Epidemic dynamics of a vector-borne disease on a villages-and-city star network with commuters

In this paper I present the genesis of R0 in demography, ecology and epidemiology, from embryo to its current adult form. I argue on why it has taken so long for the concept to mature in epidemiology when there were ample opportunities for cross-fertilisation from demography and ecology from where it reached adulthood fifty years earlier. Today, R0 is a more fully developed adult in epidemiology than in demography. In the final section I give an algorithm for its calculation in heterogeneous populations.

Mathematical models of viral transmission and control are important tools for assessing the threat posed by deliberate release of the smallpox virus and the best means of containing an outbreak. Models must balance biological realism against limitations of knowledge, and uncertainties need to be accurately communicated to policy-makers. Smallpox poses the particular challenge that key biological, social and spatial factors affecting disease spread in contemporary populations must be elucidated largely from historical studies undertaken before disease eradication in 1979. We review the use of models in smallpox planning within the broader epidemiological context set by recent outbreaks of both novel and re-emerging pathogens.

In this paper some properties of a convenient estimator, derived from a martingale estimating function, for the basic reproduction number of the general epidemic model are given for both finite and large samples. These properties give some guidelines for using this convenient estimator. It is shown that it underestimates the parameter and that the bias tends to zero when the population size and the initial number of infectives are increased simultaneously. The bias cannot be removed for a fixed number of introductory infectives. However, the estimator is asymptotically unbiased, conditional on a major outbreak. A simulation study shows that the central limit theorem applies for moderate population sizes.

A multipatch model is proposed to study the impact of travel on the spatial spread of disease between patches with different level of disease prevalence. The basic reproduction number for the ith patch in isolation is obtained along with the basic reproduction number of the system of patches, Re(0). Inequalities describing the relationship between these numbers are also given. For a two-patch model with one high prevalence patch and one low prevalence patch, results pertaining to the dependence of Re(0) on the travel rates between the two patches are obtained. For parameter values relevant for influenza, these results show that, while banning travel of infectives from the low to the high prevalence patch always contributes to disease control, banning travel of symptomatic travelers only from the high to the low prevalence patch could adversely affect the containment of the outbreak under certain ranges of parameter values. Moreover, banning all travel of infected individuals from the high to the low prevalence patch could result in the low prevalence patch becoming diseasefree, while the high prevalence patch becomes even more disease-prevalent, with the resulting number of infectives in this patch alone exceeding the combined number of infectives in both patches without border control. Under the set of parameter values used, our results demonstrate that if border control is properly implemented, then it could contribute to stopping the spatial spread of disease between patches.

Deployment of limited resources is an issue of major importance for decision-making in crisis events. This is especially true for large-scale outbreaks of infectious diseases. Little is known when it comes to identifying the most efficient way of deploying scarce resources for control when disease outbreaks occur in different but interconnected regions. The policy maker is frequently faced with the challenge of optimizing efficiency (e.g. minimizing the burden of infection) while accounting for social equity (e.g. equal opportunity for infected individuals to access treatment). For a large range of diseases described by a simple SIRS model, we consider strategies that should be used to minimize the discounted number of infected individuals during the course of an epidemic. We show that when faced with the dilemma of choosing between socially equitable and purely efficient strategies, the choice of the control strategy should be informed by key measurable epidemiological factors such as the basic reproductive number and the efficiency of the treatment measure. Our model provides new insights for policy makers in the optimal deployment of limited resources for control in the event of epidemic outbreaks at the landscape scale.

