A MULTI-PATCH MALARIA MODEL WITH LOGISTIC GROWTH POPULATIONS

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.

Dynamics and profiles of a degenerated reaction-diffusion host-pathogen model with apparent and inapparent infection period

Inapparent infection plays an important role in the disease spread, which is an infection by a pathogen that causes few or no signs or symptoms of infection in the host. Many pathogens, including HIV, typhoid fever, and coronaviruses such as COVID-19 spread in their host populations through inapparent infection. In this paper, we formulated a degenerated reaction-diffusion host-pathogen model with multiple infection period. We split the infectious individuals into two distinct classes: apparent infectious individuals and inapparent infectious individuals, coming from exposed individuals with a ratio of (1-p) and p, respectively. Some preliminary results and threshold-type results are achieved by detailed mathematical analysis. We also investigate the asymptotic profiles of the positive steady state (PSS) when the diffusion rate of susceptible individuals approaches zero or infinity. When all parameters are all constants, the global attractivity of the constant endemic equilibrium is established. It is verified by numerical simulations that spatial heterogeneity of the transmission rates can enhance the intensity of an epidemic. Especially, the transmission rate of inapparent infectious individuals significantly increases the risk of disease transmission, compared to that of apparent infectious individuals and pathogens in the environment, and we should pay special attentions to how to regulate the inapparent infectious individuals for disease control and prevention, which is consistent with the result on the sensitive analysis to the transmission rates through the normalized forward sensitivity index. We also find that disinfection of the infected environment is an important way to prevent and eliminate the risk of environmental transmission.

Dynamics analysis of an SVEIR epidemic model in a patchy environment

In this paper, we propose a multi-patch SVEIR epidemic model that incorporates vaccination of both newborns and susceptible populations. We determine the basic reproduction number $ R_{0} $ and prove that the disease-free equilibrium $ P_{0} $ is locally and globally asymptotically stable if $ R_{0} < 1, $ and it is unstable if $ R_{0} > 1. $ Moreover, we show that the disease is uniformly persistent in the population when $ R_{0} > 1. $ Numerical simulations indicate that vaccination strategies can effectively control disease spread in all patches while population migration can either intensify or prevent disease transmission within a patch.

Spatial dynamics of malaria transmission

The Ross-Macdonald model has exerted enormous influence over the study of malaria transmission dynamics and control, but it lacked features to describe parasite dispersal, travel, and other important aspects of heterogeneous transmission. Here, we present a patch-based differential equation modeling framework that extends the Ross-Macdonald model with sufficient skill and complexity to support planning, monitoring and evaluation for Plasmodium falciparum malaria control. We designed a generic interface for building structured, spatial models of malaria transmission based on a new algorithm for mosquito blood feeding. We developed new algorithms to simulate adult mosquito demography, dispersal, and egg laying in response to resource availability. The core dynamical components describing mosquito ecology and malaria transmission were decomposed, redesigned and reassembled into a modular framework. Structural elements in the framework-human population strata, patches, and aquatic habitats-interact through a flexible design that facilitates construction of ensembles of models with scalable complexity to support robust analytics for malaria policy and adaptive malaria control. We propose updated definitions for the human biting rate and entomological inoculation rates. We present new formulas to describe parasite dispersal and spatial dynamics under steady state conditions, including the human biting rates, parasite dispersal, the "vectorial capacity matrix," a human transmitting capacity distribution matrix, and threshold conditions. An [Formula: see text] package that implements the framework, solves the differential equations, and computes spatial metrics for models developed in this framework has been developed. Development of the model and metrics have focused on malaria, but since the framework is modular, the same ideas and software can be applied to other mosquito-borne pathogen systems.

Analysis of a heterogeneous SEIRS patch model with asymmetric mobility kernel

In this paper, we establish a spatial heterogeneous SEIRS patch model with asymmetric mobility kernel. The basic reproduction ratio $ \mathcal{R}_{0} $ is defined, and threshold-type results on global dynamics are investigated in terms of $ \mathcal{R}_{0} $. In certain cases, the monotonicity of $ \mathcal{R}_{0} $ with respect to the heterogeneous diffusion coefficients is established, but this is not true in all cases. Finally, when the diffusion rate of susceptible individuals approaches zero, the long-term behavior of the endemic equilibrium is explored. In contrast to most prior studies, which focused primarily on the mobility of susceptible and symptomatic infected individuals, our findings indicate the significance of the mobility of exposed and recovered persons in disease dynamics.

