A metapopulation model for malaria with transmission-blocking partial immunity in hosts

A hybrid Lagrangian-Eulerian model for vector-borne diseases

In this paper, a multi-patch and multi-group vector-borne disease model is proposed to study the effects of host commuting (Lagrangian approach) and/or vector migration (Eulerian approach) on disease spread. We first define the basic reproduction number of the model, R 0 , which completely determines the global dynamics of the model system. Namely, ifR 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable, and ifR 0 > 1 , then there exists a unique endemic equilibrium which is globally asymptotically stable. Then, we show that the basic reproduction number has lower and upper bounds which are independent of the host residence times matrix and the vector migration matrix. In particular, nonhomogeneous mixing of hosts and vectors in a homogeneous environment generally increases disease persistence and the basic reproduction number of the model attains its minimum when the distributions of hosts and vectors are proportional. Moreover, R 0 can also be estimated by the basic reproduction numbers of disconnected patches if the environment is homogeneous. The optimal vector control strategy is obtained for a special scenario. In the two-patch and two-group case, we numerically analyze the dependence of the basic reproduction number and the total number of infected people on the host residence times matrix and illustrate the optimal vector control strategy in homogeneous and heterogeneous environments.

Controlling malaria in a population accessing counterfeit antimalarial drugs

A mathematical model is developed for describing malaria transmission in a population consisting of infants and adults and in which there are users of counterfeit antimalarial drugs. Three distinct control mechanisms, namely, effective malarial drugs for treatment and insecticide-treated bednets (ITNs) and indoor residual spraying (IRS) for prevention, are incorporated in the model which is then analyzed to gain an understanding of the disease dynamics in the population and to identify the optimal control strategy. We show that the basic reproduction number, $ R_{0} $, is a decreasing function of all three controls and that a locally asymptotically stable disease-free equilibrium exists when $ R_{0} < 1 $. Moreover, under certain circumstances, the model exhibits backward bifurcation. The results we establish support a multi-control strategy in which either a combination of ITNs, IRS and highly effective drugs or a combination of IRS and highly effective drugs is used as the optimal strategy for controlling and eliminating malaria. In addition, our analysis indicates that the control strategies primarily benefit the infant population and further reveals that a high use of counterfeit drugs and low recrudescence can compromise the optimal strategy.

Relative prevalence-based dispersal in an epidemic patch model

In this paper, we propose a two-patch SIRS model with a nonlinear incidence rate: [Formula: see text] and nonconstant dispersal rates, where the dispersal rates of susceptible and recovered individuals depend on the relative disease prevalence in two patches. In an isolated environment, the model admits Bogdanov-Takens bifurcation of codimension 3 (cusp case) and Hopf bifurcation of codimension up to 2 as the parameters vary, and exhibits rich dynamics such as multiple coexistent steady states and periodic orbits, homoclinic orbits and multitype bistability. The long-term dynamics can be classified in terms of the infection rates [Formula: see text] (due to single contact) and [Formula: see text] (due to double exposures). In a connected environment, we establish a threshold [Formula: see text] between disease extinction and uniform persistence under certain conditions. We numerically explore the effect of population dispersal on disease spread when [Formula: see text] and patch 1 has a lower infection rate, our results indicate: (i) [Formula: see text] can be nonmonotonic in dispersal rates and [Formula: see text] ([Formula: see text] is the basic reproduction number of patch i) may fail; (ii) the constant dispersal of susceptible individuals (or infective individuals) between two patches (or from patch 2 to patch 1) will increase (or reduce) the overall disease prevalence; (iii) the relative prevalence-based dispersal may reduce the overall disease prevalence. When [Formula: see text] and the disease outbreaks periodically in each isolated patch, we find that: (a) small unidirectional and constant dispersal can lead to complex periodic patterns like relaxation oscillations or mixed-mode oscillations, whereas large ones can make the disease go extinct in one patch and persist in the form of a positive steady state or a periodic solution in the other patch; (b) relative prevalence-based and unidirectional dispersal can make periodic outbreak earlier.

