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### Open Access Journals

We have developed two models of sexually transmitted infections (with and without age structure) to evaluate the cost-efficacy of gender-based vaccination programs in the context of STI control. The first model ignores age structure for qualitative analysis of points (1-3), while the second refined one incorporates the age structure, reflecting the effects of immunity gained from infection of closely related strains (point 4), which is important for HSV-2 vaccination strategies. For both models, we find that the stability of the system and ultimate eradication of the disease depends explicitly on the corresponding reproduction number. We also find that vaccinating females is more cost-effective, providing a greater reduction in disease prevalence in the population and number of infected females of childbearing age. This result is counter-intuitive since vaccinating super-transmitters (males) over sub-transmitters (females) usually has the greatest impact on disease prevalence. Sensitivity analysis is implemented to investigate how the parameters affect the control reproduction numbers and infectious population sizes.

*I. scapularis*ticks. The wide geographic breeding range of

*I. scapularis*-carrying migratory birds is consistent with the widespread geographical occurrence of

*I. scapularis*in Canada. However, how important migratory birds from the United States are for the establishment and the stable endemic transmission cycle of Lyme disease in Canada remains an issue of theoretical challenge and practical significance. In this paper, we design and analyze a periodic model of differential equations with a forcing term modeling the annual bird migration to address the aforementioned issue. Our results show that ticks can establish in any migratory bird stopovers and breeding sites. Moreover, bird-transported ticks may increase the probability of

*B. burgdorferi*establishment in a tick-endemic habitat.

Ticks, including the *Ixodes ricinus* and *Ixodes scapularis* hard tick species, are regarded as the most common arthropod vectors of both human and animal diseases in Europe and the United States capable of transmitting a large number of bacteria, viruses and parasites. Since ticks in larval and nymphal stages share the same host community which can harbor multiple pathogens, they may be co-infected with two or more pathogens, with a subsequent high likelihood of co-transmission to humans or animals. This paper is devoted to the modeling of co-infection of tick-borne pathogens, with special focus on the co-infection of *Borrelia burgdorferi* (agent of Lyme disease) and *Babesia microti* (agent of human babesiosis). Considering the effect of co-infection, we illustrate that co-infection with *B. burgdorferi* increases the likelihood of *B. microti* transmission, by increasing the basic reproduction number of *B. microti* below the threshold smaller than one to be possibly above the threshold for persistence. The study confirms a mechanism of the ecological fitness paradox, the establishment of *B. microti* which has weak fitness (basic reproduction number less than one). Furthermore, co-infection could facilitate range expansion of both pathogens.

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