Nonlinear diffusion in multi-patch logistic model

We examine a multi-patch model of a population connected by nonlinear asymmetrical migration, where the population grows logistically on each patch. Utilizing the theory of cooperative differential systems, we prove the global stability of the model. In cases of perfect mixing, where migration rates approach infinity, the total population follows a logistic law with a carrying capacity that is distinct from the sum of carrying capacities and is influenced by migration terms. Furthermore, we establish conditions under which fragmentation and nonlinear asymmetrical migration can lead to a total equilibrium population that is either greater or smaller than the sum of carrying capacities. Finally, for the two-patch model, we classify the model parameter space to determine if nonlinear dispersal is beneficial or detrimental to the sum of two carrying capacities.

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.

An age-dependent immuno-epidemiological model with distributed recovery and death rates

The work is devoted to a new immuno-epidemiological model with distributed recovery and death rates considered as functions of time after the infection onset. Disease transmission rate depends on the intra-subject viral load determined from the immunological submodel. The age-dependent model includes the viral load, recovery and death rates as functions of age considered as a continuous variable. Equations for susceptible, infected, recovered and dead compartments are expressed in terms of the number of newly infected cases. The analysis of the model includes the proof of the existence and uniqueness of solution. Furthermore, it is shown how the model can be reduced to age-dependent SIR or delay model under certain assumptions on recovery and death distributions. Basic reproduction number and final size of epidemic are determined for the reduced models. The model is validated with a COVID-19 case data. Modelling results show that proportion of young age groups can influence the epidemic progression since disease transmission rate for them is higher than for other age groups.

The impact of vaccination on human papillomavirus infection with disassortative geographical mixing: a two-patch modeling study

Human papillomavirus (HPV) infection can spread between regions. What is the impact of disassortative geographical mixing on the dynamics of HPV transmission? Vaccination is effective in preventing HPV infection. How to allocate HPV vaccines between genders within each region and between regions to reduce the total infection? Here we develop a two-patch two-sex model to address these questions. The control reproduction number [Formula: see text] under vaccination is obtained and shown to provide a critical threshold for disease elimination. Both analytical and numerical results reveal that disassortative geographical mixing does not affect [Formula: see text] and only has a minor impact on the disease prevalence in the total population given the vaccine uptake proportional to the population size for each gender in the two patches. When the vaccine uptake is not proportional to the population size, sexual mixing between the two patches can reduce [Formula: see text] and mitigate the consequence of disproportionate vaccine coverage. Using parameters calibrated from the data of a case study, we find that if the two patches have the same or similar sex ratios, allocating vaccines proportionally according to the new recruits in two patches and giving priority to the gender with a smaller recruit rate within each patch will bring the maximum benefit in reducing the total prevalence. We also show that a time-variable vaccination strategy between the two patches can further reduce the disease prevalence. This study provides some quantitative information that may help to develop vaccine distribution strategies in multiple regions with disassortative mixing.

Transmission dynamics of brucellosis with patch model: Shanxi and Hebei Provinces as cases

Brucellosis is a zoonotic disease caused by Brucella, and it is an important infectious disease all over the world. The prevalence of brucellosis in the Chinese mainland has some spatial characteristics besides the temporal trend in recent years. Due to the large-scale breeding of sheep and the frequent transportation of sheep in various regions, brucellosis spreads wantonly in pastoral areas, and human brucellosis spreads from traditional pastoral areas and semi-pastoral areas in the north to non-pastoral areas with low incidence in the south. In order to study the influence of sheep immigration on the epidemic transmission, a patch dynamics model was established. In each patch, the sub-model was composed of humans, sheep and Brucella. The basic reproduction number, disease-free equilibrium and positive equilibrium of the model were discussed. On the other hand, taking Shanxi Province and Hebei Province as examples, we carried out numerical simulations. The results show that the basic reproduction numbers of Shanxi Province and Hebei Province are 0.7497 and 0.5022, respectively, which indicates that the current brucellosis in the two regions has been effectively controlled. To reduce brucellosis faster in the two provinces, there should be a certain degree of sheep immigration from high-infection area to low-infection areas, and reduce the immigration of sheep from low-infection areas to high-infection areas.

