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Hosts can activate a defensive response to clear the parasite once being infected. To explore how host survival and fecundity are affected by host-parasite coevolution for chronic parasitic diseases, in this paper, we proposed an age-structured epidemic model with infection age, in which the parasite transmission rate and parasite-induced mortality rate are structured by the infection age. By use of critical function analysis method, we obtained the existence of the host immune evolutionary singular strategy which is a continuous singular strategy (CSS). Assume that parasite-induced mortality begins at infection age τ and is constant v thereafter. We got that the value of the CSS, c ∗ , monotonically decreases with respect to infection age τ (see Case (I)), while it is non-monotone if the constant v positively depends on the immune trait c (see Case (II)). This non-monotonicity is verified by numerical simulations and implies that the direction of immune evolution depends on the initial value of immune trait. Besides that, we adopted two special forms of the parasite transmission rate to study the parasite’s virulence evolution, by maximizing the basic reproduction ratio R 0 . The values of the convergence stable parasite’s virulence evolutionary singular strategies v ∗ and k ∗ increase monotonically with respect to time lag L (i.e., the time lag between the onset of transmission and mortality). At the singular strategy v ∗ and k ∗ , we further obtained the expressions of the case mortalities χ ∗ and how they are affected by the time lag L. Finally, we only presented some preliminary results about host and parasite coevolution dynamics, including a general condition under which the coevolutionary singular strategy (c ∗ , v ∗) is evolutionarily stable. [ABSTRACT FROM AUTHOR]

In this paper, we consider global stability for a heroin model with two distributed delays. The basic reproduction number of the heroin spread is obtained, which completely determines the stability of the equilibria. Using the direct Lyapunov method with Volterra type Lyapunov function, we show that the drug use-free equilibrium is globally asymptotically stable if the basic reproduction number is less than one, and the unique drug spread equilibrium is globally asymptotically stable if the basic reproduction number is greater than one. [ABSTRACT FROM AUTHOR]

Seagrass meadows always provide a habitat for many kinds of fish. As the environment changes, some kinds of fish persist if they can evolve a positive growth rate to avoid extinction and others go to extinct in the habitat. To reveal the roles of environment and ability to evolve on the predator population persistence, in this paper, we formulate an age structured model by adopting the seagrass residence index (SRI) as the environment variable. By uniform persistence and stability analysis, we obtain the detailed dynamic behaviors of the age structured model which are mainly determined by the two thresholds η 1 ∗ and η 2 ∗ for the conversion rate η . If η < η 1 ∗ , the density of the predator population will go to zero. If η > η 1 ∗ , there is a unique positive steady state E ∗ and the predator population will be uniformly persistent. If η 1 ∗ < η < η 2 ∗ , the unique positive steady state E ∗ is stable and the predator population can establish itself in the environment. If η > η 2 ∗ , the unique positive steady state E ∗ will lose its stability and Hopf bifurcation occurs when maturation age τ = 0 . If τ > 0 , maturation age τ will play a key role in the stability criteria for the positive steady state E ∗ , and the establishment of the predator population will become more difficult and complex. Numerical simulations are presented to verify the theoretical results and reveal the evolution of predator population adaptation to the habitat. [ABSTRACT FROM AUTHOR]

We consider a discrete size-structured metapopulation model with the proportions of patches occupied by n individuals as dependent variables. Adults are territorial and stay on a certain patch. The juveniles may emigrate to enter a dispersers' pool from which they can settle on another patch and become adults. Absence of colonization and absence of emigration lead to extinction of the metapopulation. We define the basic reproduction number R0 of the metapopulation as a measure for its strength of persistence. The metapopulation is uniformly weakly persistent if R0 > 1. We identify subcritical bifurcation of persistence equilibria from the extinction equilibrium as a source of multiple persistence equilibria: it occurs, e.g., when the immigration rate, into occupied patches, exceeds the colonization rate (of empty patches). We determine that the persistence-optimal dispersal strategy which maximizes the basic reproduction number is of bang-bang type: If the number of adults on a patch is below carrying capacity all the juveniles should stay, if it is above the carrying capacity all the juveniles should leave. [ABSTRACT FROM AUTHOR]