Asymmetric host movement reshapes local disease dynamics in metapopulations

Understanding how the movement of individuals affects disease dynamics is critical to accurately predicting and responding to the spread of disease in an increasingly interconnected world. In particular, it is not yet known how movement between patches affects local disease dynamics (e.g., whether pathogen prevalence remains steady or oscillates through time). Considering a set of small, archetypal metapopulations, we find three surprisingly simple patterns emerge in local disease dynamics following the introduction of movement between patches: (1) movement between identical patches with cyclical pathogen prevalence dampens oscillations in the destination while increasing synchrony between patches; (2) when patches differ from one another in the absence of movement, adding movement allows dynamics to propagate between patches, alternatively stabilizing or destabilizing dynamics in the destination based on the dynamics at the origin; and (3) it is easier for movement to induce cyclical dynamics than to induce a steady-state. Considering these archetypal networks (and the patterns they exemplify) as building blocks of larger, more realistically complex metapopulations provides an avenue for novel insights into the role of host movement on disease dynamics. Moreover, this work demonstrates a framework for future predictive modelling of disease spread in real populations.

The Role of Movement Patterns in Epidemic Models on Complex Networks

In this paper, we analyze the influence of the usual movement variables on the spread of an epidemic. Specifically, given two spatial topologies, we can deduce which topology produces less infected individuals. In particular, we determine the topology that minimizes the overall number of infected individuals. It is worth noting that we do not assume any of the common simplifying assumptions in network theory such as all the links have the same diffusion rate or the movement of the individuals is symmetric. Our main conclusion is that the degree of mobility of the population plays a critical role in the spread of a disease. Finally, we derive theoretical insights to management of epidemics.

A model of COVID-19 transmission to understand the effectiveness of the containment measures: application to data from France

The main objective of this paper is to address the following question: are the containment measures imposed by most of the world governments effective and sufficient to stop the epidemic of COVID-19 beyond the lock-down period? In this paper, we propose a mathematical model which allows us to investigate and analyse this problem. We show by means of the reproductive number, that the containment measures appear to have slowed the growth of the outbreak. Nevertheless, these measures remain only effective as long as a very large fraction of population, p, greater than the critical value remains confined. Using French current data, we give some simulation experiments with five scenarios including: (i) the validation of model with p estimated to 93%, (ii) the study of the effectiveness of containment measures, (iii) the study of the effectiveness of the large-scale testing, (iv) the study of the social distancing and wearing masks measures and (v) the study taking into account the combination of the large-scale test of detection of infected individuals and the social distancing with linear progressive easing of restrictions. The latter scenario was shown to be effective at overcoming the outbreak if the transmission rate decreases to 75% and the number of tests of detection is multiplied by three. We also noticed that if the measures studied in our five scenarios are taken separately then the second wave might occur at least as far as the parameter values remain unchanged.

IMPACT OF ADAPTIVE MOSQUITO BEHAVIOR AND INSECTICIDE-TREATED NETS ON MALARIA PREVALENCE

Malaria prevalence in sub-Saharan Africa remains high. Kenya for example, records about 3.5 million new cases and 11 thousand deaths each year.1 Most of these cases and deaths are among children under five. The main control method in malaria endemic regions has been through the use of insecticide-treated nets (ITNs). Although this approach has been fairly successful, the gains are threatened by mosquito-resistance to pyrethroids (insecticides on nets), physical and chemical degradation of ITNs that reduce their efficacy, inconsistent and improper use by humans, etc. We present a model to investigate the effects of ITN use and mosquito-resistance and adaptation to pyrethroids used to treat bed nets on malaria prevalence and control in malaria endemic regions. The model captures the development and loss of resistance to insecticides, the effects of ITN use on malaria control in a setting where proper and consistent use is not guaranteed, as well as differentiated biting of human hosts by resistant and sensitive mosquitoes. Important thresholds, including the basic reproduction number [Formula: see text], and two parameter groupings that are important for disease control and for establishing the existence of endemic equilibria to the model are calculated. Furthermore, a global sensitivity analysis is carried out to identify important parameters such as insecticide treated bed-net coverage, ITN, the maximum biting rate of resistant mosquitoes, etc., that drive the system and that can be targeted for disease control. Threshold levels of ITN coverage and ITN efficacy required for containing the disease are identified and shown to depend on the type of insecticide-resistance. For example, when mosquito-resistance to insecticides is not permanent and is acquired only through recruitment and the efficacy of ITNs is [Formula: see text], about [Formula: see text] net coverage is required to contain malaria. However, for the same ITN efficacy, i.e., [Formula: see text], approximately [Formula: see text] net coverage is required to contain the disease when resistance to insecticides is permanent and is acquired through recruitment and mutation in mosquitoes. The model exhibits a backward bifurcation, which implies that simply reducing [Formula: see text] slightly below unity might not be enough to contain the disease. We conclude that appropriate measures to reduce or eliminate mosquito-resistance to insecticides, ensure that more people in endemic areas own and use ITNs properly, and that the efficacy of these nets remain high most of the time, as well as educating populations in malaria endemic areas on how to keep mosquito densities low and minimize mosquito bites are important for containing malaria.