Mobile Phone “Hot Spots” An obstacle to developing effective national malaria control programs is a lack of understanding of human movements, which are an important component of disease transmission. As mobile phones have become increasingly ubiquitous, it is now possible to collect individual-level, longitudinal data on human movements on a massive scale. Wesolowski et al. (p. 267 ) analyzed mobile phone call data records representing the travel patterns of 15 million mobile phone owners in Kenya over the course of a year. This was combined with a detailed malaria risk map, to estimate malaria parasite movements across the country that could be caused by human movement. This information enabled detailed analysis of parasite sources and sinks between hundreds of local settlements. Estimates were compared with hospital data from Nairobi to show that local pockets of transmission likely occur around the periphery of Nairobi, accounting for locally acquired cases, contrary to the accepted idea that there is no transmission in the capital.

For a class of epidemiological SIRS models that include public health policies, the stability at the uninfected state and the prevalence at the infected state are investigated. Backward bifurcation from the uninfected state and hysteresis effects are shown to occur for some range of parameters. In such cases, the reproduction number does not describe the necessary elimination effort; rather the effort is described by the value of the critical parameter at the turning point. An explicit expression is given for this quantity. The phenomenon of subcritical bifurcation in epidemic modeling is also discussed in terms of group models, pair formation, and macroparasite infection.

The expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population is mathematically defined as the dominant eigenvalue of a positive linear operator. It is shown that in certain special cases one can easily compute or estimate this eigenvalue. Several examples involving various structuring variables like age, sexual disposition and activity are presented.

The basic reproduction number Ro is the number of secondary cases which one case would produce in a completely susceptible population. It depends on the duration of the infectious period, the probability of infecting a susceptible individual during one contact, and the number of new susceptible individuals contacted per unit of time. Therefore Ro may vary considerably for different infectious diseases but also for the same disease in different populations. The key threshold result of epidemic theory associates the outbreaks of epidemics and the persistence of endemic levels with basic reproduction numbers greater than one. Because the magnitude of R 0 allows one to determine the amount of effort which is necessary either to prevent an epidemic or to eliminate an infection from a population, it is crucial to estimate Ro for a given disease in a particular population. The present paper gives a survey about the various estimation methods available.

Mark-release-recapture (MRR) experiments were conducted with emerging Anopheles gambiae s.l. and Anophelesfunestus Giles at Jaribuni and Mtepeni in Kilifi, along the Kenyan Coast. Of 739 and 1246 Anopheles released at Jaribuni and Mtepeni, 24.6 and 4.33% were recaptured, respectively. The daily survival probability was 0.96 for An. funestus and 0.95 for An. gambiae in Jaribuni and 0.83 and 0.95, respectively, in Mtepeni. The maximum flight distance recorded was 661 m. The high survival probability of An. gambiae and An. funestus estimated accounts for the continuous transmission of malaria along the Kenyan coast. This study also shows that the release of young, emergent female Anopheles improves the recapture rates and may be a better approach to MRR studies.

Development of strategies for mitigating the severity of a new influenza pandemic is now a top global public health priority. Influenza prevention and containment strategies can be considered under the broad categories of antiviral, vaccine and non-pharmaceutical (case isolation, household quarantine, school or workplace closure, restrictions on travel) measures. Mathematical models are powerful tools for exploring this complex landscape of intervention strategies and quantifying the potential costs and benefits of different options. Here we use a large-scale epidemic simulation to examine intervention options should initial containment of a novel influenza outbreak fail, using Great Britain and the United States as examples. We find that border restrictions and/or internal travel restrictions are unlikely to delay spread by more than 2-3 weeks unless more than 99% effective. School closure during the peak of a pandemic can reduce peak attack rates by up to 40%, but has little impact on overall attack rates, whereas case isolation or household quarantine could have a significant impact, if feasible. Treatment of clinical cases can reduce transmission, but only if antivirals are given within a day of symptoms starting. Given enough drugs for 50% of the population, household-based prophylaxis coupled with reactive school closure could reduce clinical attack rates by 40-50%. More widespread prophylaxis would be even more logistically challenging but might reduce attack rates by over 75%. Vaccine stockpiled in advance of a pandemic could significantly reduce attack rates even if of low efficacy. Estimates of policy effectiveness will change if the characteristics of a future pandemic strain differ substantially from those seen in past pandemics.

