Xiunan Wang

Rabies among dogs remains a considerable risk to humans and constitutes a serious public health concern in many parts of the world. Conventional mathematical models for rabies typically assume homogeneous environments, with a standard diffusion term for the population of rabid animals. It has recently been recognized, however, that spatial heterogeneity plays an important role in determining spatial patterns of rabies and the cost-effectiveness of vaccinations. In this paper, we develop a spatially heterogeneous dog rabies model by using the \(\theta \)-diffusion equation, where \(\theta \) reflects the way individual dogs make movement decisions in the underlying random walk. We numerically investigate the dynamics of the model in three diffusion cases: homogeneous, city-wild, and Gaussian-type. We find that the initial conditions affect whether traveling waves or epizootic waves can be observed. However, different initial conditions have little impact on steady-state solutions. An “active” interface is observed between city and wild regions, with a “ridge” on the city side and a “valley” on the wild side for the infectious dog population. In addition, the progressing speed of epizootic waves changes in heterogeneous environments. It is impossible to eliminate rabies in the entire spatial domain if vaccination is focused only in the city region or only in the wild region. When a seasonal transmission is incorporated, the dog population size approaches a positive time-periodic spatially heterogeneous state eventually.

Recent experimental study suggests that the engineered symbiotic bacteria Serratia AS1 may provide a novel, effective and sustainable biocontrol of malaria. These recombinant bacteria have been shown to be able to rapidly disseminate throughout mosquito populations and to efficiently inhibit development of malaria parasites in mosquitoes in controlled laboratory experiments. In this paper, we develop a climate-based malaria model which involves both vertical and horizontal transmissions of the engineered Serratia AS1 bacteria in mosquito population. We show that the dynamics of the model system is totally determined by the vector reproduction ratio \(R_\mathrm{v}\), and the basic reproduction ratio \(R_0\). If \(R_\mathrm{v}\le 1\), then the mosquito-free state is globally attractive. If \(R_\mathrm{v}>1\) and \(R_0\le 1\), then the disease-free periodic solution is globally attractive. If \(R_\mathrm{v}>1\) and \(R_0>1\), then the positive periodic solution is globally attractive. Numerically, we verify the obtained analytic result and evaluate the effects of releasing the engineered Serratia AS1 bacteria in field by conducting a case study for Douala, Cameroon. We find that ideally, by using Serratia AS1 alone, it takes at least 25 years to eliminate malaria from Douala. This implies that continued long-term investment is needed in the fight against malaria and confirms the necessity of integrating multiple control measures.

Insecticide-treated bed nets (ITNs) are among the most important and effective intervention measures against malaria. In order to investigate the impact of bed net use on disease control, we formulate a periodic vector-bias malaria model incorporating the juvenile stage of mosquitoes and the use of ITNs. We derive the vector reproduction ratio \(R_v\) and the basic reproduction ratio \(R_0\). We show that the global dynamics of the model is completely determined by these two reproduction ratios. More precisely, the mosquito-free periodic solution is globally attractive if \(R_v<1\); the unique disease-free periodic solution is globally attractive if \(R_v>1\) and \(R_0<1\); and the model admits a unique positive periodic solution and it is globally attractive if \(R_v>1\) and \(R_0>1\). Numerically, we study the malaria transmission case in Port Harcourt, Nigeria. Our findings show that the use of ITNs has a positive effect on reducing \(R_0\), and that malaria may be eliminated from this area if over 75% of the human population were to use ITNs. The simulation about the long term behavior of solutions has good agreement with the obtained analytic result. Moreover, we find that the ignorance of the vector-bias effect may result in underestimation of the basic reproduction ratio \(R_0\). Another notable result is that the infection risk would be underestimated if the basic reproduction ratio \([R_0]\) of the time-averaged autonomous system were used.

Malaria is an infectious disease caused by Plasmodium parasites and is transmitted among humans by female Anopheles mosquitoes. Climate factors have significant impact on both mosquito life cycle and parasite development. To consider the temperature sensitivity of the extrinsic incubation period (EIP) of malaria parasites, we formulate a delay differential equations model with a periodic time delay. We derive the basic reproduction ratio \(R_0\) and establish a threshold type result on the global dynamics in terms of \(R_0\), that is, the unique disease-free periodic solution is globally asymptotically stable if \(R_01\). Numerically, we parameterize the model with data from Maputo Province, Mozambique, and simulate the long-term behavior of solutions. The simulation result is consistent with the obtained analytic result. In addition, we find that using the time-averaged EIP may underestimate the basic reproduction ratio.