Prediction of COVID-19 spread by sliding mSEIR observer

The outbreak of COVID-19 has brought unprecedented challenges not only in China but also in the whole world. Thousands of people have lost their lives, and the social operating system has been affected seriously. Thus, it is urgent to study the determinants of the virus and the health conditions in specific populations and to reveal the strategies and measures in preventing the epidemic spread. In this study, we first adopt the long short-term memory algorithm to predict the infected population in China. However, it gives no interpretation of the dynamics of the spread process. Also the long-term prediction error is too large to be accepted. Thus, we introduce the susceptible-exposed-infected-removed (SEIR) model and further the metapopulation SEIR (mSEIR) model to capture the spread process of COVID-19. By using a sliding window algorithm, we suggest that the parameter estimation and the prediction of the SEIR populations are well performed. In addition, we conduct extensive numerical experiments to show the trend of the infected population for several provinces. The results may provide some insight into the research of epidemics and the understanding of the spread of the current COVID-19.

Habitat fragmentation promotes malaria persistence

Based on a Ross-Macdonald type model with a number of identical patches, we study the role of the movement of humans and/or mosquitoes on the persistence of malaria and many other vector-borne diseases. By using a theorem on line-sum symmetric matrices, we establish an eigenvalue inequality on the product of a class of nonnegative matrices and then apply it to prove that the basic reproduction number of the multipatch model is always greater than or equal to that of the single patch model. Biologically, this means that habitat fragmentation or patchiness promotes disease outbreaks and intensifies disease persistence. The risk of infection is minimized when the distribution of mosquitoes is proportional to that of humans. Numerical examples for the two-patch submodel are given to investigate how the multipatch reproduction number varies with human and/or mosquito movement. The reproduction number can surpass any given value whenever an appropriate travel pattern is chosen. Fast human and/or mosquito movement decreases the infection risk, but may increase the total number of infected humans.

A CHOLERA METAPOPULATION MODEL INTERLINKING MIGRATION WITH INTERVENTION STRATEGIES — A CASE STUDY OF ZIMBABWE (2008–2009)

Cholera is a water-borne disease and a major threat to human society affecting about 3–5 million people annually. A considerable number of research works have already been done to understand the disease transmission route and preventive measures in spatial or non-spatial scale. However, how the control strategies are to be linked up with the human migration in different locations in a country are not well studied. The present investigation is carried out in this direction by proposing and analyzing cholera meta-population models. The basic dynamical properties including the domain basic reproduction number are studied. Several important model parameters are estimated using cholera incidence data (2008–2009) and inter-provincial migration data from Census 2012 for the five provinces in Zimbabwe. By defining some migration index, and interlinking these indices with different cholera control strategies, namely, promotion of hand-hygiene and clean water supply and treatment, we carried out an optimal cost effectiveness analysis using optimal control theory. Our analysis suggests that there is no need to provide control measures for all the five provinces, and the control measures should be provided only to those provinces where in-migration flow is moderate. We also observe that such selective control measures which are also cost effective may reduce the overall cases and deaths.

Modeling the Potential Role of Engineered Symbiotic Bacteria in Malaria Control

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 [Formula: see text], and the basic reproduction ratio [Formula: see text]. If [Formula: see text], then the mosquito-free state is globally attractive. If [Formula: see text] and [Formula: see text], then the disease-free periodic solution is globally attractive. If [Formula: see text] and [Formula: see text], 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.

