Modelling the effects of pre-exposure and post-exposure vaccines in tuberculosis control

Epidemic control strategies alter the spread of the disease in the host population. In this paper, we describe and discuss mathematical models that can be used to explore the potential of pre-exposure and post-exposure vaccines currently under development in the control of tuberculosis. A model with bacille Calmette-Guerin (BCG) vaccination for the susceptibles and treatment for the infectives is first presented. The epidemic thresholds known as the basic reproduction numbers and equilibria for the models are determined and stabilities are investigated. The reproduction numbers for the models are compared to assess the impact of the vaccines currently under development. The centre manifold theory is used to show the existence of backward bifurcation when the associated reproduction number is less than unity and that the unique endemic equilibrium is locally asymptotically stable when the associated reproduction number is greater than unity. From the study we conclude that the pre-exposure vaccine currently under development coupled with chemoprophylaxis for the latently infected and treatment of infectives is more effective when compared to the post-exposure vaccine currently under development for the latently infected coupled with treatment of the infectives.     

A mathematical model for malaria involving differential susceptibility, exposedness and infectivity of human host

The main purpose of this article is to formulate a deterministic mathematical model for the transmission of malaria that considers two host types in the human population. The first type is called "non-immune" comprising all humans who have never acquired immunity against malaria and the second type is called "semi-immune". Non-immune are divided into susceptible, exposed and infectious and semi-immune are divided into susceptible, exposed, infectious and immune. We obtain an explicit formula for the reproductive number, R(0) which is a function of the weight of the transmission semi-immune-mosquito-semi-immune, R(0a), and the weight of the transmission non-immune-mosquito-non-immune, R(0e). Then, we study the existence of endemic equilibria by using bifurcation analysis. We give a simple criterion when R(0) crosses one for forward and backward bifurcation. We explore the possibility of a control for malaria through a specific sub-group such as non-immune or semi-immune or mosquitoes.     

Backward bifurcation,equilibrium and stability phenomena in a three-stage extended BRSV epidemic model

In this paper we consider the phenomenon of backward bifurcation in epidemic modelling illustrated by an extended model for Bovine Respiratory Syncytial Virus (BRSV) amongst cattle. In its simplest form, backward bifurcation in epidemic models usually implies the existence of two subcritical endemic equilibria for R ₀ < 1, where R ₀ is the basic reproductive number, and a unique supercritical endemic equilibrium for R ₀ > 1. In our three-stage extended model we find that more complex bifurcation diagrams are possible. The paper starts with a review of some of the previous work on backward bifurcation then describes our three-stage model. We give equilibrium and stability results, and also provide some biological motivation for the model being studied. It is shown that backward bifurcation can occur in the three-stage model for small b, where b is the common per capita birth and death rate. We are able to classify the possible bifurcation diagrams. Some realistic numerical examples are discussed at the end of the paper, both for b small and for larger values of b.      

Modeling and dynamics of Wolbachia-infected male releases and mating competition on mosquito control

Despite centuries of continuous efforts, mosquito-borne diseases (MBDs) remain enormous health threat of human life worldwide. Lately, the USA government has approved an innovative technology of releasing Wolbachia-infected male mosquitoes to suppress the wild mosquito population. In this paper we first introduce a stage-structured model for natural mosquitos, then we establish a new model considering the releasing of Wolbachia-infected male mosquitoes and the mating competition between the natural male mosquitoes and infected males on the suppression of natural mosquitoes. Dynamical analysis of the two models, including the existence and local stability of the equilibria and bifurcation analysis, reveals the existence of a forward bifurcation or a backward bifurcation with multiple attractors. Moreover, globally dynamical properties are further explored by using Lyapunov function and theory of monotone operators, respectively. Our findings suggest that infected male augmentation itself cannot always guarantee the success of population eradication, but leads to three possible levels of population suppression, so we define the corresponding suppression rate and estimate the minimum release ratio for population eradication. Furthermore, we study how the release ratio of infected males and natural ones, mating competition, the rate of cytoplasmic incompatibility and the basic offspring number affect the suppression rate of natural mosquitoes. Our results show that the successful eradication relies on assessing the reproductive capacity of natural mosquitoes, a selection of suitable Wolbachia strains and an appropriate release amount of infected males. This study will be helpful for public health authorities in designing proper strategies to control vector mosquitoes and prevent the epidemics of MBDs.