This study delves into the intricate interplay between social media platforms, interpersonal word-of-mouth communication, and the transmission dynamics associated with non-communicable diseases, with a particular emphasis on type 2 diabetes. Leveraging advanced mathematical modeling and epidemiological methodologies, our objective is to furnish a comprehensive understanding of how information dissemination through digital and interpersonal networks can impact the proliferation of such diseases within populations. We conduct sensitivity analysis to discern the pivotal model parameters that can wield a substantial influence on the dynamics of disease transmission and control. Moreover, we endeavor to explore the capacity of these model parameters to elicit stability or instability within the system. Our focus lies in the rigorous examination of Hopf and transcritical bifurcations within the system. Furthermore, we consider the influence of seasonal fluctuations in the growth rate of social media advertisements with an aim to discern its role in potentially instigating chaotic dynamics within the context of disease progression. In sum, this research seeks to offer a comprehensive and scientifically robust understanding of the patterns of type 2 diabetes and associated communicable diseases within the context of evolving digital communication landscapes. [ABSTRACT FROM AUTHOR]

We used a previously introduced HIV within-host model with sensitive and resistant strains and validated it with two data sets. The first data set is from a clinical study that investigated multi-drug treatments and measured the total CD4+ cell count and viral load. All nine patients in this data set experienced virologic failure. The second data set includes a unique patient who was treated with a unique drug and for whom both the sensitive and resistant strains were measured as well as the CD4+ cells. We studied the structural identifiability of the model with respect to each data set. With respect to the first data set, the model was structurally identifiable when the viral production rate of the sensitive strain was fixed and distinct from the viral production rate of the resistant strain. With respect to the second data set, the model was always structurally identifiable. We fit the model to the first data set using nonlinear mixed effect modeling in Monolix and estimated the population-level parameters. We inferred that the average time to emergence of a resistant strain is 844 days after treatment starts. We fit the model to the second data set and found out that the all the parameters except the mutation rate were practically identifiable. [ABSTRACT FROM AUTHOR]

In this study, we develop a mathematical model to examine the dynamics of COVID-19 with diabetes. Recognizing the increased vulnerability of diabetic patients, we proposed separate isolation classes for COVID-19 cases with and without diabetes. The model is parameterized using real data from India for the period March 2021 to September 2021. Sensitivity analysis for the basic reproduction number shows that isolation (for COVID-19 with diabetes) plays a significant role in lessening COVID-19 cases. Through numerical evaluations, we demonstrate that timely and focused care, including hospitalization/isolation, and treatment for COVID-19 patients with diabetes, can substantially reduce the disease burden by nearly three-fold. The optimal control problem with different strategies has also been discussed. We obtain that vaccination (with the least Average Cost-Effectiveness Ratio (ACER) and a combined strategy (with the highest Infection-Averted Ratio (IAR) are the most cost-saving and effective interventions to eradicate the disease. [ABSTRACT FROM AUTHOR]

In this paper we formulate a multi-scale nested immuno-epidemiological model of HIV on complex networks. The system is described by ordinary differential equations coupled with a partial differential equation. First, we prove the existence and uniqueness of the immunological model and then establish the well-posedness of the multi-scale model. We derive an explicit expression of the basic reproduction number R 0 of the immuno-epidemiological model. The system has a disease-free equilibrium and an endemic equilibrium. The disease-free equilibrium is globally stable when R 0 < 1 and unstable when R 0 > 1 . Numerical simulations suggest that R 0 increases as the number of nodes in the network increases. Further, we find that for a scale-free network the number of infected individuals at equilibrium is a hump-like function of the within-host reproduction number; however, the dependence becomes monotone if the network has predominantly low connectivity nodes or high connectivity nodes. [ABSTRACT FROM AUTHOR]

Here, we investigate a mathematical model to assess the impact of disinfectants in controlling diseases that spread in the population via direct contacts with the infected persons and also due to bacteria present in the environment. We find that the disease-free and endemic equilibria of the system are related via a transcritical bifurcation whose direction is forward. Our numerical results show that controlling the transmissions of disease through direct contacts and bacteria present in the environment can help in reducing the disease prevalence. Moreover, fostering the recovery rate and the death rate of bacteria play significant roles in disease eradication. Our numerical observations convey that reducing the bacterial density at the source discharged by the infected population through the use of chemicals has prominent effect in disease control. Overall, our findings manifest that the disinfectants of high quality can completely control the bacterial density and the disease outbreak. [ABSTRACT FROM AUTHOR]