Assessing the effects of daily commuting in two-patch dengue dynamics: A case study of Cali, Colombia

There are many infectious diseases that can be spread by daily commuting of people and dengue fever is one of them. The absence of vaccine and irregularities in ongoing vector control programs make this disease the most frequent and persistent in many tropical and subtropical regions of the world. This paper targets to access the effects of daily commuting on dengue transmission dynamics by using a deterministic two-patch model fitted to observed data gathered in Cali, Colombia where dengue fever is highly persistent and exhibits endemo-epidemic patterns. The two-patch dengue transmission model with daily communing of human residents between patches (that is, between the city and its suburban areas) is presented using the concept of residence times, which certainly affect the disease transmission rates by inducing variability in human population sizes and vectorial densities at each patch. The same modeling framework is applied to two primary scenarios (epidemic outbreaks and endemic persistence of the disease) and for each scenario two coupling cases (one-way and asymmetric commuting) with different inflow and outflow intensities are analyzed. The concept of effective vectorial density, introduced in this paper, allows to explain in very simple terms why the daily commuting affects quite differently the dengue morbidity among human residents in both patches. In particular, residents of the patch with a greater share of incoming than outgoing commuters may actually "benefit" from inflow of daily commuter by avoiding a considerable number of infections. However, residents of the patch with a greater share of outgoing than incoming commuters, especially those who stay at home patch, incur more risk of getting infected. Additionally, the model shows that daily commuting enhance the total number of human infections acquired in both patches and may even provoke an epidemic outbreak in one patch while moderately lowering the level of the disease persistence in another patch.

Frequent implementation of interventions may increase HIV infections among MSM in China

Intervention measures among men who have sex with men (MSM) are usually designed to reduce the frequency of high risk behaviors (within-community level), but unfortunately may change the contact network and consequently increase the opportunity for them to have sex with new partners (between-community level). A multi-community periodic model on complex network is proposed to study the two-side effects of interventions on HIV transmission among MSM in China, in which the wanning process of the impacts of interventions are modelled. The basic reproduction number for the multi-community periodic system is defined and calculated numerically. Based on the number of annual reported HIV/AIDS cases among MSM in China, the unknown parameters are estimated by using MCMC method and the basic reproduction number is estimated as 3.56 (95%CI [3.556, 3.568]). Our results show that strong randomness of the community-connection networks leads to more new infections and more HIV/AIDS cases. Moreover, main conclusion indicates that implementation of interventions may induce more new infections, depending on relative level of between- and within-community impacts, and the frequency of implementation of interventions. The findings can help to guide the policy maker to choose the appropriate intervention measures, and to implement the interventions with proper frequency.

A climate-based malaria model with the use of bed nets

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 [Formula: see text] and the basic reproduction ratio [Formula: see text]. 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 [Formula: see text]; the unique disease-free periodic solution is globally attractive if [Formula: see text] and [Formula: see text]; and the model admits a unique positive periodic solution and it is globally attractive if [Formula: see text] and [Formula: see text]. 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 [Formula: see text], 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 [Formula: see text]. Another notable result is that the infection risk would be underestimated if the basic reproduction ratio [Formula: see text] of the time-averaged autonomous system were used.

Spatio-temporal spread of infectious pathogens of humans

Spatio-temporal aspects in the propagation of infectious pathogens of humans are reviewed. Mathematical modelling of these issues using metapopulation models is presented.

Vector-borne diseases models with residence times - A Lagrangian perspective

A multi-patch and multi-group modeling framework describing the dynamics of a class of diseases driven by the interactions between vectors and hosts structured by groups is formulated. Hosts' dispersal is modeled in terms of patch-residence times with the nonlinear dynamics taking into account the effective patch-host size. The residence times basic reproduction number R₀ is computed and shown to depend on the relative environmental risk of infection. The model is robust, that is, the disease free equilibrium is globally asymptotically stable (GAS) if R₀≤1 and a unique interior endemic equilibrium is shown to exist that is GAS whenever R₀>1 whenever the configuration of host-vector interactions is irreducible. The effects of patchiness and groupness, a measure of host-vector heterogeneous structure, on the basic reproduction number R₀, are explored. Numerical simulations are carried out to highlight the effects of residence times on disease prevalence.

Assessing potential countermeasures against the dengue epidemic in non-tropical urban cities

Background

Dengue is a common mosquito-borne viral disease epidemic especially in tropical and sub-tropical regions where water sanitation is not substantially controlled. However, dengue epidemics sometimes occur in non-tropical urban cities with substantial water sanitary control. Using a mathematical model, we investigate what conditions can be important for a dengue epidemic to occur in an urban city such as Tokyo, where vectors are active only in summer and there are little number of vectors around hosts.

Methods

The model, which is a modified Ross-Macdonald model, consists of two sets of host-vector compartments. The two sets correspond to high-risk and low-risk areas, and only hosts can move between them. Assuming that mosquitoes have constant activity for only 90 days, we assess five potential countermeasures: (1) restricted movement between the two areas, (2) insecticide application, (3) use of repellents, (4) vector control, and (5) isolation of the infected.