     

The transmission dynamics of a within-and between-hosts malaria model

In this paper, we developed a novel deterministic coupled model tying together the effects of within-host and population level dynamics on malaria transmission dynamics. We develop within-host and within-vector dynamic models, population level between-hosts models, and a nested coupled model combining these levels. The unique feature of this work is the way the coupling and feedback for the model use the various life stages of the malaria parasite both in the human host and the mosquito vector. Analysis of the coupled and the within-human host models indicate the existence of locally asymptotically stable infection- and parasite-free equilibria when the associated reproduction numbers are less than one. The population-level model, on the other hand, exhibits backward bifurcation, where the stable disease-free equilibrium co-exists with a stable endemic equilibrium. A global sensitivity analysis was carried out to measure the effects of the sensitivity and uncertainty in the various model parameters estimates. The results indicate that the most important parameters driving the pathogen level within an infected human are the production rate of the red blood cells from the bone marrow, the infection rate, the immunogenicity of the infected red blood cells, merozoites and gametocytes, and the immunosensitivity of the merozoites and gametocytes. The key parameters identified at the population level are the human recovery rate, the death rate of the mosquitoes, the recruitment rate of susceptible humans into the population, the mosquito biting rate, the transmission probabilities per contact in mosquitoes and in humans, and the parasite production and clearance rates in the mosquitoes. Defining the feedback functions as a linear function of the mosquito biting rate, numerical exploration of the coupled model reveals oscillations in the parasite populations within a human host in the presence of the host immune response. These oscillations dampen as the mosquito biting rate increases. We also observed that the oscillation and damping effect seen in the within-human host dynamics fed back into the population level dynamics; this in turn amplifies the oscillations in the parasite population within the mosquito-host.     

Backward bifurcations in dengue transmission dynamics

A deterministic model for the transmission dynamics of a strain of dengue disease, which allows transmission by exposed humans and mosquitoes, is developed and rigorously analysed. The model, consisting of seven mutually-exclusive compartments representing the human and vector dynamics, has a locally-asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number(R(0)) is less than unity. Further, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. The epidemiological consequence of this phenomenon is that the classical epidemiological requirement of making R(0) less than unity is no longer sufficient, although necessary, for effectively controlling the spread of dengue in a community. The model is extended to incorporate an imperfect vaccine against the strain of dengue. Using the theory of centre manifold, the extended model is also shown to undergo backward bifurcation. In both the original and the extended models, it is shown, using Lyapunov function theory and LaSalle Invariance Principle, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. In other words, in addition to establishing the presence of backward bifurcation in models of dengue transmission, this study shows that the use of standard incidence in modelling dengue disease causes the backward bifurcation phenomenon of dengue disease.     

Backward Bifurcation and Optimal Control in Transmission Dynamics of West Nile Virus

The paper considers a deterministic model for the transmission dynamics of West Nile virus (WNV) in the mosquito-bird-human zoonotic cycle. The model, which incorporates density-dependent contact rates between the mosquito population and the hosts (birds and humans), is rigorously analyzed using dynamical systems techniques and theories. These analyses reveal the existence of the phenomenon of backward bifurcation (where the stable disease-free equilibrium of the model co-exists with a stable endemic equilibrium when the reproduction number of the disease is less than unity) in WNV transmission dynamics. The epidemiological consequence of backward bifurcation is that the classical requirement of having the reproduction number less than unity, while necessary, is no longer sufficient for WNV elimination from the population. It is further shown that the model with constant contact rates can also exhibit this phenomenon if the WNV-induced mortality in the avian population is high enough. The model is extended to assess the impact of some anti-WNV control measures, by re-formulating the model as an optimal control problem with density-dependent demographic parameters. This entails the use of two control functions, one for mosquito-reduction strategies and the other for personal (human) protection, and redefining the demographic parameters as density-dependent rates. Appropriate optimal control methods are used to characterize the optimal levels of the two controls. Numerical simulations of the optimal control problem, using a set of reasonable parameter values, suggest that mosquito reduction controls should be emphasized ahead of personal protection measures.     

