Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission

A deterministic ordinary differential equation model for the dynamics of malaria transmission that explicitly integrates the demography and life style of the malaria vector and its interaction with the human population is developed and analyzed. The model is different from standard malaria transmission models in that the vectors involved in disease transmission are those that are questing for human blood. Model results indicate the existence of nontrivial disease free and endemic steady states, which can be driven to instability via a Hopf bifurcation as a parameter is varied in parameter space. Our model therefore captures oscillations that are known to exist in the dynamics of malaria transmission without recourse to external seasonal forcing. Additionally, our model exhibits the phenomenon of backward bifurcation. Two threshold parameters that can be used for purposes of control are identified and studied, and possible reasons why it has been difficult to eradicate malaria are advanced.     

Effectiveness of zooprophylaxis in malaria control: a theoretical inquiry, with a model for mosquito populations with two bloodmeal hosts

A model for a vector mosquito population with two bloodmeal hosts (man and a domestic animal) was developed to study the influences of domestic animals on the frequency of mosquito bites on man and the endemicity of human malaria. The vector population model, including blood-feeding success in the adult stage (depending on host density and biting efficiency) and density-dependent regulation in the larval stage, was combined with the Ross-Macdonald malaria transmission model. Model analyses suggested that introduction of domestic animals easily fed upon by mosquitoes increases mosquito density and, in some situations, frequency of mosquito bites on man and the infection rate of malaria through increased success of blood-feeding. Extinction of malaria was predicted only when an extremely large number of easily accessible (as compared to man) domestic animals are introduced. Limitations in the concept of zooprophylaxis and problems of livestock management in malaria control are discussed.     

A simple SIS epidemic model with a backward bifurcation

It is shown that an SIS epidemic model with a non-constant contact rate may have multiple stable equilibria, a backward bifurcation and hysteresis. The consequences for disease control are discussed. The model is based on a Volterra integral equation and allows for a distributed infective period. The analysis includes both local and global stability of equilibria.     

New repellent effective against African malaria mosquito Anopheles gambiae: implications for vector control

Anopheles gambiae Giles sensu stricto (Diptera: Culicidae) is a vector for Plasmodium, the causative agent of malaria. Current control strategies to reduce the impact of malaria focus on reducing the frequency of mosquito attacks on humans, thereby decreasing Plasmodium transmission. A need for new repellents effective against Anopheles mosquitoes has arisen because of changes in vector behaviour as a result of control strategies and concern over the health impacts of current repellents. The response of A. gambiae to potential repellents was investigated through an electroantennogram screen and the most promising of these candidates (1‐allyloxy‐4‐propoxybenzene, 3c {3,6}) chosen for behavioural testing. An assay to evaluate the blood‐host seeking behaviour of A. gambiae towards a simulated host protected with this repellent was then performed. The compound 3c {3,6} was shown to be an effective repellent, causing mosquitoes to reduce their contact with a simulated blood‐host and probe less at the host odour. Thus, 3c {3,6} may be an effective repellent for the control of A. gambiae.     

Backward bifurcation and oscillations in a nested immuno-eco-epidemiological model

This paper introduces a novel partial differential equation immuno-eco-epidemiological model of competition in which one species is affected by a disease while another can compete with it directly and by lowering the first species' immune response to the infection, a mode of competition termed stress-induced competition. When the disease is chronic, and the within-host dynamics are rapid, we reduce the partial differential equation model (PDE) to a three-dimensional ordinary differential equation (ODE) model. The ODE model exhibits backward bifurcation and sustained oscillations caused by the stress-induced competition. Furthermore, the ODE model, although not a special case of the PDE model, is useful for detecting backward bifurcation and oscillations in the PDE model. Backward bifurcation related to stress-induced competition allows the second species to persist for values of its invasion number below one. Furthermore, stress-induced competition leads to destabilization of the coexistence equilibrium and sustained oscillations in the PDE model. We suggest that complex systems such as this one may be studied by appropriately designed simple ODE models.     

