Re-infection-induced backward bifurcation in the transmission dynamics of Chlamydia trachomatis

Abstract

A new two-group deterministic model for Chlamydia trachomatis is designed and analyzed to gain insights into its transmission dynamics. The model is shown to exhibit the phenomenon of backward bifurcation, where a stable disease-free equilibrium (DFE) co-exists with one or more stable endemic equilibria when the associated reproduction number is less than unity. It is further shown that the backward bifurcation dynamic is caused by the re-infection of individuals who recovered from the disease. The epidemiological implication of this result is that the classical requirement of the reproduction number being less than unity becomes only a necessary, but not sufficient, condition for disease elimination. The basic model is extended to incorporate the use of treatment for infectious individuals (including those who show disease symptoms and those who do not). Rigorous analysis of the treatment model reveals that the use of treatment could have positive or negative population-level impact, depending on the sign of a certain epidemiological threshold. The treatment model is used to evaluate various treatment strategies, namely treating every infected individual showing symptoms of Chlamydia (universal strategy), treating only infectious males showing Chlamydia symptoms (male-only strategy) and treating only infectious females showing symptoms of Chlamydia (female-only strategy). Numerical simulations show that the implementation of the male-only or female-only strategy can induce an indirect benefit of saving new cases of Chlamydia infection in the opposite sex. Further, the universal strategy gives the highest reduction in the cumulative number of new cases of infection.

Cross-immunity-induced backward bifurcation for a model of transmission dynamics of two strains of influenza

“…In other words, the solutions of the model (1) with positive initial data will remain positive for all t ≥ 0. The following result can be proven (see, for instance, [8,22,24,37,40]) …”

Global Dynamics of Modeling Flocculation of Microorganism

“…In this section, the center manifold theory is used on Model (4) to obtain the rigorous result (see, for example, [40,47,48]). …”

“…Using the methods similar to [47], we can give the sufficient conditions of the existence of the positive equilibria of Model (4). The following results (Theorem 3) follow from the various possibilities enumerated in Table A1 (see Appendix A): (4) has a unique positive equilibrium if R 0 < 1 and Conditions (9) hold and whenever Cases 1, 9, 13, 15 and 16 in Table A1 are satisfied; (ii) Model (4) could have more than one positive equilibrium if R 0 < 1 and Conditions (9) hold and whenever Cases 2-8, 10-12 and 14 in Table A1 are satisfied; (iii) Model (4) could have five positive equilibria at most if R 0 > 1 and Conditions (9) hold and whenever Cases 1-16 in Table A1 are satisfied.…”

“…The existence of multiple positive equilibria of Model (4) when R 0 < 1 (shown in Table A1; see Appendix A) indicates the possibility of the existence of backward bifurcation (see, for example, [40,47,48]), where the stable boundary equilibrium co-exists with a stable positive equilibrium. This can be explored below via numerical simulations (see Figure 2a,b).…”

“…In this section, let us apply the Descartes rule of signs to the classifications of the positive roots of f (X * ) [47]. Let m represent the number of sign changes of the coefficients a, b, c, d, e, f of f (X * ) and n represent the number of the positive roots.…”