Human immunodeficiency virus (HIV) infection is a major public health concern with 1.2 million people living with HIV in the United States. The role of nutrition in general, and albumin/globulin in particular in HIV progression has long been recognized. However, no mathematical models exist to describe the interplay between HIV and albumin/globulin. In this paper, we present a family of models of HIV and the two protein components albumin and globulin. We use albumin, globulin, viral load and target cell data from simian immunodeficiency virus (SIV)-infected monkeys to perform model selection on the family of models. We discover that the simplest model accurately and uniquely describes the data. The selection of the simplest model leads to the observation that albumin and globulin do not impact the infection rate of target cells by the virus and the clearance of the infected target cells by the immune system. Moreover, the recruitment of target cells and immune cells are modeled independently of globulin in the selected model. Mathematical analysis of the selected model reveals that the model has an infection-free equilibrium and a unique infected equilibrium when the immunological reproduction number is above one. The infection-free equilibrium is locally stable when the immunological reproduction number is below one, and unstable when the immunological reproduction number is greater than one. The infection equilibrium is locally stable whenever it exists. To determine the parameters of the best fitted model we perform structural and practical identifiability analysis. The structural identifiability analysis reveals that the model is identifiable when the immune cell infection rate is fixed at a value obtained from the literature. Practical identifiability reveals that only seven of the sixteen parameters are practically identifiable with the given data. Practical identifiability of parameters performed with synthetic data sampled a lot more frequently reveals that only two parameters are practically unidentifiable. We conclude that experiments that will improve the quality of the data can help improve the parameter estimates and lead to better understanding of the interplay of HIV and albumin-globulin metabolism. [ABSTRACT FROM AUTHOR]

Syphilis and HIV infections form a dangerous combination. In this paper, we propose an epidemic model of HIV-syphilis coinfection. The model always has a unique disease-free equilibrium, which is stable when both reproduction numbers of syphilis and HIV are less than 1. If the reproduction number of syphilis (HIV) is greater than 1, there exists a unique boundary equilibrium of syphilis (HIV), which is locally stable if the invasion number of HIV (syphilis) is less than 1. Coexistence equilibrium exists and is stable when all reproduction numbers and invasion numbers are greater than 1. Using data of syphilis cases and HIV cases from the US, we estimated that both reproduction numbers for syphilis and HIV are slightly greater than 1, and the boundary equilibrium of syphilis is stable. In addition, we observed competition between the two diseases. Treatment for primary syphilis is more important in mitigating the transmission of syphilis. However, it might lead to increase of HIV cases. The results derived here could be adapted to other multi-disease scenarios in other regions. [ABSTRACT FROM AUTHOR]

In this paper, a model for the use of heroin with treat-age is formulated based on the principles of mathematical epidemiology. The model accounts for relapse rate that depends on how long the host has been in treatment for heroin addiction. An explicit formula for the reproductive number of the heroin spread is obtained. By using the method of Lyapunov functional, we established the dynamical properties of the heroin epidemic model, and the results show that the global dynamics of the model is completely determined by the basic reproduction number. It is shown that the drug-free equilibrium is locally and globally asymptotically stable if the basic reproduction number is less than one. In addition, the heroin spread system is uniform persistence and the unique drug spread equilibrium is locally and globally asymptotically stable if the basic reproduction number is greater than one. [ABSTRACT FROM AUTHOR]

In this paper, a two-strain model with coinfection that links immunological and epidemiological dynamics across scales is formulated. On the with-in host scale, the two strains eliminate each other with the strain having the larger immunological reproduction number persisting. However, on the population scale coinfection is a common occurrence. Individuals infected with strain one can become coinfected with strain two and similarly for individuals originally infected with strain two. The immunological reproduction numbers R j , the epidemiological reproduction numbers R j and invasion reproduction numbers R j i are computed. Besides the disease-free equilibrium, there are strain one and strain two dominance equilibria. The disease-free equilibrium is locally asymptotically stable when the epidemiological reproduction numbers R j are smaller than one. In addition, each strain dominance equilibrium is locally asymptotically stable if the corresponding epidemiological reproduction number is larger than one and the invasion reproduction number of the other strain is smaller than one. The coexistence equilibrium exists when all the reproduction numbers are greater than one. Simulations suggest that when both invasion reproduction numbers are smaller than one, bistability occurs with one of the strains persisting or the other, depending on initial conditions. [ABSTRACT FROM AUTHOR]

