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  • 1. Cai, Liming
    et al.
    Li, Xuezhi
    Tuncer, Necibe
    Martcheva, Maia
    Lashari, Abid Ali
    Stockholm University, Faculty of Science, Department of Mathematics.
    Optimal control of a malaria model with asymptomatic class and superinfection2017In: Mathematical Biosciences, ISSN 0025-5564, E-ISSN 1879-3134, Vol. 288, p. 94-108Article in journal (Refereed)
    Abstract [en]

    In this paper, we introduce a malaria model with an asymptomatic class in human population and exposed classes in both human and vector populations. The model assumes that asymptomatic individuals can get re-infected and move to the symptomatic class. In the case of an incomplete treatment, symptomatic individuals move to the asymptomatic class. If successfully treated, the symptomatic individuals recover and move to the susceptible class. The basic reproduction number, R0,R0, is computed using the next generation approach. The system has a disease-free equilibrium (DFE) which is locally asymptomatically stable when R0<1,R0<1, and may have up to four endemic equilibria. The model exhibits backward bifurcation generated by two mechanisms; standard incidence and superinfection. If the model does not allow for superinfection or deaths due to the disease, then DFE is globally stable which suggests that backward bifurcation is no longer possible. Simulations suggest that total prevalence of malaria is the highest if all individuals show symptoms upon infection, but then undergoes an incomplete treatment and the lowest when all the individuals first move to the symptomatic class then treated successfully. Total prevalence is average if more individuals upon infection move to the asymptomatic class. We study optimal control strategies applied to bed-net use and treatment as main tools for reducing the total number of symptomatic and asymptomatic individuals. Simulations suggest that the optimal control strategies are very dynamic. Although they always lead to decrease in the symptomatic infectious individuals, they may lead to increase in the number of asymptomatic infectious individuals. This last scenario occurs if a large portion of newly infected individuals move to the symptomatic class but many of them do not complete treatment or if they all complete treatment but the superinfection rate of asymptomatic individuals is average.

  • 2.
    Lashari, Abid Ali
    National University of Sciences & Technology, Pakistan .
    Comment on Transmission Model of Hepatitis B Virus with the Migration Effect2015In: BioMed Research International, ISSN 2314-6133, E-ISSN 2314-6141, article id 469240Article in journal (Refereed)
    Abstract [en]

    Some consequences of erroneous results concerning eigenvalues in the recent literature of mathematical biology are highlighted. Furthermore, an improved stability criterion and the true value of the basic reproduction number is presented.

  • 3.
    Lashari, Abid Ali
    Stockholm University, Faculty of Science, Department of Mathematics. National University of Sciences and Technology, Pakistan.
    Comments and an improved result on "mathematical analysis of typhoid model with saturated incidence rate"2015In: Advanced Studies in Biology, ISSN 1313-9495, E-ISSN 1314-7668, Vol. 7, no 8, p. 389-392Article in journal (Refereed)
  • 4.
    Lashari, Abid Ali
    Stockholm University, Faculty of Science, Department of Mathematics. National University of Sciences and Technology, Pakistan.
    Optimal Control of an SIR Epidemic Model with a Saturated Treatment2016In: Applied Mathematics & Information Sciences, ISSN 1935-0090, E-ISSN 2325-0399, Vol. 10, no 1, p. 185-191Article in journal (Refereed)
    Abstract [en]

    In this paper, we formulate an optimal control problem for an SIR epidemic model with saturated incidence and saturated treatment. Two main efforts, namely treatment and vaccination are considered to limit the disease transmission. The impacts of vaccination and treatment on the disease transmission are discussed through the basic reproduction number. Then to achieve control of the disease, a control problem is formulated and the existence of the control is shown. Two control functions are used, one for vaccinating the susceptible and the other for treatment efforts for infectious individuals. Appropriate optimal control methods are used to characterize the optimal levels of the two controls. The effectiveness of the proposed control solution is shown by comparing the behavior of controlled and uncontrolled systems. Numerical results show the impacts of two controls in decreasing both susceptible and infectious members of the population.