Vector-host epidemic model with direct transmission in random environment

This paper studies a stochastic vector-host epidemic model with direct transmission in random environment, governed by a system of stochastic differential equations with regime-switching diffusion. We first examine the existence and uniqueness of a positive global solution. Then, we investigate stability properties of the solution, including almost sure and pth moment exponential stability and stochastic asymptotic stability. Moreover, we study conditions for the existence and uniqueness of a stationary distribution. Numerical simulations are presented to illustrate the theoretical results.

Impact of State-Dependent Dispersal on Disease Prevalence

Based on a susceptible-infected-susceptible patch model, we study the influence of dispersal on the disease prevalence of an individual patch and all patches at the endemic equilibrium. Specifically, we estimate the disease prevalence of each patch and obtain a weak order-preserving result that correlated the patch reproduction number with the patch disease prevalence. Then we assume that dispersal rates of the susceptible and infected populations are proportional and derive the overall disease prevalence, or equivalently, the total infection size at no dispersal or infinite dispersal as well as the right derivative of the total infection size at no dispersal. Furthermore, for the two-patch submodel, two complete classifications of the model parameter space are given: one addressing when dispersal leads to higher or lower overall disease prevalence than no dispersal, and the other concerning how the overall disease prevalence varies with dispersal rate. Numerical simulations are performed to further investigate the effect of movement on disease prevalence.

Global dynamics of COVID-19 epidemic model with recessive infection and isolation

In this paper, we present an SEIIaHR epidemic model to study the influence of recessive infection and isolation in the spread of COVID-19. We first prove that the infection-free equilibrium is globally asymptotically stable with condition R₀<1 and the positive equilibrium is uniformly persistent when the condition R₀>1. By using the COVID-19 data in India, we then give numerical simulations to illustrate our results and carry out some sensitivity analysis. We know that asymptomatic infections will affect the spread of the disease when the quarantine rate is within the range of [0.3519, 0.5411]. Furthermore, isolating people with symptoms is important to control and eliminate the disease.

Dynamical analysis of an SIS epidemic model with migration and residence time

Migration can be divided into temporary and permanent migration, which is related to the residence time of people in the patch, thus we consider an SIS epidemic model with migration and residence time in a patchy environment. If [Formula: see text], the disease-free equilibrium is globally asymptotically stable and the disease dies out. With the same migration rate of susceptible and infectious individuals and without disease-induced death, when [Formula: see text], the endemic equilibrium is unique and globally asymptotically stable. Numerical simulations are carried out to show the effects of residence time and the migration rate on disease prevalence.

Disease Emergence in Multi-Patch Stochastic Epidemic Models with Demographic and Seasonal Variability

Factors such as seasonality and spatial connectivity affect the spread of an infectious disease. Accounting for these factors in infectious disease models provides useful information on the times and locations of greatest risk for disease outbreaks. In this investigation, stochastic multi-patch epidemic models are formulated with seasonal and demographic variability. The stochastic models are used to investigate the probability of a disease outbreak when infected individuals are introduced into one or more of the patches. Seasonal variation is included through periodic transmission and dispersal rates. Multi-type branching process approximation and application of the backward Kolmogorov differential equation lead to an estimate for the probability of a disease outbreak. This estimate is also periodic and depends on the time, the location, and the number of initial infected individuals introduced into the patch system as well as the magnitude of the transmission and dispersal rates and the connectivity between patches. Examples are given for seasonal transmission and dispersal in two and three patches.

Assessing the role of human mobility on malaria transmission

South Sudan accounts for a large proportion of all annual malaria cases in Africa. In recent years, the country has witnessed an unprecedented number of people on the move, refugees, internally displaced people, people who have returned to their counties or areas of origin, stateless people and other populations of concern, posing challenges to malaria control. Thus, one can claim that human mobility is one of the contributing factors to the resurgence of malaria. The aim of this paper is to assess the impact of human mobility on the burden of malaria disease in South Sudan. For this, we formulate an SIR-type model that describes the transmission dynamics of malaria disease between multiple patches. The proposed model is a system of stochastic differential equations consisting of ordinary differential equations perturbed by a stochastic Wiener process. For the deterministic part of the model, we calculate the basic reproduction number. Concerning the whole stochastic model, we use the maximum likelihood approach to fit the model to weekly malaria data of 2011 from Central Equatoria State, Western Bahr El Ghazal State and Warrap State. Using the parameters estimated on the fitted model, we simulate the future observation of the disease pattern. The disease was found to persist in the low transmission patches when there is human inflow in these patches and although the intervention coverage reaches 75%.