The spatial structure of populations is a key element in the understanding of the large-scale spreading of epidemics. Motivated by the recent empirical evidence on the heterogeneous properties of transportation and commuting patterns among urban areas, we present a thorough analysis of the behavior of infectious diseases in metapopulation models characterized by heterogeneous connectivity and mobility patterns. We derive the basic reaction-diffusion equations describing the metapopulation system at the mechanistic level and derive an early stage dynamics approximation for the subpopulation invasion dynamics. The analytical description uses a homogeneous assumption on degree block variables that allows us to take into account arbitrary degree distribution of the metapopulation network. We show that along with the usual single population epidemic threshold the metapopulation network exhibits a global threshold for the subpopulation invasion. We find an explicit analytic expression for the invasion threshold that determines the minimum number of individuals traveling among subpopulations in order to have the infection of a macroscopic number of subpopulations. The invasion threshold is a function of factors such as the basic reproductive number, the infectious period and the mobility process and it is found to decrease for increasing network heterogeneity. We provide extensive mechanistic numerical Monte Carlo simulations that recover the analytical finding in a wide range of metapopulation network connectivity patterns. The results can be useful in the understanding of recent data driven computational approaches to disease spreading in large transportation networks and the effect of containment measures such as travel restrictions.

Abstract The development of mathematical models has been very useful in the study of the dynamics of infectious diseases. However, there is a scarcity of models which take into account the spatial heterogeneity of the environment. The few models developed indicate that, when the environment is fragmented, an increase in mobility between the patches enhances the persistence of the disease. We develop a model which describes the dynamics of malaria in time in a heterogeneous environment. As expected, fragmenting the environment decreases the likelihood of establishment of the disease in comparison with a homogeneous environment. Two types of mobility patterns are considered in the fragmented environment: migration between patches with no return, and visitation in which the individuals return to their patch of origin after visiting other patches. When compared with the fragmented environment without mobility, the time elapsed to reach the equilibrium decreases with both patterns of mobility; however, the chances of establishment and the equilibrium prevalence are the same with the migration pattern and higher with the visitation pattern. It is concluded that the effects of mobility in heterogeneous environments is not a simple matter, and depends on the patterns of mobility. The relevance of these findings to the design of control strategies is discussed.

The prevalence of Plasmodium falciparum malaria in Zanzibar has reached historic lows. Improving control requires quantifying malaria importation rates, identifying high-risk travelers, and assessing onwards transmission.Estimates of Zanzibar's importation rate were calculated through two independent methodologies. First, mobile phone usage data and ferry traffic between Zanzibar and mainland Tanzania were re-analyzed using a model of heterogeneous travel risk. Second, a dynamic mathematical model of importation and transmission rates was used.Zanzibar residents traveling to malaria endemic regions were estimated to contribute 1-15 times more imported cases than infected visitors. The malaria importation rate was estimated to be 1.6 incoming infections per 1,000 inhabitants per year. Local transmission was estimated too low to sustain transmission in most places.Malaria infections in Zanzibar largely result from imported malaria and subsequent transmission. Plasmodium falciparum malaria elimination appears feasible by implementing control measures based on detecting imported malaria cases and controlling onward transmission.

