Theoretical analysis of mixed Plasmodium malariae and Plasmodium falciparum infections with partial cross-immunity

Abstract

A deterministic model for assessing the dynamics of mixed species malaria infections in a human population is presented to investigate the effects of dual infection with Plasmodium malariae and Plasmodium falciparum. Qualitative analysis of the model including positivity and boundedness is performed. In addition to the disease free equilibrium, we show that there exists a boundary equilibrium corresponding to each species. The isolation reproductive number of each species is computed as well as the reproductive number of the full model. Conditions for global stability of the disease free equilibrium as well as local stability of the boundary equilibria are derived. The model has an interior equilibrium which exists if at least one of the isolation reproductive numbers is greater than unity. Among the interesting dynamical behaviours of the model, the phenomenon of backward bifurcation where a stable boundary equilibrium coexists with a stable interior equilibrium, for a certain range of the associated invasion reproductive number less than unity is observed. Results from analysis of the model show that, when cross-immunity between the two species is weak, there is a high probability of coexistence of the two species and when cross-immunity is strong, competitive exclusion is high. Further, an increase in the reproductive number of species i increases the stability of its boundary equilibrium and its ability to invade an equilibrium of species j. Numerical simulations support our analytical conclusions and illustrate possible behaviour scenarios of the model.

Introduction

Malaria is the world's most prevalent vector borne disease and it still remains among the most devastating diseases occurring in the world. It represents 10% of Africa's overall disease burden (World Malaria Report, 2005). Children under five years of age are particularly vulnerable to Plasmodium falciparum infection. There were 881 000 [610 000–1 212 000] estimated malaria deaths in 2006, of which 91% were in Africa and 85% were of children under the age of five (World Malaria Report, 2008). The symptoms and epidemiological manifestations from this micro-organism are highly variable and geographically determined by a balanced interplay of the parasite with human host and vector. Despite persistent control efforts set up since the end of the fifties, the emergence of resistance of the parasite to drugs and of the mosquito vector to insecticides, combined with the difficulties in implementing and maintaining effective control schemes have led to a resurgence of the disease in many parts of the world (Hayton and Su, 2008, Faulde et al., 2007). In addition, despite over a century of research, much of the biology of malaria parasites and how they interact with their human host and with each other remains to be discovered (Mayxay et al., 2004).

Among the numerous Plasmodium species that infect reptiles, birds and mammals, four of them are human specific: P. falciparum, Plasmodium vivax, Plasmodium malariae and Plasmodium ovale. P. falciparum is the most virulent agent responsible for 200–300 million infections and 1–3 million deaths annually, mainly in Africa (Volkman et al., 2001). Mixed-species infections are frequently observed, and almost all combinations of species have been found within human populations and individuals (Mason et al., 1999). Mixed-species malaria infections are often not recognized or underestimated (Mayxay et al., 2004). P. falciparum and P. malariae are malaria species that occur endemically in many parts of sub-Saharan Africa (Bousema et al., 2008). The co-occurrence of P. falciparum and P. malariae is often higher than would be expected on the basis of their individual parasite prevalence, and one parasite species may influence the infection dynamics of the other (Arez et al., 2003). Experimental studies indicated that different parasite species seem to interact, affecting mortality, pathology and infection dynamics (Richie, 1988, Collins and Jeffery, 1999).

At least seven species of Anopheles have been shown to carry more than one species of human Plasmodium in the field (McKenzie and Bossert, 1999) and all four malaria species of humans can be carried by Anopheles gambiae (Fonenille et al., 1992). Even though numerous articles are published every year about the parasite and the disease (Ollomo et al., 2009), a few mathematical studies have been devoted to analysing the effects of mixed infections within a host (Mason and McKenzie, 1999, Mason et al., 1999) by considering the blood-stage population dynamics of a dual infection with P. malariae and P. falciparum and P. vivax and P. falciparum. To the best of our knowledge, there are no mathematical models developed up to date for fully assessing the impact of mixed malaria infections in a human population.

Interaction between different human Plasmodium species when simultaneously infecting the same host (vertebrate or vector) also may have an effect on the dynamics of transmission of each species, but such studies are scarce. In this work, we explore the dynamics of dual malaria infections with P. falciparum and P. malariae in a naive human population by incorporating immunity raised by one species, or stage of parasite which also acts against the other species or stage (Richie, 1988). A comprehensive qualitative assessment of the population-level implications of the dual infection with the two species in a community is carried out. The paper is organized as follows. In the following section, we formulate a deterministic mathematical model for mixed species malaria infections, which incorporates the key epidemiological and biological features of malaria infection. In Section 3, we analyse the model by computing the reproductive numbers of the two species in isolation, invasion reproductive numbers and determining stability of the boundary equilibria. Conditions for existence of an interior equilibrium and its local stability are established in this section. Numerical simulations follow in Section 4 and finally, in Section 5, we summarize and discuss the results of the study.