An almost periodic Ross-Macdonald model with structured vector population in a patchy environment

An almost periodic Ross-Macdonald model with age structure for the vector population in a patchy environment is considered. The basic reproduction ratio [Formula: see text] for this model is derived and a threshold-type result on its global dynamics in terms of [Formula: see text] is established. It is shown that the disease is uniformly persistent if [Formula: see text], while the disease will die out if [Formula: see text]. Numerical simulations show that the biting rate greatly affects the disease transmission, and human migration sometimes could reduce the transmission risk. We further obtain a condition numerically to determine whether a control strategy on migration is necessary. Moreover, numerical results indicate that prolonging the length of maturation period of vector is beneficial to the disease control, and the threshold length of the maturation period for disease outbreak can be computed. Finally, the comparison between the almost periodic and periodic models shows that the periodic model may overestimate or underestimate the disease transmission risk.

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.

An SIR epidemic model with vaccination in a patchy environment

In this paper, an SIR patch model with vaccination is formulated to investigate the effect of vaccination coverage and the impact of human mobility on the spread of diseases among patches. The control reproduction number Rv is derived. It shows that the disease-free equilibrium is unique and is globally asymptotically stable if Rv< 1, and unstable if Rv>1. The sufficient condition for the local stability of boundary equilibria and the existence of equilibria are obtained for the case n=2. Numerical simulations indicate that vaccines can control and prevent the outbreak of infectious in all patches while migration may magnify the outbreak size in one patch and shrink the outbreak size in other patch.

Modeling the importation and local transmission of vector-borne diseases in Florida: The case of Zika outbreak in 2016

Chikungunya, dengue, and Zika viruses are all transmitted by Aedes aegypti and Aedes albopictus mosquito species, had been imported to Florida and caused local outbreaks. We propose a deterministic model to study the importation and local transmission of these mosquito-borne diseases. The purpose is to model and mimic the importation of these viruses to Florida via travelers, local infections in domestic mosquitoes by imported travelers, and finally non-travel related transmissions to local humans by infected local mosquitoes. As a case study, the model will be used to simulate the accumulative Zika cases in Florida. Since the disease system is driven by a continuing input of infections from outside sources, orthodox analytic methods based on the calculation of the basic reproduction number are inadequate to describe and predict their behavior. Via steady-state analysis and sensitivity analysis, effective control and prevention measures for these mosquito-borne diseases are tested.

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.

Simple multi-scale modeling of the transmission dynamics of the 1905 plague epidemic in Bombay

The first few disease generations of an infectious disease outbreak is the most critical phase to implement control interventions. The lack of accurate data and information during the early transmission phase hinders the application of complex compartmental models to make predictions and forecasts about important epidemic quantities. Thus, simpler models are often times better tools to understand the early dynamics of an outbreak particularly in the context of limited data. In this paper we mechanistically derive and fit a family of logistic models to spatial-temporal data of the 1905 plague epidemic in Bombay, India. We systematically compare parameter estimates, reproduction numbers, model fit, and short-term forecasts across models at different spatial resolutions. At the same time, we also assess the presence of sub-exponential growth dynamics at different spatial scales and investigate the role of spatial structure and data resolution (district level data and city level data) using simple structured models. Our results for the 1905 plague epidemic in Bombay indicates that it is possible for the growth of an epidemic in the early phase to be sub-exponential at sub-city level, while maintaining near exponential growth at an aggregated city level. We also show that the rate of movement between districts can have a significant effect on the final epidemic size.

Two-patch model for the spread of West Nile virus

A two-patch model for the spread of West Nile virus between two discrete geographic regions is established to incorporate a mobility process which describes how contact transmission occurs between individuals from and between two regions. In the mobility process, we assume that the host birds can migrate between regions, but not the mosquitoes. The basic reproduction number [Formula: see text] is computed by the next generation matrix method. We prove that if [Formula: see text], then the disease-free equilibrium is globally asymptotically stable. If [Formula: see text], the endemic equilibrium is globally asymptotically stable for any nonnegative nontrivial initial data. Using the perturbation theory, we obtain the concrete expression of the endemic equilibrium of the model with a mild restriction of the birds movement rate between patches. Finally, numerical simulations demonstrate that the disease becomes endemic in both patches when birds move back and forth between the two regions. Some numerical simulations for [Formula: see text] in terms of the birds movement rate are performed which show that the impacts could be very complicated.

