basic reproduction number to be R 0 = 1.2036 . Some graphical results for the effective parameters are drawn for the disease elimination. In addition, a variable-order model is introduced, and so as to handle the outbreak effectively and efficiently, a genetic algorithm is used to produce high-quality control. Numerical simulations clearly show that decision-makers may develop helpful and practical strategies to manage future waves by implementing optimum policies.

The effect of incidence functions on the dynamics of a quarantine/isolation model with time delay.

The problem of the asymptotic dynamics of a quarantine/isolation model with time delay is considered, subject to two incidence functions, namely standard incidence and the Holling type II (saturated) incidence function. Rigorous qualitative analysis of the model shows that it exhibits essentially the same (equilibrium) dynamics regardless of which of the two incidence functions is used. In particular, for each of the two incidence functions, the model has a globally asymptotically stable disease-free equilibrium whenever the associated reproduction threshold quantity is less than unity. Further, it has a unique endemic equilibrium when the threshold quantity exceeds unity. For the case with the Holling type II incidence function, it is shown that the unique endemic equilibrium of the model is globally asymptotically stable for a special case. The permanence of the disease is also established for the model with the Holling type II incidence function. Furthermore, it is shown that adding time delay to and/or replacing the standard incidence function with the Holling type II incidence function in the corresponding autonomous quarantine/isolation model with standard incidence (considered in Safi and Gumel (2010) [10]) does not alter the qualitative dynamics of the autonomous system (with respect to the elimination or persistence of the disease). Finally, numerical simulations of the model with standard incidence show that the disease burden decreases with increasing time delay (incubation period). Furthermore, models with time delay seem to be more suitable for modeling the 2003 SARS outbreaks than those without time delay.

Qualitative study of a quarantine/isolation model with multiple disease stages.

Recent studies suggest that, for disease transmission models with latent and infectious periods, the use of gamma distribution assumption seems to provide a better fit for the associated epidemiological data in comparison to the use of exponential distribution assumption. The objective of this study is to carry out a rigorous mathematical analysis of a communicable disease transmission model with quarantine (of latent cases) and isolation (of symptomatic cases), in which the waiting periods in the infected classes are assumed to have gamma distributions. Rigorous analysis of the model reveals that it has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. The model has a unique endemic equilibrium when the threshold quantity exceeds unity. The endemic equilibrium is shown to be locally and globally-asymptotically stable for special cases. Numerical simulations, using data related to the 2003 SARS outbreaks, show that the cumulative number of disease-related mortality increases with increasing number of disease stages. Furthermore, the cumulative number of new cases is higher if the asymptomatic period is distributed such that most of the period is spent in the early stages of the asymptomatic compartments in comparison to the cases where the average time period is equally distributed among the associated stages or if most of the time period is spent in the later (final) stages of the asymptomatic compartments. Finally, it is shown that distributing the average sojourn time in the infectious (asymptomatic) classes equally or unequally does not effect the cumulative number of new cases.

Dynamics of a model with quarantine-adjusted incidence and quarantine of susceptible individuals.

A new deterministic model for the spread of a communicable disease that is controllable using mass quarantine is designed. Unlike in the case of the vast majority of prior quarantine models in the literature, the new model includes a quarantine-adjusted incidence function for the infection rate and the quarantine of susceptible individuals suspected of being exposed to the disease (thereby making it more realistic epidemiologically). The earlier quarantine models tend to only explicitly consider individuals who are already infected, but show no clinical symptoms of the disease (i.e., those latently-infected), in the quarantine class (while ignoring the quarantine of susceptible individuals). In reality, however, the vast majority of people in quarantine (during a disease outbreak) are susceptible. Rigorous analysis of the model shows that the assumed imperfect nature of quarantine (in preventing the infection of quarantined susceptible individuals) induces the phenomenon of backward bifurcation when the associated reproduction threshold is less than unity (thereby making effective disease control difficult). For the case when the efficacy of quarantine to prevent infection during quarantine is perfect, the disease-free equilibrium is globally-asymptotically stable when the reproduction threshold is less than unity. Furthermore, the model has a unique endemic equilibrium when the reproduction threshold exceeds unity (and the disease persists in the population in this case).

Global stability analysis of SEIR model with holling type II incidence function.

A deterministic model for the transmission dynamics of a communicable disease is developed and rigorously analysed. The model, consisting of five mutually exclusive compartments representing the human dynamics, has a globally asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number (ℛ0), is less than unity; in such a case the endemic equilibrium does not exist. On the other hand, when the reproduction number is greater than unity, it is shown, using nonlinear Lyapunov function of Goh-Volterra type, in conjunction with the LaSalle's invariance principle, that the unique endemic equilibrium of the model is globally asymptotically stable under certain conditions. Furthermore, the disease is shown to be uniformly persistent whenever ℛ0 > 1.

Threshold dynamics of a non-autonomous SEIRS model with quarantine and isolation.

A model for assessing the effect of periodic fluctuations on the transmission dynamics of a communicable disease, subject to quarantine (of asymptomatic cases) and isolation (of individuals with clinical symptoms of the disease), is considered. The model, which is of a form of a non-autonomous system of non-linear differential equations, is analysed qualitatively and numerically. It is shown that the disease-free solution is globally-asymptotically stable whenever the associated basic reproduction ratio of the model is less than unity, and the disease persists in the population when the reproduction ratio exceeds unity. This study shows that adding periodicity to the autonomous quarantine/isolation model developed in Safi and Gumel (Discret Contin Dyn Syst Ser B 14:209-231, 2010) does not alter the threshold dynamics of the autonomous system with respect to the elimination or persistence of the disease in the population.