The seasonal dynamics and spatial distributions of Anopheles mosquitoes and Plasmodium falciparum parasites were studied for one year at 30 villages in Malindi, Kilifi, and Kwale Districts along the coast of Kenya. Anopheline mosquitoes were sampled inside houses at each site once every two months and malaria parasite prevalence in local school children was determined at the end of the entomologic survey. A total of 5,476 Anopheles gambiae s.l. and 3,461 An. funestus were collected. Species in the An. gambiae complex, identified by a polymerase chain reaction, included 81.9% An. gambiae s.s., 12.8% An. arabiensis, and 5.3% An. merus. Anopheles gambiae s.s. contributed most to the transmission of P. falciparum along the coast as a whole, while An. funestus accounted for more than 50% of all transmission in Kwale District. Large spatial heterogeneity of transmission intensity (< 1 up to 120 infective bites per person per year) resulted in correspondingly large and significantly related variations in parasite prevalence (range = 38-83%). Thirty-two percent of the sites (7 of 22 sites) with malaria prevalences ranging from 38% to 70% had annual entomologic inoculation rates (EIR) less than five infective bites per person per year. Anopheles gambiae s.l. and An. funestus densities in Kwale were not significantly influenced by rainfall. However, both were positively correlated with rainfall one and three months previously in Malindi and Kilifi Districts, respectively. These unexpected variations in the relationship between mosquito populations and rainfall suggest environmental heterogeneity in the predominant aquatic habitats in each district. One important conclusion is that the highly non-linear relationship between EIRs and prevalence indicates that the consistent pattern of high prevalence might be governed by substantial variation in transmission intensity measured by entomologic surveys. The field-based estimate of entomologic parameters on a district level does not provide a sensitive indicator of transmission intensity in this study.

There is much historical evidence of the spread of disease through human mobility. Today in spite of medical advances and international health measures there is still much cause for concern. There is now more mobility, facilitated by modern transport and sometimes precipitated by major natural and man-made disasters. Redistribution of population is occuring in the developing world, particularly massive rural-urban movements. Population mobility has contributed to the transmission of malaria and prejudiced programmes for control and eradication; but mobility and other human factors have not been adequately studied. Parasites and vectors receive more attention than do people. Epidemiological studies need to pay greater attention to the nature and variety of population movements and to their differing impacts upon disease and health. It is essential to distinguish between migration (involving change of residence) and circulation (movement away from residence with sebsequent return). In tropical Africa various spatial and temporal dimensions can be applied to differentiate within these two major categories of mobility. In turn there are various associated physical and psychological health hazards.

A precise definition of the basic reproduction number, R0, is presented for a general compartmental disease transmission model based on a system of ordinary differential equations. It is shown that, if R0<1, then the disease free equilibrium is locally asymptotically stable; whereas if R0>1, then it is unstable. Thus, R0 is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super- and sub-threshold endemic equilibria for R0 near one. This criterion, together with the definition of R0, is illustrated by treatment, multigroup, staged progression, multistrain and vector-host models and can be applied to more complex models. The results are significant for disease control.

The basic reproduction number ℛ 0 is arguably the most important quantity in infectious disease epidemiology. The next-generation matrix (NGM) is the natural basis for the definition and calculation of ℛ 0 where finitely many different categories of individuals are recognized. We clear up confusion that has been around in the literature concerning the construction of this matrix, specifically for the most frequently used so-called compartmental models. We present a detailed easy recipe for the construction of the NGM from basic ingredients derived directly from the specifications of the model. We show that two related matrices exist which we define to be the NGM with large domain and the NGM with small domain . The three matrices together reflect the range of possibilities encountered in the literature for the characterization of ℛ 0 . We show how they are connected and how their construction follows from the basic model ingredients, and establish that they have the same non-zero eigenvalues, the largest of which is the basic reproduction number ℛ 0 . Although we present formal recipes based on linear algebra, we encourage the construction of the NGM by way of direct epidemiological reasoning, using the clear interpretation of the elements of the NGM and of the model ingredients. We present a selection of examples as a practical guide to our methods. In the appendix we present an elementary but complete proof that ℛ 0 defined as the dominant eigenvalue of the NGM for compartmental systems and the Malthusian parameter r , the real-time exponential growth rate in the early phase of an outbreak, are connected by the properties that ℛ 0 &gt; 1 if and only if r &gt; 0, and ℛ 0 = 1 if and only if r = 0.

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.