Non-linear Dynamics of Two-Patch Model Incorporating Secondary Dengue Infection

In this paper, the impact of human migration on the dynamics of dengue epidemic has been discussed. The vector-host model considers two patches with different dengue serotype in each patch. The model considers the constant rate of migration in susceptible and recovered class from one patch to other. Recovered migrants from prior infection are exposed to secondary infection in the patch where different serotype is present. The basic reproduction number is computed and analyzed in terms of migration parameters. The model is analyzed for the existence and local stability of various equilibrium states in terms of migration parameters. The numerical simulations for the choice of relevant data from literature have been performed to verify analytical results and to further explore the dynamics of the system. The sensitivity analysis of basic reproduction number with respect to migration parameters is carried out. It is found that immigration in a patch increases the basic reproduction in respective patch and vice-versa. The basic reproduction number has been estimated for the two states of Brazil which verifies the occurrence of severe epidemic in one of the states of Brazil.

A periodic two-patch SIS model with time delay and transport-related infection

In this paper, we propose a periodic SIS epidemic model with time delay and transport-related infection in a patchy environment. The basic reproduction number R₀ is derived which determines the global dynamics of the model system: if R₀ < 1, the disease-free periodic state is globally attractive while there exists at least one positive periodic state and the disease persists if R₀ > 1. Numerical simulations are performed to confirm the analytical results and to explore the dependence of R₀ on the transport-related infection parameters and the amplitude of fluctuations.

Modeling the transmission dynamics and control of rabies in China

Human rabies was first recorded in ancient China in about 556 BC and is still one of the major public-health problems in China. From 1950 to 2015, 130,494 human rabies cases were reported in Mainland China with an average of 1977 cases per year. It is estimated that 95% of these human rabies cases are due to dog bites. The purpose of this article is to provide a review about the models, results, and simulations that we have obtained recently on studying the transmission of rabies in China. We first construct a basic susceptible, exposed, infectious, and recovered (SEIR) type model for the spread of rabies virus among dogs and from dogs to humans and use the model to simulate the human rabies data in China from 1996 to 2010. Then we modify the basic model by including both domestic and stray dogs and apply the model to simulate the human rabies data from Guangdong Province, China. To study the seasonality of rabies, in Section 4 we further propose a SEIR model with periodic transmission rates and employ the model to simulate the monthly data of human rabies cases reported by the Chinese Ministry of Health from January 2004 to December 2010. To understand the spatial spread of rabies, in Section 5 we add diffusion to the dog population in the basic SEIR model to obtain a reaction-diffusion equation model and determine the minimum wave speed connecting the disease-free equilibrium to the endemic equilibrium. Finally, in order to investigate how the movement of dogs affects the geographically inter-provincial spread of rabies in Mainland China, in Section 6 we propose a multi-patch model to describe the transmission dynamics of rabies between dogs and humans and use the two-patch submodel to investigate the rabies virus clades lineages and to simulate the human rabies data from Guizhou and Guangxi, Hebei and Fujian, and Sichuan and Shaanxi, respectively. Some discussions are provided in Section 7.

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.

MODELING THE SPREAD OF HOOKWORM DISEASE AND ASSESSING CHEMOTHERAPY PROGRAMS: MATHEMATICAL ANALYSIS AND COMPARISON WITH SURVEILLANCE DATA

Soil-transmitted helminthes including hookworm infections result in a major disease burden worldwide. Periodic chemotherapy treatments with anthelminthic drugs remain major intervention strategy in highly endemic areas and currently there is no approved human vaccine. We developed a model to study the spread of the hookworm infection in the population level having a variety of parasite development stages. We investigated long-term effectiveness of selective and mass chemotherapy, scenarios in which the therapy becomes less effective, and is interrupted. We analyzed the mathematical model, which includes determining the basic reproductive number and steady states, defining model initial conditions, proving positiveness and boundedness of solutions, local and global stability of the disease-free and endemic steady states, and uniform persistence of the system. We fitted our model using field data from hookworm control programs in endemic areas in China and Zimbabwe. The model provided accurate predictions of disease prevalence for all considered locations for multiple years of utilizing deworming programs. We demonstrated that administration of the deworming therapy is an effective method in terms of controlling the hookworm infection. Nevertheless, the chemotherapy programs do not eradicate the disease in the villages and the infection occurrence promptly increases once the program is interrupted, thus ultimately leading to the pre-treatment levels. Our modeling results suggest that the deworming programs should be maintained at least once per year to control the infection.

