In this work we obtain sufficient conditions for the exponential stability of equilibrium points of the nonautonomous difference equations by means of weak contraction arguments.

Two common transmission pathways for the spread of COVID‐19 virus are direct and indirect. The direct pathway refers to the person‐to‐person transmission between susceptibles and infectious individuals. Infected individuals shed virus on the objects, and new infections arise through touching a contaminated surface; this refers to the indirect transmission pathway. We model the direct and indirect transmission pathways with a S A D O I R ode model. Our proposal explicitly includes compartments for the contaminated objects, susceptible individuals, asymptomatic infectious, detected infectious, and recovered individuals. We compute the basic reproduction number and epidemic growth rate of the model and determine how these fundamental quantities relate to the transmission rate of the pathways. We further study the relationship between the rate of loss of immunity and the occurrence of backward bifurcation. An efficient statistical framework is introduced to estimate the parameters of the model. We show the performance of the model in the simulation scenarios and the real data from the COVID‐19 daily cases in South Korea.

For a locally compact group G, we investigate topological inner invariant means on L8(G) and its subspaces. In particular, we characterize strict inner amenability of L1(G) to study the relation between this notion and strict inner amenability of G.

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In the paper, the authors put two drug strains under various levels of competition and establish that cross‐immunity and host isolation lead to periodic epidemic outbreaks in the multistrain system. For the model, by the reproduction number of strain one, scriptR1, and the reproduction number of strain two, scriptR2, they show local and global stability and backward bifurcation under suitable conditions.