Progression age enhanced backward bifurcation in an epidemic model with super-infection

 We consider a model for a disease with a progressing and a quiescent exposed class and variable susceptibility to super-infection. The model exhibits backward bifurcations under certain conditions, which allow for both stable and unstable endemic states when the basic reproduction number is smaller than one. Received: 11 October 2001 / Revised version: 17 September 2002 / Published online: 17 January 2003 Present address: Department of Biological Statistics and Computational Biology, 434 Warren Hall, Cornell University, Ithaca, NY 14853-7801 This author was visiting Arizona State University when most of the research was done. Research partially supported by NSF grant DMS-0137687. This author's research was partially supported by NSF grant DMS-9706787. Key words or phrases: Backward bifurcation – Multiple endemic equilibria – Alternating stability – Break-point density – Super-infection – Dose-dependent latent period – Progressive and quiescent latent stages – Progression age structure – Threshold type disease activation – Operator semigroups – Hille-Yosida operators – Dynamical systems – Persistence – Global compact attractor     

Influence of backward bifurcation in a model of hepatitis B and C viruses

In the 1990s, liver transplantation for hepatitis B and C virus (HBV and HCV) related-liver diseases was a very controversial issue since recurrent infection of the graft was inevitable. Significant progress has been made in the prophylaxis and treatment of recurrent hepatitis B/C (or HBV/HCV infection) after liver transplantation. In this paper, we propose a mathematical model of ordinary differential equations describing the dynamics of the HBV/HCV and its interaction with both liver and blood cells. A single model is used to describe infection of either virus since the dynamics in-host (infected of the liver) are similar. Analyzing the model, we observe that the system shows either a transcritical or a backward bifurcation. Explicit conditions on the model parameters are given for the backward bifurcation to be present. Consequently, we investigate possible factors that are responsible for HBV/HCV infection and assess control strategies to reduce HBV/HCV reinfection and improve graft survival after liver transplantation.     

Large-scale use of mosquito larval source management for malaria control in Africa: a cost analysis

Background

At present, large-scale use of two malaria vector control methods, long-lasting insecticidal nets (LLINs) and indoor residual spraying (IRS) is being scaled up in Africa with substantial funding from donors. A third vector control method, larval source management (LSM), has been historically very successful and is today widely used for mosquito control globally, except in Africa. With increasing risk of insecticide resistance and a shift to more exophilic vectors, LSM is now under re-evaluation for use against afro-tropical vector species. Here the costs of this intervention were evaluated.

Methods

The 'ingredients approach' was used to estimate the economic and financial costs per person protected per year (pppy) for large-scale LSM using microbial larvicides in three ecologically diverse settings: (1) the coastal metropolitan area of Dar es Salaam in Tanzania, (2) a highly populated Kenyan highland area (Vihiga District), and (3) a lakeside setting in rural western Kenya (Mbita Division). Two scenarios were examined to investigate the cost implications of using alternative product formulations. Sensitivity analyses on product prices were carried out.

Results

The results show that for programmes using the same granular formulation larviciding costs the least pppy in Dar es Salaam (US$0.94), approximately 60% more in Vihiga District (US$1.50) and the most in Mbita Division (US$2.50). However, these costs are reduced substantially if an alternative water-dispensable formulation is used; in Vihiga, this would reduce costs to US$0.79 and, in Mbita Division, to US$1.94. Larvicide and staff salary costs each accounted for approximately a third of the total economic costs per year. The cost pppy depends mainly on: (1) the type of formulation required for treating different aquatic habitats, (2) the human population density relative to the density of aquatic habitats and (3) the potential to target the intervention in space and/or time.

Conclusion

Costs for LSM compare favourably with costs for IRS and LLINs, especially in areas with moderate and focal malaria transmission where mosquito larval habitats are accessible and well defined. LSM presents an attractive tool to be integrated in ongoing malaria control effort in such settings. Further data on the epidemiological health impact of larviciding is required to establish cost effectiveness.     