Limiting the spread of disease through altered migration patterns

We consider a model for an epidemic in a population that occupies geographically distinct locations. The disease is spread within subpopulations by contacts between infective and susceptible individuals, and is spread between subpopulations by the migration of infected individuals. We show how susceptible individuals can act collectively to limit the spread of disease during the initial phase of an epidemic by specifying the distribution that minimises the growth rate of the epidemic when the infectives are migrating so as to maximise the growth rate. We also give an explicit strategy that minimises the basic reproduction number, which is also shown be optimal in terms of the probability of extinction and total size of the epidemic.

A validated agent-based model to study the spatial and temporal heterogeneities of malaria incidence in the rainforest environment

BACKGROUND

The Amazon environment has been exposed in the last decades to radical changes that have been accompanied by a remarkable rise of both Plasmodium falciparum and Plasmodium vivax malaria. The malaria transmission process is highly influenced by factors such as spatial and temporal heterogeneities of the environment and individual-based characteristics of mosquitoes and humans populations. All these determinant factors can be simulated effectively trough agent-based models.

METHODS

This paper presents a validated agent-based model of local-scale malaria transmission. The model reproduces the environment of a typical riverine village in the northern Peruvian Amazon, where the malaria transmission is highly seasonal and apparently associated with flooding of large areas caused by the neighbouring river. Agents representing humans, mosquitoes and the two species of Plasmodium (P. falciparum and P. vivax) are simulated in a spatially explicit representation of the environment around the village. The model environment includes: climate, people houses positions and elevation. A representation of changes in the mosquito breeding areas extension caused by the river flooding is also included in the simulation environment.

RESULTS

A calibration process was carried out to reproduce the variations of the malaria monthly incidence over a period of 3 years. The calibrated model is also able to reproduce the spatial heterogeneities of local scale malaria transmission. A "what if" eradication strategy scenario is proposed: if the mosquito breeding sites are eliminated through mosquito larva habitat management in a buffer area extended at least 200 m around the village, the malaria transmission is eradicated from the village.

CONCLUSIONS

The use of agent-based models can reproduce effectively the spatiotemporal variations of the malaria transmission in a low endemicity environment dominated by river floodings like in the Amazon.

From within host dynamics to the epidemiology of infectious disease: Scientific overview and challenges

Since their earliest days, humans have been struggling with infectious diseases. Caused by viruses, bacteria, protozoa, or even higher organisms like worms, these diseases depend critically on numerous intricate interactions between parasites and hosts, and while we have learned much about these interactions, many details are still obscure. It is evident that the combined host-parasite dynamics constitutes a complex system that involves components and processes at multiple scales of time, space, and biological organization. At one end of this hierarchy we know of individual molecules that play crucial roles for the survival of a parasite or for the response and survival of its host. At the other end, one realizes that the spread of infectious diseases by far exceeds specific locales and, due to today's easy travel of hosts carrying a multitude of organisms, can quickly reach global proportions. The community of mathematical modelers has been addressing specific aspects of infectious diseases for a long time. Most of these efforts have focused on one or two select scales of a multi-level disease and used quite different computational approaches. This restriction to a molecular, physiological, or epidemiological level was prudent, as it has produced solid pillars of a foundation from which it might eventually be possible to launch comprehensive, multi-scale modeling efforts that make full use of the recent advances in biology and, in particular, the various high-throughput methodologies accompanying the emerging -omics revolution. This special issue contains contributions from biologists and modelers, most of whom presented and discussed their work at the workshop From within Host Dynamics to the Epidemiology of Infectious Disease, which was held at the Mathematical Biosciences Institute at Ohio State University in April 2014. These contributions highlight some of the forays into a deeper understanding of the dynamics between parasites and their hosts, and the consequences of this dynamics for the spread and treatment of infectious diseases.