In this paper, to understand the impact of spatial heterogeneity of treatment and movement of individuals on the persistence and extinction of heroin spread, we propose a new diffusive heroin transmission model with treatment-dependent age-structure. The basic reproduction number in heterogenous environment R 0 of the system is defined, which is consistent with the one deduced from the next generation operator approach R (x). The threshold dynamics in terms of the basic reproduction number is established: if R 0 ≤ 1 , the drug-free steady state is globally asymptotically stable, if R 0 > 1 , heroin transmission is uniformly persistent if it is present initially. In particular, when the environment is homogeneous and R 0 > 1 , our system has a unique space-independent drug spread steady state and it is globally asymptotically stable. Finally, some numerical simulations are carried out to illustrate the main results. [ABSTRACT FROM AUTHOR]

COVID-19, induced by the SARS-CoV-2 infection, has caused an unprecedented pandemic in the world. New variants of the virus have emerged and dominated the virus population. In this paper, we develop a multi-strain model with asymptomatic transmission to study how the asymptomatic or pre-symptomatic infection influences the transmission between different strains and control strategies that aim to mitigate the pandemic. Both analytical and numerical results reveal that the competitive exclusion principle still holds for the model with the asymptomatic transmission. By fitting the model to the COVID-19 case and viral variant data in the US, we show that the omicron variants are more transmissible but less fatal than the previously circulating variants. The basic reproduction number for the omicron variants is estimated to be 11.15, larger than that for the previous variants. Using mask mandate as an example of non-pharmaceutical interventions, we show that implementing it before the prevalence peak can significantly lower and postpone the peak. The time of lifting the mask mandate can affect the emergence and frequency of subsequent waves. Lifting before the peak will result in an earlier and much higher subsequent wave. Caution should also be taken to lift the restriction when a large portion of the population remains susceptible. The methods and results obtained her e may be applied to the study of the dynamics of other infectious diseases with asymptomatic transmission using other control measures. • We developed a multi-strain model with asymptomatic transmission. • The model studies the competition between different strains. • Real data in the US are used for model parameterization and simulation. • The results improve the understanding of multi-strain disease spread dynamics. • The results help to formulate guidelines of control strategies. [ABSTRACT FROM AUTHOR]

In this study, we apply optimal control theory to an immuno-epidemiological model of HIV and opioid epidemics. For the multi-scale model, we used four controls: treating the opioid use, reducing HIV risk behaviour among opioid users, entry inhibiting antiviral therapy, and antiviral therapy which blocks the viral production. Two population-level controls are combined with two within-host-level controls. We prove the existence and uniqueness of an optimal control quadruple. Comparing the two population-level controls, we find that reducing the HIV risk of opioid users has a stronger impact on the population who is both HIV-infected and opioid-dependent than treating the opioid disorder. The within-host-level antiviral treatment has an effect not only on the co-affected population but also on the HIV-only infected population. Our findings suggest that the most effective strategy for managing the HIV and opioid epidemics is combining all controls at both within-host and between-host scales.

In this paper, we investigate an optimal harvesting problem of a spatially explicit fishery model that was previously analyzed. On the surface, this problem looks innocent, but if parameters are set to where a singular arc occurs, two complex questions arise. The first question pertains to Fuller's phenomenon (or chattering), a phenomenon in which the optimal control possesses a singular arc that cannot be concatenated with the bang-bang arcs without prompting infinite oscillations over a finite region. 1) How do we numerically assess whether or not a problem chatters in cases when we cannot analytically prove such a phenomenon? The second question focuses on implementation of an optimal control. 2) When an optimal control has regions that are difficult to implement, how can we find alternative strategies that are both suboptimal and realistic to use? Although the former question does not apply to all optimal harvesting problems, most fishery managers should be concerned about the latter. Interestingly, for this specific problem, our techniques for answering the first question results in an answer to the the second. Our methods involve using an extended version of the switch point algorithm (SPA), which handles control problems having initial and terminal conditions on the states. In our numerical experiments, we obtain strong empirical evidence that the harvesting problem chatters, and we find three alternative harvesting strategies with fewer switches that are realistic to implement and near optimal.

