Okosun K O, Ouifki Rachid, Marcus Nizar
Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa.
Biosystems. 2011 Nov;106(2-3):136-45. doi: 10.1016/j.biosystems.2011.07.006. Epub 2011 Aug 5.
We derive and analyse a deterministic model for the transmission of malaria disease with mass action form of infection. Firstly, we calculate the basic reproduction number, R(0), and investigate the existence and stability of equilibria. The system is found to exhibit backward bifurcation. The implication of this occurrence is that the classical epidemiological requirement for effective eradication of malaria, R(0)<1, is no longer sufficient, even though necessary. Secondly, by using optimal control theory we derive the conditions under which it is optimal to eradicate the disease and examine the impact of a possible combined vaccination and treatment strategy on the disease transmission. When eradication is impossible, we derive the necessary conditions for optimal control of the disease using Pontryagin's Maximum Principle. The results obtained from the numerical simulations of the model show that a possible vaccination combined with effective treatment regime would reduce the spread of the disease appreciably.
我们推导并分析了一个具有感染的质量作用形式的疟疾传播确定性模型。首先，我们计算基本再生数(R(0))，并研究平衡点的存在性和稳定性。发现该系统呈现后向分支。这种情况的含义是，即使有效根除疟疾的经典流行病学要求(R(0)<1)是必要的，但它已不再充分。其次，通过使用最优控制理论，我们推导了根除疾病的最优条件，并研究了可能的联合疫苗接种和治疗策略对疾病传播的影响。当根除不可能时，我们使用庞特里亚金极大值原理推导疾病最优控制的必要条件。从该模型的数值模拟中获得的结果表明，一种可能的疫苗接种与有效的治疗方案相结合将显著减少疾病的传播。