Models for the effects of host movement in vector-borne disease systems

Host and/or vector movement patterns have been shown to have significant effects in both empirical studies and mathematical models of vector-borne diseases. The processes of economic development and globalization seem likely to make host movement even more important in the future. This article is a brief survey of some of the approaches that have been used to study the effects of host movement in analytic mathematical models for vector-borne diseases. It describes the formulation and interpretation of various types of spatial models and describes a few of the conclusions that can be drawn from them. It is not intended to be comprehensive but rather to provide sufficient background material and references to the literature to serve as an entry point into this area of research for interested readers.

Modeling the geographic spread of rabies in China

In order to investigate how the movement of dogs affects the geographically inter-provincial spread of rabies in Mainland China, we propose a multi-patch model to describe the transmission dynamics of rabies between dogs and humans, in which each province is regarded as a patch. In each patch the submodel consists of susceptible, exposed, infectious, and vaccinated subpopulations of both dogs and humans and describes the spread of rabies among dogs and from infectious dogs to humans. The existence of the disease-free equilibrium is discussed, the basic reproduction number is calculated, and the effect of moving rates of dogs between patches on the basic reproduction number is studied. To investigate the rabies virus clades lineages, the two-patch submodel is used to simulate the human rabies data from Guizhou and Guangxi, Hebei and Fujian, and Sichuan and Shaanxi, respectively. It is found that the basic reproduction number of the two-patch model could be larger than one even if the isolated basic reproduction number of each patch is less than one. This indicates that the immigration of dogs may make the disease endemic even if the disease dies out in each isolated patch when there is no immigration. In order to reduce and prevent geographical spread of rabies in China, our results suggest that the management of dog markets and trades needs to be regulated, and transportation of dogs has to be better monitored and under constant surveillance.

WITHDRAWN: From within host dynamics to the epidemiology of infectious disease: Scientific overview and challenges

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Global dynamics of a general class of multi-group epidemic models with latency and relapse

A multi-group model is proposed to describe a general relapse phenomenon of infectious diseases in heterogeneous populations. In each group, the population is divided into susceptible, exposed, infectious, and recovered subclasses. A general nonlinear incidence rate is used in the model. The results show that the global dynamics are completely determined by the basic reproduction number R0. In particular, a matrix-theoretic method is used to prove the global stability of the disease-free equilibrium when R0 ≤ 1, while a new combinatorial identity (Theorem 3.3 in Shuai and van den Driessche) in graph theory is applied to prove the global stability of the endemic equilibrium when R0 > 1. We would like to mention that by applying the new combinatorial identity, a graph of 3n (or 2n+m) vertices can be converted into a graph of n vertices in order to deal with the global stability of the endemic equilibrium in this paper.

A PERIODIC ROSS-MACDONALD MODEL IN A PATCHY ENVIRONMENT

Based on the classical Ross-Macdonald model, in this paper we propose a periodic malaria model to incorporate the effects of temporal and spatial heterogeneity on disease transmission. The temporal heterogeneity is described by assuming that some model coefficients are time-periodic, while the spatial heterogeneity is modeled by using a multi-patch structure and assuming that individuals travel among patches. We calculate the basic reproduction number [Formula: see text] and show that either the disease-free periodic solution is globally asymptotically stable if [Formula: see text] or the positive periodic solution is globally asymptotically stable if [Formula: see text]. Numerical simulations are conducted to confirm the analytical results and explore the effect of travel control on the disease prevalence.

Optimal seasonal timing of oral azithromycin for malaria

Mass administration of azithromycin for trachoma has been shown to reduce malarial parasitemia. However, the optimal seasonal timing of such distributions for antimalarial benefit has not been established. We performed numerical analyses on a seasonally forced epidemic model (of Ross-Macdonald type) with periodic impulsive annual mass treatment to address this question. We conclude that when azithromycin-based trachoma elimination programs occur in regions of seasonal malaria transmission, such as Niger, the optimal seasonal timing of mass drug administration (MDA) may not occur during the season of maximum transmission.