The rapid increase in the world's urban population has major implications for the epidemiology of malaria. A review of malaria transmission in sub-Saharan African cities shows the strong likelihood of transmission occurring within these sprawling cities, whatever the size or characteristics of their bioecologic environment. A meta-analysis of results from studies of malaria transmission in sub-Saharan Africa shows a loose linear negative relationship between mean annual entomologic inoculation rates (EIR) and the level of urbanicity. Few studies have failed to find entomologic evidence of some transmission. Our results show mean annual EIRs of 7.1 in the city centers, 45.8 in periurban areas, and 167.7 in rural areas. The impact of urbanization in reducing transmission is more marked in areas where the mean rainfall is low and seasonal. Considerable variation in the level of transmission exists among cities and within different districts in the same city. This article presents evidence from past literature to build a conceptual framework to begin to explain this heterogeneity. The potential for malaria epidemics owing to decreasing levels of natural immunity may be offset by negative impacts of urbanization on the larval ecology of anopheline mosquitoes. Malaria control in urban environments may be simpler as a result of urbanization; however, much of what we know about malaria transmission in rural environments might not hold in the urban context.

In metropolitan areas people travel frequently and extensively but often in highly structured commuting patterns. We investigate the role of this type of human movement in the epidemiology of vector-borne pathogens such as dengue. Analysis is based on a metapopulation model where mobile humans connect static mosquito subpopulations. We find that, due to frequency dependent biting, infection incidence in the human and mosquito populations is almost independent of the duration of contact. If the mosquito population is not uniformly distributed between patches the transmission potential of the pathogen at the metapopulation level, as summarized by the basic reproductive number, is determined by the size of the largest subpopulation and reduced by stronger connectivity. Global extinction of the pathogen is less likely when increased human movement enhances the rescue effect but, in contrast to classical theory, it is not minimized at an intermediate level of connectivity. We conclude that hubs and reservoirs of infection can be places people visit frequently but briefly and the relative importance of human and mosquito populations in maintaining the pathogen depends on the distribution of the mosquito population and the variability in human travel patterns. These results offer an insight in to the paradoxical observation of resurgent urban vector-borne disease despite increased investment in vector control and suggest that successful public health intervention may require a dual approach. Prospective studies can be used to identify areas with large mosquito populations that are also visited by a large fraction of the human population. Retrospective studies can be used to map recent movements of infected people, pinpointing the mosquito subpopulation from which they acquired the infection and others to which they may have transmitted it.

Little is known about how best to deploy scarce resources for disease control when epidemics occur in different but interconnected regions. We use a combination of optimal control methods and epidemiological theory for metapopulations to address this problem. We consider what strategy should be used if the objective is to minimize the discounted number of infected individuals during the course of an epidemic. We show, for a system with two interconnected regions and an epidemic in which infected individuals recover and can be reinfected, that equalizing infection in the two regions is the worst possible strategy in minimizing the total level of infection. Treatment should instead be preferentially directed at the region with the lower level of infection, treating the other subpopulation only when there is resource left over. The same strategy holds with preferential treatments of regions with lower levels of infection when quarantine is introduced.

Mathematical models of epidemics have a long history of contributing to the understanding of the impact of vaccination programmes. Simple, one-line models can predict target vaccination coverage that will eradicate an infectious agent, whilst other questions require complex simulations of stochastic processes in space and time. This review introduces some simple ordinary differential equation models of mass vaccination that can be used to address important questions about the predicted impact of vaccination programmes. We show how to calculate the threshold vaccination coverage rate that will eradicate an infection, explore the impact of vaccine-induced immunity that wanes through time, and study the competitive interactions between vaccine susceptible and vaccine resistant strains of infectious agent.