Results

The basic reproduction number R₀ and the cumulative number of infected hosts for 90 days are evaluated for each of the five countermeasures. In the cases of Measures 2–5, the cumulative number of the infected for 90 days can be reduced substantially for small R₀ even if R₀>1. Although R₀ for Measure 1 monotonically decreases with the mobility rates, the cumulative number of the infected for 90 days has a maximum at a moderate mobility rate. If the mobility rate is sufficiently small, the restricted movement effectively increases the number density of vectors in the high-risk area, and the epidemic starts earlier in the high-risk area than in the low-risk one, while the growth of infections is slow.

Conclusions

Measures 2–5 are more or less effective. However, Measure 1 can have the opposite effect, depending on the mobility rates. The restricted movement results in the formation of a kind of core population, which can promote the epidemic in the entire population.

Hitting a Moving Target: A Model for Malaria Elimination in the Presence of Population Movement

South Africa is committed to eliminating malaria with a goal of zero local transmission by 2018. Malaria elimination strategies may be unsuccessful if they focus only on vector biology, and ignore the mobility patterns of humans, particularly where the majority of infections are imported. In the first study in Mpumalanga Province in South Africa designed for this purpose, a metapopulation model is developed to assess the impact of their proposed elimination-focused policy interventions. A stochastic, non-linear, ordinary-differential equation model is fitted to malaria data from Mpumalanga and neighbouring Maputo Province in Mozambique. Further scaling-up of vector control is predicted to lead to a minimal reduction in local infections, while mass drug administration and focal screening and treatment at the Mpumalanga-Maputo border are predicted to have only a short-lived impact. Source reduction in Maputo Province is predicted to generate large reductions in local infections through stemming imported infections. The mathematical model predicts malaria elimination to be possible only when imported infections are treated before entry or eliminated at the source suggesting that a regionally focused strategy appears needed, for achieving malaria elimination in Mpumalanga and South Africa.

Predicting the impact of border control on malaria transmission: a simulated focal screen and treat campaign

Background

South Africa is one of many countries committed to malaria elimination with a target of 2018 and all malaria-endemic provinces, including Mpumalanga, are increasing efforts towards this ambitious goal. The reduction of imported infections is a vital element of an elimination strategy, particularly if a country is already experiencing high levels of imported infections. Border control of malaria is one tool that may be considered.

Methods

A metapopulation, non-linear stochastic ordinary differential equation model is used to simulate malaria transmission in Mpumalanga and Maputo province, Mozambique (the source of the majority of imported infections) to predict the impact of a focal screen and treat campaign at the Mpumalanga–Maputo border. This campaign is simulated by nesting an individual-based model for the focal screen and treat campaign within the metapopulation transmission model.

Results

The model predicts that such a campaign, simulated for different levels of resources, coverage and take-up rates with a variety of screening tools, will not eliminate malaria on its own, but will reduce transmission substantially. Making the campaign mandatory decreases transmission further though sub-patent infections are likely to remain undetected if the diagnostic tool is not adequately sensitive. Replacing screening and treating with mass drug administration results in substantially larger decreases as all (including sub-patent) infections are treated before movement into Mpumalanga.

Conclusions

The reduction of imported cases will be vital to any future malaria control or elimination strategy. This simulation predicts that FSAT at the Mpumalanga–Maputo border will be unable to eliminate local malaria on its own, but may still play a key role in detecting and treating imported infections before they enter the country. Thus FSAT may form part of an integrated elimination strategy where a variety of interventions are employed together to achieve malaria elimination.

Electronic supplementary material

The online version of this article (doi:10.1186/s12936-015-0776-2) contains supplementary material, which is available to authorized users.

The role of residence times in two-patch dengue transmission dynamics and optimal strategies