Global analysis for spread of infectious diseases via transportation networks

We formulate an epidemic model for the spread of an infectious disease along with population dispersal over an arbitrary number of distinct regions. Structuring the population by the time elapsed since the start of travel, we describe the infectious disease dynamics during transportation as well as in the regions. As a result, we obtain a system of delay differential equations. We define the basic reproduction number R(0) as the spectral radius of a next generation matrix. For multi-regional systems with strongly connected transportation networks, we prove that if R(0) ≤ 1 then the disease will be eradicated from each region, while if R(0) > 1 there is a globally asymptotically stable equilibrium, which is endemic in every region. If the transportation network is not strongly connected, then the model analysis shows that numerous endemic patterns can exist by admitting a globally asymptotically stable equilibrium, which may be disease free in some regions while endemic in other regions. We provide a procedure to detect the disease free and the endemic regions according to the network topology and local reproduction numbers. The main ingredients of the mathematical proofs are the inductive applications of the theory of asymptotically autonomous semiflows and cooperative dynamical systems. We visualise stability boundaries of equilibria in a parameter plane to illustrate the influence of the transportation network on the disease dynamics. For a system consisting of two regions, we find that due to spatial heterogeneity characterised by different local reproduction numbers, R(0) may depend non-monotonically on the dispersal rates, thus travel restrictions are not always beneficial.

Assessing the optimal virulence of malaria-targeting mosquito pathogens: a mathematical study of engineered Metarhizium anisopliae

Background

Metarhizium anisopliae is a naturally occurring fungal pathogen of mosquitoes. Recently, Metarhizium has been engineered to act against malaria by directly killing the disease agent within mosquito vectors and also effectively blocking onward transmission. It has been proposed that efforts should be made to minimize the virulence of the fungal pathogen, in order to slow the development of resistant mosquitoes following an actual deployment.

Results

Two mathematical models were developed and analysed to examine the efficacy of the fungal pathogen. It was found that, in many plausible scenarios, the best effects are achieved with a reduced or minimal pathogen virulence, even if the likelihood of resistance to the fungus is negligible. The results for both models depend on the interplay between two main effects: the ability of the fungus to reduce the mosquito population, and the ability of fungus‐infected mosquitoes to compete for resources with non‐fungus‐infected mosquitoes.

Conclusions

The results indicate that there is no obvious choice of virulence for engineered Metarhizium or similar pathogens, and that all available information regarding the population ecology of the combined mosquito‐fungus system should be carefully considered. The models provide a basic framework for examination of anti‐malarial mosquito pathogens that should be extended and improved as new laboratory and field data become available.

Probability of a disease outbreak in stochastic multipatch epidemic models

Environmental heterogeneity, spatial connectivity, and movement of individuals play important roles in the spread of infectious diseases. To account for environmental differences that impact disease transmission, the spatial region is divided into patches according to risk of infection. A system of ordinary differential equations modeling spatial spread of disease among multiple patches is used to formulate two new stochastic models, a continuous-time Markov chain, and a system of stochastic differential equations. An estimate for the probability of disease extinction is computed by approximating the Markov chain model with a multitype branching process. Numerical examples illustrate some differences between the stochastic models and the deterministic model, important for prevention of disease outbreaks that depend on the location of infectious individuals, the risk of infection, and the movement of individuals.

The network level reproduction number for infectious diseases with both vertical and horizontal transmission

A wide range of infectious diseases are both vertically and horizontally transmitted. Such diseases are spatially transmitted via multiple species in heterogeneous environments, typically described by complex meta-population models. The reproduction number, R0, is a critical metric predicting whether the disease can invade the meta-population system. This paper presents the reproduction number for a generic disease vertically and horizontally transmitted among multiple species in heterogeneous networks, where nodes are locations, and links reflect outgoing or incoming movement flows. The metapopulation model for vertically and horizontally transmitted diseases is gradually formulated from two species, two-node network models. We derived an explicit expression of R0, which is the spectral radius of a matrix reduced in size with respect to the original next generation matrix. The reproduction number is shown to be a function of vertical and horizontal transmission parameters, and the lower bound is the reproduction number for horizontal transmission. As an application, the reproduction number and its bounds for the Rift Valley fever zoonosis, where livestock, mosquitoes, and humans are the involved species are derived. By computing the reproduction number for different scenarios through numerical simulations, we found the reproduction number is affected by livestock movement rates only when parameters are heterogeneous across nodes. To summarize, our study contributes the reproduction number for vertically and horizontally transmitted diseases in heterogeneous networks. This explicit expression is easily adaptable to specific infectious diseases, affording insights into disease evolution.