Revisiting the basic reproductive number for malaria and its implications for malaria control

The prospects for the success of malaria control depend, in part, on the basic reproductive number for malaria, R0. Here, we estimate R0 in a novel way for 121 African populations, and thereby increase the number of R0 estimates for malaria by an order of magnitude. The estimates range from around one to more than 3,000. We also consider malaria transmission and control in finite human populations, of size H. We show that classic formulas approximate the expected number of mosquitoes that could trace infection back to one mosquito after one parasite generation, Z0(H), but they overestimate the expected number of infected humans per infected human, R0(H). Heterogeneous biting increases R0 and, as we show, Z0(H), but we also show that it sometimes reduces R0(H); those who are bitten most both infect many vectors and absorb infectious bites. The large range of R0 estimates strongly supports the long-held notion that malaria control presents variable challenges across its transmission spectrum. In populations where R0 is highest, malaria control will require multiple, integrated methods that target those who are bitten most. Therefore, strategic planning for malaria control should consider R0, the spatial scale of transmission, human population density, and heterogeneous biting.     

Delayed action insecticides and their role in mosquito and malaria control

There is considerable interest in the management of insecticide resistance in mosquitoes. One possible approach to slowing down the evolution of resistance is to use late-life-acting (LLA) insecticides that selectively kill only the old mosquitoes that transmit malaria, thereby reducing selection pressure favoring resistance. In this paper we consider an age-structured compartmental model for malaria with two mosquito strains that differ in resistance to insecticide, using an SEI approach to model malaria in the mosquitoes and thereby incorporating the parasite developmental times for the two strains. The human population is modeled using an SEI approach. We consider both conventional insecticides that target all adult mosquitoes, and LLA insecticides that target only old mosquitoes. According to linearised theory the potency of the insecticide affects mainly the speed of evolution of resistance. Mutations that confer resistance can also affect other parameters such as mean adult life span and parasite developmental time. For both conventional and LLA insecticides the stability of the malaria-free equilibrium, with only the resistant mosquito strain present, depends mainly on these other parameters. This suggests that the main long term role of an insecticide could be to induce genetic changes that have a desirable effect on a vital parameter such as adult life span. However, when this equilibrium is unstable, numerical simulations suggest that a potent LLA insecticide can slow down the spread of malaria in humans but that the timing of its action is very important.     

Patterns of nucleotide and haplotype diversity at ICAM-1 across global human populations with varying levels of malaria exposure

Malaria is one of the strongest selective pressures in recent human evolution. African populations have been and continue to be at risk for malarial infections. However, few studies have re-sequenced malaria susceptibility loci across geographically and genetically diverse groups in Africa. We examined nucleotide diversity at Intercellular adhesion molecule-1 (ICAM-1), a malaria susceptibility candidate locus, in a number of human populations with a specific focus on diverse African ethnic groups. We used tests of neutrality to assess whether natural selection has impacted this locus and tested whether SNP variation at ICAM-1 is correlated with malaria endemicity. We observe differing patterns of nucleotide and haplotype variation in global populations and higher levels of diversity in Africa. Although we do not observe a deviation from neutrality based on the allele frequency distribution, we do observe several alleles at ICAM-1, including the ICAM-1 ^Kilifi allele, that are correlated with malaria endemicity. We show that the ICAM-1 ^Kilifi allele, which is common in Africa and Asia, exists on distinct haplotype backgrounds and is likely to have arisen more recently in Asia. Our results suggest that correlation analyses of allele frequencies and malaria endemicity may be useful for identifying candidate functional variants that play a role in malaria resistance and susceptibility.     

Observance of period-doubling bifurcation and chaos in an autonomous ODE model for malaria with vector demography

We illustrate that an autonomous ordinary differential equation model for malaria transmission can exhibit period-doubling bifurcations leading to chaos when ecological aspects of malaria transmission are incorporated into the model. In particular, when demography, feeding, and reproductive patterns of the mosquitoes that transmit the malaria-causing parasite are explicitly accounted for, the resulting model exhibits subcritical bifurcations, period-doubling bifurcations, and chaos. Vectorial and disease reproduction numbers that regulate the size of the vector population at equilibrium and the endemicity of the malaria disease, respectively, are identified and used to simulate the model to show the different bifurcations and chaotic dynamics. A subcritical bifurcation is observed when the disease reproduction number is less than unity. This highlights the fact that malaria control efforts need to be long lasting and sustained to drive the infectious populations to levels below the associated saddle-node bifurcation point at which control is feasible. As the disease reproduction number increases beyond unity, period-doubling cascades that develop into chaos closely followed by period-halving sequences are observed. The appearance of chaos suggests that characterization of the physiological status of disease vectors can provide a pathway toward understanding the complex phenomena that are known to characterize the dynamics of malaria and other indirectly transmitted infections of humans. To the best of our knowledge, there is no known unforced continuous time deterministic host-vector transmission malaria model that has been shown to exhibit chaotic dynamics. Our results suggest that malaria data may need to be critically examined for complex dynamics.     

