Abstract
A discrete tuberculosis model with direct progression and treatment of latently infected individuals is presented. The model does not consider the drug-resistant TB, and it assumes that latently infected individuals develop the active disease only because of being endogenous reactive, and a small fraction of infected individuals is assumed to develop the active disease soon after infection. The global stability of a disease-free equilibrium, the persistence of system, and the local stability of endemic equilibrium are discussed. The basic reproductive numbers with different control measures are determined and analyzed, and we give the critical value of probability of successful detection and treatment of infectious individuals. If a treatment only of infectious individuals cannot control TB transmission, the treatment of latent TB individuals should be carried out, and we give the critical value of the probability of treatment of infectious individuals. Numerical simulations are done to demonstrate the complex dynamics of the model.
Highlights
Differential equations and difference equations are widely applied in epidemiological modeling
6 Conclusion and discussion In this paper, we analyze a class of discrete SLIS models with direct progression and chemoprophylaxis for latent TB individuals
We study the global stability of the disease-free equilibrium as R
Summary
Differential equations and difference equations are widely applied in epidemiological modeling. The fact < p < and the last equation in ( ) implies that N∗ = –p is the unique equilibrium of ( ), and N∗ is globally asymptotically stable, i.e., for any solution N(t) of ( ) with positive initial value N( ), limt→∞ N(t) = N∗ holds. It implies that N(t) is bounded and all solutions starting in the region approach, enter or stay in. The basic reproductive number R is defined mathematically as the spectral radius of the generation matrix in [ ]; each term in R has a clear epidemiological interpretation. /( – p( – γ )) is the average infection period. ( – m)pα/( – p( – km) + pα( – m)) is the proportion of latent individuals that become infectious by natural progression. pβ/( – p( – γ )) is the average of new cases generated by a typical infectious member in the entire infection period, where qpβ/( – p( – γ )) is the average of new cases generated by a typical infectious member who enters the infectious compartment by natural progression in the entire infection period, ( – q)pβ/( – p( – γ )) is the average of new cases generated by a typical infectious member who enters the infectious compartment by direct progression in the entire infection period
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