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

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

Transmission dynamics of West Nile virus in mosquitoes and corvids and non-corvids

There are more than 300 avian species that can transmit West Nile virus (WNv). In general, the corvid and non-corvid families of birds have different responses to the virus, with corvids suffering a higher disease-induced mortality rate. By taking both corvids and non-corvids as the primary reservoir hosts and mosquitoes as vectors; we formulate and study a system of ordinary differential equations to model a single season of the transmission dynamics of WNv in the mosquito–bird cycle. We calculate the basic reproduction number and analyze the existence and stability of the equilibria. The existence of a backward bifurcation gives a further sub-threshold condition beyond the basic reproduction number for the spread of the virus. We also discuss the role of corvids and non-corvids in spreading the virus. We conclude that knowledge of the relative abundance of corvid bird species and other mammals assist us in accurate estimation of the epidemic of WNv.     

Endemic threshold results in an age-duration-structured population model for HIV infection

In this paper we consider an age-duration-structured population model for HIV infection in a homosexual community. First we investigate the invasion problem to establish the basic reproduction ratio R(0) for the HIV/AIDS epidemic by which we can state the threshold criteria: The disease can invade into the completely susceptible population if R(0)>1, whereas it cannot if R(0)<1. Subsequently, we examine existence and uniqueness of endemic steady states. We will show sufficient conditions for a backward or a forward bifurcation to occur when the basic reproduction ratio crosses unity. That is, in contrast with classical epidemic models, for our HIV model there could exist multiple endemic steady states even if R(0) is less than one. Finally, we show sufficient conditions for the local stability of the endemic steady states.     

Dynamic phenomena arising from an extended Core Group model

In order to obtain a reasonably accurate model for the spread of a particular infectious disease through a population, it may be necessary for this model to possess some degree of structural complexity. Many such models have, in recent years, been found to exhibit a phenomenon known as backward bifurcation, which generally implies the existence of two subcritical endemic equilibria. It is often possible to refine these models yet further, and we investigate here the influence such a refinement may have on the dynamic behaviour of a system in the region of the parameter space near R₀=1.We consider a natural extension to a so-called Core Group model for the spread of a sexually transmitted disease, arguing that this may in fact give rise to a more realistic model. From the deterministic viewpoint we study the possible shapes of the resulting bifurcation diagrams and the associated stability patterns. Stochastic versions of both the original and the extended models are also developed so that the probability of extinction and time to extinction may be examined, allowing us to gain further insights into the complex system dynamics near R₀=1. A number of interesting phenomena are observed, for which heuristic explanations are provided.     

Persistence thresholds for phocine distemper virus infection in harbour seal Phoca vitulina metapopulations

1. This paper explores the concept of the critical community size for persistence of infection in wildlife populations. We use as a case study the 1988 epidemic of phocine distemper virus in the North Sea population of harbour seals, Phoca vitulina .
2. We summarize the available data on this epidemic and use it to parameterize a stochastic compartmental model for an infection spreading through a spatial array of patches coupled by nearest-neighbour mixing, with replacement of susceptibles occurring as a discrete annual event.
3. A combination of analytical and simulation techniques is used to show that the high levels of transmission between different seal subpopulations, combined with the small annual birth cohort, act to make persistence of infection impossible in this harbour seal population at realistic population levels. The well known mechanisms by which metapopulation structures may act to promote persistence can be seen to have an effect only at weaker levels of spatial coupling, and higher levels of host recruitment, than those empirically observed.     