In this paper we develop a four compartment within-host model of nutrition and HIV. We show that the model has two equilibria: an infection-free equilibrium and infection equilibrium. The infection free equilibrium is locally asymptotically stable when the basic reproduction number $ \mathcal{R}&#95;0 < 1 $, and unstable when $ \mathcal{R}&#95;0 > 1 $. The infection equilibrium is locally asymptotically stable if $ \mathcal{R}&#95;0 > 1 $ and an additional condition holds. We show that the within-host model of HIV and nutrition is structured to reveal its parameters from the observations of viral load, CD4 cell count and total protein data. We then estimate the model parameters for these 3 data sets. We have also studied the practical identifiability of the model parameters by performing Monte Carlo simulations, and found that the rate of clearance of the virus by immunoglobulins is practically unidentifiable, and that the rest of the model parameters are only weakly identifiable given the experimental data. Furthermore, we have studied how the data frequency impacts the practical identifiability of model parameters.

HIV continues to be a major global health issue, having claimed millions of lives in the last few decades. While several empirical studies support the fact that proper nutrition is useful in the fight against HIV, very few studies have focused on developing and using mathematical modelling approaches to assess the association between HIV, human immune response to the disease, and nutrition. We develop a within-host model for HIV that captures the dynamic interactions between HIV, the immune system and nutrition. We find that increased viral activity leads to increased serum protein levels. We also show that the viral production rate is positively correlated with HIV viral loads, as is the enhancement rate of protein by virus. Although our numerical simulations indicate a direct correlation between dietary protein intake and serum protein levels in HIV-infected individuals, further modelling and clinical studies are necessary to gain comprehensive understanding of the relationship.

In this paper, we introduce a novel multi-scale network model of two epidemics: HIV infection and opioid addiction. The HIV infection dynamics is modeled on a complex network. We determine the basic reproduction number of HIV infection, $ \mathcal{R}&#95;{v} $, and the basic reproduction number of opioid addiction, $ \mathcal{R}&#95;{u} $. We show that the model has a unique disease-free equilibrium which is locally asymptotically stable when both $ \mathcal{R}&#95;{u} $ and $ \mathcal{R}&#95;{v} $ are less than one. If $ \mathcal{R}&#95;{u} > 1 $ or $ \mathcal{R}&#95;{v} > 1 $, then the disease-free equilibrium is unstable and there exists a unique semi-trivial equilibrium corresponding to each disease. The unique opioid only equilibrium exist when the basic reproduction number of opioid addiction is greater than one and it is locally asymptotically stable when the invasion number of HIV infection, $ \mathcal{R}^{1}&#95;{v&#95;i} $ is less than one. Similarly, the unique HIV only equilibrium exist when the basic reproduction number of HIV is greater than one and it is locally asymptotically stable when the invasion number of opioid addiction, $ \mathcal{R}^{2}&#95;{u&#95;i} $ is less than one. Existence and stability of co-existence equilibria remains an open problem. We performed numerical simulations to better understand the impact of three epidemiologically important parameters that are at the intersection of two epidemics: $ q&#95;v $ the likelihood of an opioid user being infected with HIV, $ q&#95;u $ the likelihood of an HIV-infected individual becoming addicted to opioids, and $ \delta $ recovery from opioid addiction. Simulations suggest that as the recovery from opioid use increases, the prevalence of co-affected individuals, those who are addicted to opioids and are infected with HIV, increase significantly. We demonstrate that the dependence of the co-affected population on $ q&#95;u $ and $ q&#95;v $ are not monotone.

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., (© 2022. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

Vaccination is effective in preventing human papillomavirus (HPV) infection. It is imperative to investigate who should be vaccinated and what the best vaccine distribution strategy is. In this paper, we use a dynamic model to assess HPV vaccination strategies in a heterosexual population combined with gay, bisexual, and other men who have sex with men (MSM). The basic reproduction numbers for heterosexual females, heterosexual males and MSM as well as their average for the total population are obtained. We also derive a threshold parameter, based on basic reproduction numbers, for model analysis. From the analysis and numerical investigations, we have several conclusions. (1) To eliminate HPV infection, the priority of vaccination should be given to MSM, especially in countries that have already achieved high coverage in females. The heterosexual population gets great benefit but MSM only get minor benefit from vaccinating heterosexual females or males. (2) The best vaccination strategy is to vaccinate MSM firstly as many as possible, then heterosexual females, lastly heterosexual males. (3) Given a fixed vaccination coverage of MSM, distributing the remaining vaccines to only heterosexual females or males leads to a similar prevalence in the total population. This prevalence is lower than that when vaccines are distributed to both genders. The evener the distribution, the higher the prevalence in the total population. (4) Vaccination becomes less effective in reducing the prevalence as more vaccines are given. It is more effective to allocate vaccines to a region with lower vaccination coverage. This study provides information that may help policymakers formulate guidelines for vaccine distribution to reduce HPV prevalence on the basis of vaccine availability and prior vaccination coverage. Whether these guidelines are affected when the objective is to reduce HPV-associated cancer incidence remains to be further studied.