  • 5.
    Lashari, Abid Ali
    Stockholm University, Faculty of Science, Department of Mathematics.
    Stochastic epidemics on random networks2019Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis considers stochastic epidemic models for the spread of epidemics in structured populations. The asymptotic behaviour of the models is analysed by using branching process approximations. The thesis contains four manuscripts.

    Paper I is concerned with the study of the spread of sexually transmitted infections, or any other infectious diseases on a dynamic network. The model we investigate is about the spread of an SI (Susceptible → Infectious) type infectious disease in a population where partnerships are dynamic. We derive explicit formulas for the probability of extinction and the threshold parameter R0 using two branching process approximations for the model. In the first approximation some dependencies between infected individuals are ignored while the second branching process approximation is asymptotically exact and only defined if every individual in the population can have at most one partner at a time. By comparing the two approximations, we show that ignoring subtle dependencies in the dynamic epidemic model leads to wrong prediction of the probability of a large outbreak.

    In paper II, we study a stochastic SIR (Susceptible → Infectious → Removed) epidemic model for the spread of an epidemic in populations structured through configuration model random graphs. We study the asymptotic (properly scaled) time until the end of an epidemic. This paper heavily relies on the theory of branching processes in continuous time.

    In paper III, the effect of vaccination strategies on the duration of an epidemic in a large population is investigated. We consider three vaccination strategies: uniform vaccination, leaky vaccination and acquaintance vaccination.

    In paper IV, we present a stochastic model for two successive SIR epidemics in the same network structured population. Individuals infected during the first epidemic might have (partial) immunity for the second one. The first epidemic is analysed through a bond percolation model, while the second epidemic is approximated by a three-type branching process in which the types of individuals depend on their status in the percolation clusters used for the analysis of the first epidemic. This branching process approximation enables us to calculate a threshold parameter and the probability of a large outbreak for the second epidemic. We use two special cases of acquired immunity for further evaluation.

  • 6.
    Lashari, Abid Ali
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Lee, Kwang Sung
    STABILITY ANALYSIS OF A HOST-VECTOR TRANSMISSION MODEL FOR PINE WILT DISEASE WITH ASYMPTOMATIC CARRIER TREES2017In: Journal of the Korean Mathematical Society, ISSN 0304-9914, E-ISSN 2234-3008, Vol. 54, no 3, p. 987-997Article in journal (Refereed)
    Abstract [en]

    A deterministic model for the spread of pine wilt disease with asymptomatic carrier trees in the host pine population is designed and rigorously analyzed. We have taken four different classes for the trees, namely susceptible, exposed, asymptomatic carrier and infected, and two different classes for the vector population, namely susceptible and infected. A complete global analysis of the model is given, which reveals that the global dynamics of the disease is completely determined by the associated basic reproduction number, denoted by R-0. If R-0 is less than one, the disease-free equilibrium is globally asymptotically stable, and in such a case, the endemic equilibrium does not exist. If R-0 is greater than one, the disease persists and the unique endemic equilibrium is globally asymptotically stable.

  • 7.
    Lashari, Abid Ali
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Branching process approach for epidemics in dynamic partnership network2018In: Journal of Mathematical Biology, ISSN 0303-6812, E-ISSN 1432-1416, Vol. 76, no 1-2, p. 265-294Article in journal (Refereed)
    Abstract [en]