The reemergence and geographical dispersal of vector-borne diseases challenge global health experts around the world and in particular, dengue poses increasing difficulties in the Americas, due in part to explosive urban and semi-urban growth, increases of within and between region mobility, the absence of a vaccine, and the limited resources available for public health services. In this work, a simple deterministic two-patch model is introduced to assess the impact of dengue transmission dynamics in heterogeneous environments. The two-patch system models the movement (e.g. urban versus rural areas residence times) of individuals between and within patches/environments using residence-time matrices with entries that budget within and between host patch relative residence times, under the assumption that only the human budgets their residence time across regions. Three scenarios are considered: (i) resident hosts in Patch i visit patch j, where i≠j but not the other way around, a scenario referred to as unidirectional motion; (ii) symmetric bi-directional motion; and (iii) asymmetric bi-directional motion. Optimal control theory is used to identify and evaluate patch-specific control measures aimed at reducing dengue prevalence in humans and vectors at a minimal cost. Optimal policies are computed under different residence-matrix configurations mentioned above as well as transmissibility scenarios characterized by the magnitude of the basic reproduction number. Optimal patch-specific polices can ameliorate the impact of epidemic outbreaks substantially when the basic reproduction number is moderate. The final patch-specific epidemic size variation increases as the residence time matrix moves away from the symmetric case (asymmetry). As expected, the patch where individuals spend most of their time or in the patch where transmissibility is higher tend to support larger patch-specific final epidemic sizes. Hence, focusing on intervention that target areas where individuals spend "most" time or where transmissibility is higher turn out to be optimal. Therefore, reducing traffic is likely to take a host-vector system into the world of manageable outbreaks.

Schistosomiasis japonica: modelling as a tool to explore transmission patterns

Modelling is an important tool for the exploration of Schistosoma japonicum transmission patterns. It provides a general theoretical framework for decision-makers and lends itself specifically to assessing the progress of the national control programme by following the outcome of surveys. The challenge of keeping up with the many changes of social, ecological and environmental factors involved in control activities is greatly facilitated by modelling that can also indicate which activities are critical and which are less important. This review examines the application of modelling tools in the epidemiological study of schistosomiasis japonica during the last 20 years and explores the application of enhanced models for surveillance and response. Updated and timely information for decision-makers in the national elimination programme is provided but, in spite of the new modelling techniques introduced, many questions remain. Issues on application of modelling are discussed with the view to improve the current situation with respect to schistosomiasis japonica.

The time distribution of sulfadoxine-pyrimethamine protection from malaria

Sulfadoxine-pyrimethamine (SP) has been one of the most widely used antimalarial treatments world-wide, and is also used prophylactically in vulnerable populations. In this paper, we develop a mathematical model which allows us to infer the time distribution of SP protection from drug-trial data. Fitting our model to data from a controlled field study in Mali, we find that SP provided protection from malaria for an average of 37.9 days in this pediatric population. We demonstrate that the duration of SP protection is not well described by an exponential distribution, and in fact has a much narrower dispersal about the mean; the best-fit standard deviation predicted by our model was only 17.0 days, as opposed to 41.8 days for the exponential model. We estimate the monthly entomological inoculation rate and the basic reproductive number for malaria in this population, and demonstrate that extremely high SP treatment rates would be necessary to maintain an effective reproductive number below one throughout a single rainy season. These results have implications for further efforts to model the impact of SP treatment, or for investigations of the optimal timing of prophylactic SP.

A vaccination model for a multi-city system

A modelling approach is used for studying the effects of population vaccination on the epidemic dynamics of a set of n cities interconnected by a complex transportation network. The model is based on a sophisticated mover-stayer formulation of inter-city population migration, upon which is included the classical SIS dynamics of disease transmission which operates within each city. Our analysis studies the stability properties of the Disease-Free Equilibrium (DFE) of the full n-city system in terms of the reproductive number R(0). Should vaccination reduce R(0) below unity, the disease will be eradicated in all n-cities. We determine the precise conditions for which this occurs, and show that disease eradication by vaccination depend on the transportation structure of the migration network in a very direct manner. Several concrete examples are presented and discussed, and some counter-intuitive results found.

A MULTI-PATCH MALARIA MODEL WITH LOGISTIC GROWTH POPULATIONS

In this paper, we propose a multi-patch model to study the effects of population dispersal on the spatial spread of malaria between patches. The basic reproduction number [Formula: see text] is derived and it is shown that the disease-free equilibrium is locally asymptotically stable if [Formula: see text] and unstable if [Formula: see text]. Bounds on the disease-free equilibrium and [Formula: see text] are given. A sufficient condition for the existence of an endemic equilibrium when [Formula: see text] is obtained. For the two-patch submodel, the dependence of [Formula: see text] on the movement of exposed, infectious, and recovered humans between the two patches is investigated. Numerical simulations indicate that travel can help the disease to become endemic in both patches, even though the disease dies out in each isolated patch. However, if travel rates are continuously increased, the disease may die out again in both patches.