The ongoing COVID-19 pandemic has affected most of the countries on Earth. It has become a pandemic outbreak with more than 50 million confirmed infections and above 1 million deaths worldwide. In this study, we consider a mathematical model on COVID-19 transmission with the prosocial awareness effect. The proposed model can have four equilibrium states based on different parametric conditions. The local and global stability conditions for awareness-free, disease-free equilibrium are studied. Using Lyapunov function theory and LaSalle invariance principle, the disease-free equilibrium is shown globally asymptotically stable under some parametric constraints. The existence of unique awareness-free, endemic equilibrium and unique endemic equilibrium is presented. We calibrate our proposed model parameters to fit daily cases and deaths from Colombia and India. Sensitivity analysis indicates that the transmission rate and the learning factor related to awareness of susceptibles are very crucial for reduction in disease-related deaths. Finally, we assess the impact of prosocial awareness during the outbreak and compare this strategy with popular control measures. Results indicate that prosocial awareness has competitive potential to flatten the COVID-19 prevalence curve., Competing Interests: Conflict of interestThe authors declare that they have no conflict of interest., (© The Author(s), under exclusive licence to Springer Nature B.V. 2021.)

The transmission of avian influenza between humans is extremely rare, and it mostly affects individuals who are in contact with infected family member. Although this scenario is uncommon, there have been multiple outbreaks that occur in small infection clusters in Asia with relatively lowtransmissibility, and thus are too weak to cause an epidemic. Still, subcritical transmission from stut-tering chain data is vital for determining whether avian influenza is close to the threshold of ℜ₀ &gt; 1.In this article, we will explore two methods of estimating ℜ₀ using transmission chains and parameterestimation through data fitting. We found that ℜ₀ = 0.2205 when calculating the ℜ₀ using the maxi-mum likelihood method. When we computed the reproduction number for human to human transmis-sion through differential equations and fitted the model to data from the cumulative cases, cumulativedeaths, and cumulative secondary cases, we estimated ℜ₀ = 0.1768. To avoid violating the assumptionof the least square method, we fitted the model to incidence data to obtain ℜ₀ = 0.1520. We tested thestructural and practical identifiability of the model, and concluded that the model is identifiable undercertain assumptions. We further use two more methods to estimate ℜ₀ : by the ℜ₀ definition whichgives an overestimate of 0.28 and by Ferguson approach which yields ℜ₀ = 0.1586. We conclude that ℜ₀ for human to human transmission was about 0.2.

Based on the development of heroin vaccine, in this paper, we propose an age structured heroin transmission model with treatment and vaccination. The model allows the drug reuse rate of the individuals in treatment to depend on a treatment-age and the vaccine waning rate of the vaccinated to depend on a vaccination age. Meanwhile, the model allows that the heroin vaccine provides an imperfect protection (i.e., the vaccinated individuals can also become drug addicted). We derive the basic reproduction number which dependents on vaccination. The basic reproduction number completely determines the persistence and extinction of heroin spread, i.e., if the basic reproduction number is less than one the drug-free steady state is globally asymptotically stable (i.e., the heroin spread dies out), if the basic reproduction number is larger than one, there exists an unique positive steady state and it is locally and globally stable in some special cases. Finally, some numerical simulations are carried out to illustrate the stability of the positive steady state.

This article focuses on the study of an age-structured two-strain model with super-infection. The explicit expression of basic reproduction numbers and the invasion reproduction numbers corresponding to strain one and strain two are obtained. It is shown that the infection-free steady state is globally stable if the basic reproductive number R(0) is below one. Existence of strain one and strain two exclusive equilibria is established. Conditions for local stability or instability of the exclusive equilibria of the strain one and strain two are established. Existence of coexistence equilibrium is also obtained under the condition that both invasion reproduction numbers are larger than one.