The McKendrick partial differential equation and its uses in epidemiology and population study

Keyfitz, Barbara Lee; Keyfitz, Nathan

doi:10.1016/s0895-7177(97)00165-9

Cited by 120 publications

(88 citation statements)

References 10 publications

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“…This comment applies to the discrete-time system (2.6) as well. [56,57] or models for blowfly [58], respectively. These models fall beyond the scope of (2.32) and require more sophisticated mathematical treatment, but their semidiscretizations (leading to systems of ordinary differential equations) may result in models of the form (2.32).…”

Section: Continuous-timementioning

confidence: 99%

Simple Adaptive Control for Positive Linear Systems with Applications to Pest Management

Guiver¹,

Edholm²,

Jin³

et al. 2016

SIAM J. Appl. Math.

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Abstract. Pest management is vitally important for modern arable farming, but models for pest species are often highly uncertain. In the context of pest management, control actions are naturally described by a nonlinear feedback that is generally unknown, which thus motivates a robust control approach. We argue that adaptive approaches are well suited for the management of pests and propose a simple high-gain adaptive tuning mechanism so that the nonlinear feedback achieves exponential stabilization. Furthermore, a switched adaptive controller is proposed, cycling through a set of given control actions, that also achieves global asymptotic stability. Such a model in practice allows for the possibility of rotating between different courses of management action. In developing our control strategies we appeal to comparison and monotonicity arguments. Interestingly, componentwise nonnegativity of the model, combined with an irreducibility assumption, implies that several issues typically associated with high-gain adaptive controllers do not arise and usual high-gain structural assumptions are not required.

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Section: Continuous-timementioning

confidence: 99%

Simple Adaptive Control for Positive Linear Systems with Applications to Pest Management

Guiver¹,

Edholm²,

Jin³

et al. 2016

SIAM J. Appl. Math.

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“…Assume that the dynamics of the population are described by the McKendrick equation (see Keyfitz and Keyfitz (1997)). …”

Section: Optimal Trade-off Between Consumption and Health Expenditurementioning

confidence: 99%

“…susceptible vs. infected individuals in section 2.2) and is a distributed state of the same form as Y i (a, t). The dynamics are described by a McKendrick type equation (see Keyfitz and Keyfitz (1997)), i.e.…”

Section: Boundary Condition) Is Modeled By Y (0 T) = ϕ(T B(t) Q(t)mentioning

confidence: 99%

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The reproductive value in distributed optimal control models

Wrzaczek

Kühn

Prskawetz

et al. 2010

Theoretical Population Biology

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may The Reproductive Value in Distributed Optimal Control Models AbstractWe show that in a large class of distributed optimal control models (DOCM), where population is described by a McKendrick type equation with an endogenous number of newborns, the reproductive value of Fisher shows up as part of the shadow price of the population. Depending on the objective function, the reproductive value may be negative. Moreover, we show results of the reproductive value for changing vital rates. To motivate and demonstrate the general framework, we provide examples in health economics, epidemiology, and population biology.

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“…There are two types of population growth models: continuous-time and continuousage models (Sharpe and Lotka, 1911;McKendrick., 1926;Von Foerster, 1959) and discrete time-variable and discrete-age scale model (Bernadelli, 1941;Leslie 1945). Keyfitz and Keyfitz (1997) compare the McKendrick-von Foerster equation with discrete (i.e. Leslie-type) models and show some advantages of the continuous model.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a Two Subpopulations System Including Immigration

Zincenko

Petrovskii

2017

Math. Model. Nat. Phenom.

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Abstract. The phenomenon of replacement migration into declining population prompts development of multicomponent models in population dynamics. We propose a simple model of population including resident and migrant components with migration flow as an external input. The main assumption is that offspring that are born to migrants will have the same vital rates as the resident population. The proposed model is based on partial differential equation to take into account the age structure of the population. The formulae for exact solutions are derived. We focus on the case when native population declines in the absence of migration. Assuming a sufficiently large constant growth rate of migration we obtain asymptotic solutions as t → ∞. Using the asymptotic solutions, we have calculated "critical" value of growth rate of migrant inflow that is the value that provides, as time tends to infinity, the equal number of residents and migrants in the population. We provide numerical illustrations using demographic data for Germany in 2010.

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The McKendrick partial differential equation and its uses in epidemiology and population study

Cited by 120 publications

References 10 publications

Simple Adaptive Control for Positive Linear Systems with Applications to Pest Management

Simple Adaptive Control for Positive Linear Systems with Applications to Pest Management

The reproductive value in distributed optimal control models

Dynamics of a Two Subpopulations System Including Immigration

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