Evolutionary dynamics of giant viruses and their virophages

Giant viruses contain large genomes, encode many proteins atypical for viruses, replicate in large viral factories, and tend to infect protists. The giant virus replication factories can in turn be infected by so called virophages, which are smaller viruses that negatively impact giant virus replication. An example is Mimiviruses that infect the protist Acanthamoeba and that are themselves infected by the virophage Sputnik. This study examines the evolutionary dynamics of this system, using mathematical models. While the models suggest that the virophage population will evolve to increasing degrees of giant virus inhibition, it further suggests that this renders the virophage population prone to extinction due to dynamic instabilities over wide parameter ranges. Implications and conditions required to avoid extinction are discussed. Another interesting result is that virophage presence can fundamentally alter the evolutionary course of the giant virus. While the giant virus is predicted to evolve toward increasing its basic reproductive ratio in the absence of the virophage, the opposite is true in its presence. Therefore, virophages can not only benefit the host population directly by inhibiting the giant viruses but also indirectly by causing giant viruses to evolve toward weaker phenotypes. Experimental tests for this model are suggested.     

To Cut or Not to Cut: A Modeling Approach for Assessing the Role of Male Circumcision in HIV Control

A recent randomized controlled trial shows a significant reduction in women-to-men transmission of HIV due to male circumcision. Such development calls for a rigorous mathematical study to ascertain the full impact of male circumcision in reducing HIV burden, especially in resource-poor nations where access to anti-retroviral drugs is limited. First of all, this paper presents a compartmental model for the transmission dynamics of HIV in a community where male circumcision is practiced. In addition to having a disease-free equilibrium, which is locally-asymptotically stable whenever a certain epidemiological threshold is less than unity, the model exhibits the phenomenon of backward bifurcation, where the disease-free equilibrium coexists with a stable endemic equilibrium when the threshold is less than unity. The implication of this result is that HIV may persist in the population even when the reproduction threshold is less than unity. Using partial data from South Africa, the study shows that male circumcision at 60% efficacy level can prevent up to 220,000 cases and 8,200 deaths in the country within a year. Further, it is shown that male circumcision can significantly reduce, but not eliminate, HIV burden in a community. However, disease elimination is feasible if male circumcision is combined with other interventions such as ARVs and condom use. It is shown that the combined use of male circumcision and ARVs is more effective in reducing disease burden than the combined use of male circumcision and condoms for a moderate condom compliance rate.     

Low level viral persistence after infection with LCMV: a quantitative insight through numerical bifurcation analysis

Many important viruses persist at very low levels in the body in the face of host immunity, and may influence the maintenance of this state of 'infection immunity'. To analyse low level viral persistence in quantitative terms, we use a mathematical model of antiviral cytotoxic T lymphocyte (CTL) response to lymphocytic choriomeningitis virus (LCMV).This model, described by a non-linear system of delay differential equations (DDEs), is studied using numerical bifurcation analysis techniques for DDEs. Domains where low level LCMV coexistence with CTL memory is possible, either as an equilibrium state or an oscillatory pattern, are identified in spaces of the model parameters characterising the interaction between virus and CTL populations. Our analysis suggests that the coexistence of replication competent virus below the conventional detection limit (of about 100 pfu per spleen) in the immune host as an equilibrium state requires the per day relative growth rate of the virus population to decrease at least 5-fold compared to the acute phase of infection. Oscillatory patterns in the dynamics of persisting LCMV and CTL memory, with virus population varying between 1 and 100 pfu per spleen, are possible within quite narrow intervals of the rates of virus growth and precursor CTL population death. Whereas the virus replication rate appears to determine the stability of the low level virus persistence, it does not affect the steady-state level of the viral population, except for very low values.     

