Research On Dynamics Of Heterogeneous Epidemic Model And Assessment Of Epidemic Control Effect

Heterogeneity plays an important role in the transmission and persistence of many infectious diseases. In this dissertation, a research has been made by using mathematical models on the impact of heterogeneity on the dynamics of many diseases, such as HSV-2, HIV, SARS and malaria. Multi-group model is formulated to describe the diversity and dispersion of the population for these four diseases. The dynamics of these four diseases are discussed. The effects of control measures are assessed further. This dissertation gives a preliminary study on the epidemic dynamics and control with heterogeneity factors. The main idea of this dissertation can be summarized as follows:(1) A9-dimensional ordinary differential equation model is formulated for the trans-mission dynamics of HSV-2. The model incorporates heterosexual interactions in which a single male population and two groups of female populations with different activity levels are considered. The method of global Lyapunov functions as well as the LaSalle Invariance Principle are used to show that the basic reproduction number provides a sharp threshold which completely determines the global dynamics of the model. Then, the method is extended to a general case in which the HSV-2model includes l groups males and n groups of females. The result also shows that the model dynamics is com-pletely determined by the basic reproduction number. That is, in the case when the production number is less than or equal to unity, the disease-free equilibrium is globally asymptotically stable; whereas in the case when the reproduction number is greater than unity, a unique endemic equilibrium is globally asymptotically stable in the interior of the feasible region and the disease will persist at the endemic equilibrium if it is initially present. This gets us a big step closer to one type of mechanism for how HSV-2virus spreads among diverse groups of population. The result also reveals that the basic re-production number can be a good index of the way investigators assess the impacts of epidemic control measures.(2) To more fully understand the epidemiological synergy between HIV and HSV-2, a deterministic compartmental model is formulated that describes the transmission dynamics of both pathogens. Unlike earlier models, ours incorporates sexual structure and heterogeneous mixing between activity groups. Explicit expressions for the reproduction numbers of HSV-2and HIV are obtained via next generation matrices. A systematic qualitative analysis of the system includes the existence of equilibria and local and global behavior. Simulations reinforce these analytical results and demonstrate epidemiological synergy between HSV-2and HIV. In particular, numerical results show that HSV-2favors the invasion of HIV, may dramatically increase the peak and accelerate the time-to-peak HIV prevalence, and almost certainly has exacerbated HIV epidemics. The potential population-level impact of HSV-2on HIV is demonstrated by calculating the fraction of HIV infections attributable to HSV-2and the excess HIV prevalence. The potential impact of HSV-2therapy on HIV control is demonstrated by comparing HIV prevalence with and without therapy. Numerical simulation suggests that HSV-2therapy for male should be given top priority, closely followed by HSV-2therapy for high-risk female and low-risk female.(3) Spital heterogeneity is an important characteristic of the transmission of epi-demics. A multi-regional deterministic compartmental model is proposed to describe the spital heterogeneity and assess the effectiveness and implications of non-pharmaceutical interventions. The reproduction number is determined as the spectral radius of a nonneg-ative matrix product. Comparisons are made using the reproduction number, epidemic peaks and cumulative number of infections and mortality as indexes. Simulation re-sult for SARS epidemics shows that quarantine of suspected cases and isolation of cases with symptom are effective in reducing disease burden for multiple regions. Using entry screening strategy leads to a moderate time delay for epidemic peaks, but is of no help for preventing an epidemic breaking out. Furthermore, the combination epidemic con-trol strategy is investigated for multiple regions by applying optimal control theory. The effects of intra-regional and inter-regional non-pharmaceutical interventions are assessed under the spatial configurations of the four generic meta-population structures with four patches. Numerical simulation results show that a combination epidemic control strat-egy should give priority to intra-regional control interventions, and be supplemented by inter-regional non-pharmaceutical interventions if allowed.(4) Vector control and pharmaceutical treatments are currently the main method-s of malaria control. To assess their impacts on disease transmission and prevalence, sensitivity and optimal control analysis are performed respectively on a mathematical malaria model. Comparisons are made between the result of sensitivity analysis and that of optimal control analysis. The forward-backward sweep method is used the implemen-tation of optimal control numerically. Numerical simulation shows that optimal control strategy is available and cost-efficient. The simulating results further suggests that the results of sensitivity analysis by calculating sensitivity indices can not help policy-makers to formulate a more effective optimal control programme. Numerical result also indicates that vector control is always much more beneficial than other anti-malaria measures in an optimal control programme.