Much recent work in Africa has focused on the impacts of structural adjustment on prevailing economic, social and political structures, and especially so in the urban areas. Rather less has focused on the resulting land-use changes in and around African cities. This paper focuses on recent land-use changes in the peri-urban zone of Dar es Salaam, Tanzania, in the context of structural adjustment programmes experienced in the 1990s. Recent land-use development in Dar es Salaam's peri-urban zone has been characterised by infill rather than further linear expansion along routeways; by densification of existing linear settlement; and by limited spatial growth to the south of the city. Much of this can be explained as a response to new economic conditions resulting from macroeconomic reform. Access has improved due to greater competition in the public transport sector and increased access to private transport. Rapid capital accumulation, based on trade liberalisation and rent-seeking activities on the part of some individuals, has resulted in investment opportunities in the peri-urban zone. However, lack of confidence in future state policies and uncertainties over land ownership have resulted in investment being made in housing stock in particular, or spread over a range of small-scale enterprises, rather than in larger-scale productive investment.

Highly pathogenic H5N1 influenza A viruses are now endemic in avian populations in Southeast Asia, and human cases continue to accumulate. Although currently incapable of sustained human-to-human transmission, H5N1 represents a serious pandemic threat owing to the risk of a mutation or reassortment generating a virus with increased transmissibility. Identifying public health interventions that might be able to halt a pandemic in its earliest stages is therefore a priority. Here we use a simulation model of influenza transmission in Southeast Asia to evaluate the potential effectiveness of targeted mass prophylactic use of antiviral drugs as a containment strategy. Other interventions aimed at reducing population contact rates are also examined as reinforcements to an antiviral-based containment policy. We show that elimination of a nascent pandemic may be feasible using a combination of geographically targeted prophylaxis and social distancing measures, if the basic reproduction number of the new virus is below 1.8. We predict that a stockpile of 3 million courses of antiviral drugs should be sufficient for elimination. Policy effectiveness depends critically on how quickly clinical cases are diagnosed and the speed with which antiviral drugs can be distributed.

With the recent resurgence of vector-borne diseases due to urbanization and development there is an urgent need to understand the dynamics of vector-borne diseases in rapidly changing urban environments. For example, many empirical studies have produced the disturbing finding that diseases continue to persist in modern city centers with zero or low rates of transmission. We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian models that mimic human commuting behavior and (ii) Eulerian models that mimic human migration. We determine the basic reproduction number R(0) for both modeling approaches. We show that for both approaches that if the disease-free equilibrium is stable (R(0)<1) then it is globally stable and if the disease-free equilibrium is unstable (R(0)>1) then there exists a unique positive (endemic) equilibrium that is globally stable among positive solutions. Finally, we prove in general that Lagrangian and Eulerian modeling approaches are not equivalent. The modeling approaches presented provide a framework to explore spatial vector-borne disease dynamics and control in heterogeneous environments. As an example, we consider two patches in which the disease dies out in both patches when there is no movement between them. Numerical simulations demonstrate that the disease becomes endemic in both patches when humans move between the two patches.

This study investigated whether Anopheles gambiae s.l. and Anopheles funestus Giles mosquito populations were distributed randomly among houses on the coast of Kenya. Sample means and variances of mosquitoes were estimated from bimonthly pyrethrum spray collections at 30 villages from July 1997 through May 1998. In total, 5,476 An. gambiae s.l. and 3,461 An. funestus were collected. The number of An. gambiae s.l. collected was highest in November/December and lowest in May. The number of An. funestus collected was highest during September/October and lowest during May. As the density of mosquitoes decreased, there was a tendency toward randomness in the distribution. The proportion of An. gambiae s.l. and An. funestus mosquitoes collected per house for each sampling period also showed patterns of clustering, with 80% of An. gambiae s.l. collected from <30% of the houses and 80% of An. funestus collected from <20% of the total houses. The total number of mosquitoes collected from any one house ranged from 0 to 121 for An. gambiae s.l. and from 0 to 152 for An. funestus. This coupled with the results of the variance to mean ratio plots suggests extensive clustering in the distribution of An. gambiae s.l. and An. funestus mosquito populations throughout the year along the coast of Kenya.