Modelling the dynamics of LCMV infection in mice: II. Compartmental structure and immunopathology

In this study, we develop a mathematical model for analysis of the compartmental aspects and immunopathology of lymphocytic choriomeningitis virus (LCMV) infection in mice. We used sets of original and published data on systemic (extrasplenic) virus distribution to estimate the parameters of virus growth and elimination for spleen and other anatomical compartments, such as the liver, kidney, thymus and lung as well as transfer rates between blood and the above organs. A mathematical model quantitatively integrating the virus distribution kinetics in the host, the specific cytotoxic T lymphocyte (CTL) response in spleen and the re-circulation of effector CTL between spleen, blood and liver is advanced to describe the CTL-mediated immunopathology (hepatitis) in mice infected with LCMV. For intravenous and "peripheral" routes of infection we examine the severity of the liver disease, as a function of the virus dose and the host's immune status characterized by the numbers of precursor and/or cytolytic effector CTL. The model is used to predict the efficacy of protection against virus persistence and disease in a localized viral infection as a function of the composition of CTL population. The modelling analysis suggests quantitative demands to CTL memory for maximal protection against a wide range of doses of infection with a primarily peripheral site of virus replication without the risk of favoring immunopathology. It specifies objectives for CTL vaccination to ensure virus elimination with minimal immunopathology vs. vaccination for disease.     

Evolutionary Dynamics of the Prokaryotic Adaptive Immunity System CRISPR-Cas in an Explicit Ecological Context

A stochastic, agent-based mathematical model of the coevolution of the archaeal and bacterial adaptive immunity system, CRISPR-Cas, and lytic viruses shows that CRISPR-Cas immunity can stabilize the virus-host coexistence rather than leading to the extinction of the virus. In the model, CRISPR-Cas immunity does not specifically promote viral diversity, presumably because the selection pressure on each single proto-spacer is too weak. However, the overall virus diversity in the presence of CRISPR-Cas grows due to the increase of the host and, accordingly, the virus population size. Above a threshold value of total viral diversity, which is proportional to the viral mutation rate and population size, the CRISPR-Cas system becomes ineffective and is lost due to the associated fitness cost. Our previous modeling study has suggested that the ubiquity of CRISPR-Cas in hyperthermophiles, which contrasts its comparative low prevalence in mesophiles, is due to lower rates of mutation fixation in thermal habitats. The present findings offer a complementary, simpler perspective on this contrast through the larger population sizes of mesophiles compared to hyperthermophiles, because of which CRISPR-Cas can become ineffective in mesophiles. The efficacy of CRISPR-Cas sharply increases with the number of proto-spacers per viral genome, potentially explaining the low information content of the proto-spacer-associated motif (PAM) that is required for spacer acquisition by CRISPR-Cas because a higher specificity would restrict the number of spacers available to CRISPR-Cas, thus hampering immunity. The very existence of the PAM might reflect the tradeoff between the requirement of diverse spacers for efficient immunity and avoidance of autoimmunity.     

Competitive exclusion in a vector-host model for the dengue fever

 We study a system of differential equations that models the population dynamics of an SIR vector transmitted disease with two pathogen strains. This model arose from our study of the population dynamics of dengue fever. The dengue virus presents four serotypes each induces host immunity but only certain degree of cross-immunity to heterologous serotypes. Our model has been constructed to study both the epidemiological trends of the disease and conditions that permit coexistence in competing strains. Dengue is in the Americas an epidemic disease and our model reproduces this kind of dynamics. We consider two viral strains and temporary cross-immunity. Our analysis shows the existence of an unstable endemic state (‘saddle’ point) that produces a long transient behavior where both dengue serotypes cocirculate. Conditions for asymptotic stability of equilibria are discussed supported by numerical simulations. We argue that the existence of competitive exclusion in this system is product of the interplay between the host superinfection process and frequency-dependent (vector to host) contact rates. Received 4 December 1995; received in revised form 5 March 1996