DCDS-B
Cost-effectiveness evaluation of gender-based vaccination programs against sexually transmitted infections
Jane M. Heffernan Yijun Lou Marc Steben Jianhong Wu
The ultimate goal of a vaccination program is to interrupt pathogen transmission so as to eradicate the disease from the population in the future, and/or to decrease morbidity and mortality due to the disease in the short term. For sexually transmitted infections (STI) the determination of an optimal vaccination program is not straightforward since (1) the transmission probabilities between two different sexes are normally unequal (weighted to a greater probability from males to females than vice versa), (2) demographic parameters between the two sexes are unequal, (3) the prevalence of disease in one sex may have a greater impact on the morbidity and mortality of the next generation (transmission to the neonate) and, (4) the existence of pathogens closely related to the STI in question (i.e. herpes - HSV-1 vs. HSV-2, different strains of Chlamydia trachomatis, different strains of Neisseria which cause Gonorrhea, and others) may induce immunity in individuals that render a vaccine ineffective.
    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.
keywords: vaccine mathematical model Sexually transmitted infection cost-effectiveness.
DCDS-B
Threshold dynamics in a time-delayed periodic SIS epidemic model
Yijun Lou Xiao-Qiang Zhao
The global dynamics of a periodic SIS epidemic model with maturation delay is investigated. We first obtain sufficient conditions for the single population growth equation to admit a globally attractive positive periodic solution. Then we introduce the basic reproduction ratio $\mathcal{R}_0$ for the epidemic model, and show that the disease dies out when $\mathcal{R}_0<1$, and the disease remains endemic when $\mathcal{R}_0>1$. Numerical simulations are also provided to confirm our analytic results.
keywords: Periodic epidemic model Periodic solutions Uniform persistence. Basic reproduction ratio Maturation delay
MBE
Stability and persistence in ODE models for populations with many stages
Guihong Fan Yijun Lou Horst R. Thieme Jianhong Wu
A model of ordinary differential equations is formulated for populations which are structured by many stages. The model is motivated by ticks which are vectors of infectious diseases, but is general enough to apply to many other species. Our analysis identifies a basic reproduction number that acts as a threshold between population extinction and persistence. We establish conditions for the existence and uniqueness of nonzero equilibria and show that their local stability cannot be expected in general. Boundedness of solutions remains an open problem though we give some sufficient conditions.
keywords: persistence uniqueness extinction Basic reproduction number boundedness equilibria (existence Lyapunov functions and stability).
DCDS-B
Range expansion of Ixodes scapularis ticks and of Borrelia burgdorferi by migratory birds
Jane M. Heffernan Yijun Lou Jianhong Wu
Recent studies have suggested that the risk of exposure to Lyme disease is emerging in Canada because of the expanding range of 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.
keywords: ticks bird migration Lyme disease spatial expansion periodic system.
MBE
Modeling co-infection of Ixodes tick-borne pathogens
Yijun Lou Li Liu Daozhou Gao

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.

keywords: Co-infection tick-borne pathogens mathematical model
DCDS-B
A periodic Ross-Macdonald model in a patchy environment
Daozhou Gao Yijun Lou Shigui Ruan
Based on the classical Ross-Macdonald model, in this paper we propose a periodic malaria model to incorporate the effects of temporal and spatial heterogeneity on disease transmission. The temporal heterogeneity is described by assuming that some model coefficients are time-periodic, while the spatial heterogeneity is modeled by using a multi-patch structure and assuming that individuals travel among patches. We calculate the basic reproduction number $\mathcal{R}_0$ and show that either the disease-free periodic solution is globally asymptotically stable if $\mathcal{R}_0\le 1$ or the positive periodic solution is globally asymptotically stable if $\mathcal{R}_0>1$. Numerical simulations are conducted to confirm the analytical results and explore the effect of travel control on the disease prevalence.
keywords: basic reproduction number. patch model threshold dynamics Malaria seasonality

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