Forward hysteresis and backward bifurcation caused by culling in an avian influenza model

The emerging threat of a human pandemic caused by the H5N1 avian influenza virus strain magnifies the need for controlling the incidence of H5N1 infection in domestic bird populations. Culling is one of the most widely used control measures and has proved effective for isolated outbreaks. However, the socio-economic impacts of mass culling, in the face of a disease which has become endemic in many regions of the world, can affect the implementation and success of culling as a control measure. We use mathematical modeling to understand the dynamics of avian influenza under different culling approaches. We incorporate culling into an SI model by considering the per capita culling rates to be general functions of the number of infected birds. Complex dynamics of the system, such as backward bifurcation and forward hysteresis, along with bi-stability, are detected and analyzed for two distinct culling scenarios. In these cases, employing other control measures temporarily can drastically change the dynamics of the solutions to a more favorable outcome for disease control.     

Malaria mortality rates in South Asia and in Africa: implications for malaria control

Malaria mortality in human populations varies greatly under different circumstances. The intense malaria transmission conditions found in many parts of tropical Africa, the much lower malaria inoculation rates currently sustained in areas of southern Asia, and the epidemic outbreaks of malaria occasionally seen on both continents, present highly contrasting patterns of malaria-related mortality. Here Harsha Alles, Kamini Mendis and Richard Carter examine malaria-related mortality under different circumstances and discuss implications for the management of malaria in these settings. They emphasize the power of rapid case treatment to save lives at risk under virtually all circumstances of malaria transmission.     

Blood-feeding behaviour of the malarial mosquito Anopheles arabiensis: implications for vector control

Feeding behaviour of the malaria vector Anopheles arabiensis Patton (Diptera: Culicidae) was monitored for 12 months (March 2003-February 2004) in the Konso District of southern Ethiopia (5 degrees 15'N, 37 degrees 28'E). More than 45 000 An. arabiensis females were collected by host-baited sampling methods (light-traps, human landing catches, cattle-baited traps) and from resting sites (huts and pit shelters). In the village of Fuchucha, where the ratio of cattle : humans was 0.6 : 1, 51% of outdoor-resting mosquitoes and 66% of those collected indoors had fed on humans, human baits outdoors caught > 2.5 times more mosquitoes than those indoors and the mean catch of mosquitoes from pit shelters was about five times that from huts. Overall, the vast majority of feeding and resting occurred outdoors. In the cattle camps of Konso, where humans slept outdoors close to their cattle, approximately 46% of resting mosquitoes collected outdoors had fed on humans despite the high cattle : human ratio (17 : 1). In both places, relatively high proportions of bloodmeals were mixed cow + human: 22-25% at Fuchucha and 37% in the cattle camps. Anthropophily was also gauged experimentally by comparing the numbers of mosquitoes caught in odour-baited entry traps baited with either human or cattle odour. The human-baited trap caught about five times as many mosquitoes as the cattle-baited one. Notwithstanding the potential pitfalls of using standard sampling devices to analyse mosquito behaviour, the results suggest that the An. arabiensis population is inherently anthropophagic, but this is counterbalanced by exophagic and postprandial exophilic tendencies. Consequently, the population feeds sufficiently on humans to transmit malaria (sporozoite rates: 0.3% for Plasmodium falciparum and 0.5% for P. vivax, by detection of circumsporozoite antigen) but also takes a high proportion of meals from non-human hosts, with 59-91% of resting mosquitoes containing blood from cattle. Hence, classical zooprophylaxis is unlikely to have a significant impact on the malaria vectorial capacity of An. arabiensis in Konso, whereas treating cattle with insecticide might do.     