    We study the spread of sexually transmitted infections (STIs) and other infectious diseases on a dynamic network by using a branching process approach. The nodes in the network represent the sexually active individuals, while connections represent sexual partnerships. This network is dynamic as partnerships are formed and broken over time and individuals enter and leave the sexually active population due to demography. We assume that individuals enter the sexually active network with a random number of partners, chosen according to a suitable distribution and that the maximal number of partners that an individual can have at a time is finite. We discuss two different branching process approximations for the initial stages of an outbreak of the STI. In the first approximation we ignore some dependencies between infected individuals. We compute the offspring mean of this approximating branching process and discuss its relation to the basic reproduction number R0. The second branching process approximation is asymptotically exact, but only defined if individuals can have at most one partner at a time. For this model we compute the probability of a minor outbreak of the epidemic starting with one or few initial cases. We illustrate complications caused by dependencies in the epidemic model by showing that if individuals have at most one partner at a time, the probabilities of extinction of the two approximating branching processes are different. This implies that ignoring dependencies in the epidemic model leads to a wrong prediction of the probability of a large outbreak. Finally, we analyse the first branching process approximation if the number of partners an individual can have at a given time is unbounded. In this model we show that the branching process approximation is asymptomatically exact as the population size goes to infinity.

  • 8.
    Lashari, Abid
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Serafimović, Ana
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    The duration of an SIR epidemic on a configuration modelManuscript (preprint) (Other academic)
    Abstract [en]

    We consider the spread of a stochastic SIR (Susceptible, Infec-tious, Recovered) epidemic on a configuration model random graph.We focus especially on the final stages of the outbreak and providelimit results for the duration of the entire epidemic, while we allowfor non-exponential distributions of the infectious period and for bothfinite and infinite variance of the asymptotic degree distribution in thegraph.

    Our analysis relies on the analysis of some subcritical continuoustime branching processes and on ideas from first-passage percolation.

  • 9.
    Lashari, Abid
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Effect of vaccination on the duration of an SIR epidemic in homogeneously mixing and structured populationsManuscript (preprint) (Other academic)
    Abstract [en]

    This paper is concerned with the effects of vaccination on the properly scaled dura-tion of the stochastic SIR (Susceptible → Infected → Recovered/Removed) epidemicboth in homogeneous mixing populations and in populations structured through config-uration model random graphs. We examine uniform vaccination and leaky vaccinationin both homogeneous and structured populations. Furthermore, we consider acquain-tance vaccination on the configuration model. For these vaccination schemes, we studythe asymptotic time until the end of the epidemic and study the effect of the vaccinationon this duration.

    We show that, depending on the degree distribution, uniform vaccination witha perfect vaccine may lead to both an increase and decrease in the duration of anepidemic in structured populations, whereas in homogeneously mixing populations,vaccination with a perfect vaccine either prevents or prolongs an epidemic in the largepopulation limit. In homogeneously mixing populations, the leaky vaccine has a similareffect as uniform vaccination on the duration of the epidemic, whereas in structuredpopulations, we conjecture that the leaky vaccine always increases the duration of anepidemic. For the acquaintance vaccination scheme we give, through the derivationof the effective degree distribution of unvaccinated individuals, a recipe to obtain theasymptotic duration of an epidemic and show that acquaintance vaccination may bothdecrease and increase the duration of an epidemic.

  • 10.
    Lashari, Abid
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Trapman, Pieter
    Stockholm University, Faculty of Science, Department of Mathematics.
    Ball, Frank
    Sirl, David
    Modeling the spread of two successive SIR epidemics on a configuration model networkManuscript (preprint) (Other academic)
    Abstract [en]

    We present a stochastic model for two successive SIR (Susceptible, Infectious, Recov-ered) epidemics in the same network structured population. Individuals infected duringthe first epidemic might have (partial) immunity for the second one. The first epidemic isanalysed through a bond percolation model, while the second epidemic is approximated bya three-type branching process in which the types of individuals depend on their position inthe percolation clusters used for the first epidemic. This branching process approximationenables us to calculate a threshold parameter and the probability of a large outbreak for thesecond epidemic.

    We illustrate our results through two examples. In the first example individuals infectedby the first epidemic are independently either completely susceptible or completely immuneto the second epidemic. The probability of being completely immune is the same for allindividuals infected in the first epidemic. In the second example the recovered individual inthe first epidemic have reduced susceptibility and infectivity for the second epidemic.

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