Insecticide treated mosquito nets for malaria control in India-experience from a tribal area on operational feasibility and uptake

The study assessed the operational feasibility and acceptability of insecticide-treated mosquito nets (ITNs) in one Primary Health Centre (PHC) in a falciparum malaria endemic district in the state of Orissa, India, where 74% of the people are tribes and DDT indoor residual spraying had been withdrawn and ITNs introduced by the National Vector Borne Disease Control Programme. To a population of 63,920, 24,442 ITNs were distributed free of charge through 101 treatment centers during July-August 2002. Interview of 1,130, 1,012 and 126 respondents showed that the net use rates were 80%, 74% and 55% in the cold, rainy and summer seasons, respectively. Since using ITNs, 74.5-76.6% of the respondents observed reduction of mosquito bites and 7.2-32.1% reduction of malaria incidence; 37% expressed willingness to buy ITNs if the cost was lower and they were affordable. Up to ten months post-treatment, almost 100% mortality of vector mosquitoes was recorded on unwashed and washed nets (once or twice). Health workers re-treated the nets at the treatment centers eight months after distribution on a cost-recovery basis. The coverage reported by the PHC was only 4.2%, mainly because of unwillingness of the people to pay for re-treatment and to go to the treatment centers from their villages. When the re-treatment was continued at the villages involving personnel from several departments, the coverage improved to about 90%.Interview of 126 respondents showed that among those who got their nets re-treated, 81.4% paid cash for the re-treatment and the remainder were reluctant to pay. Majority of those who paid said that they did so due to the fear that if they did not do so they would lose benefits from other government welfare schemes. The 2nd re-treatment was therefore carried out free of charge nine months after the 1st re-treatment and thus achieved coverage of 70.4%. The study showed community acceptance to use ITNs as they perceived the benefit. Distribution and re-treatment of nets was thus possible through the PHC system, if done free of charge and when personnel from different departments, especially those at village level, were involved.     

Genetic and dietary adaptation to malaria in human populations

Plasmodial invasion places a severe oxidant stress on parasitized erythrocytes which can result in red cell damage and removal within the reticuloendothelial system or lysis, thus interrupting the parasitic cycle. The basis of a number of genetic adaptations to malaria--including the hemoglobin variants, the thalassemias, and glucose-6-phosphate dehydrogenase deficiency--is an increased sensitivity of the variant erythrocytes to the oxidant stress of plasmodial parasitization. It is suggested that dietary adaptations of traditional cusines in human populations living in areas where malaria is endemic augment this oxidant stress. It appears that there are three components of this adaptive dietary pattern in most tropical populations: the consumption of 'oxidant fuels', moderate to high iron intake, and limitation of dietary antioxidant intake or storage. It is argued that this dietary pattern maximizes iron-mediated free radical production in parasitized erythrocytes and thus provides a form of diet-mediated antimalarial prophylaxis. African pastoral populations that are heavy consumers of milk appear to manifest a different adaptive pattern involving low intakes of para-aminobenzoic acid, vitamin E, and iron. Periodic food restriction may also contribute to this antimalarial effect.     

Dynamics and bifurcation of a model for hormonal control of the menstrual cycle with inhibin delay

A system of 13 ordinary differential equations with 42 parameters is presented to model hormonal regulation of the menstrual cycle. For an excellent fit to clinical data, the model requires a 36 h time delay for the effect of inhibin on the synthesis of follicle stimulating hormone. Biological and mathematical reasons for this delay are discussed. Bifurcations with respect to changes in three important parameters are examined. One parameter represents the level of estradiol adequate for significant synthesis of luteinizing hormone. Bifurcation diagrams with respect to this parameter reveal an interval of parameter values for which a unique stable periodic solution exists and this solution represents a menstrual cycle during which ovulation occurs. The second parameter measures mass transfer between the first two stages of ovarian development and is indicative of healthy follicular growth. The third parameter is the time delay. Changes in the second parameter and the time delay affect the size of the uniqueness interval defined with respect to the first parameter. Saddle-node, transcritical and degenerate Hopf bifurcations are studied.