arXiv subjects

Jonathan Hoseana

Publications and source records attributed to Jonathan Hoseana.

12 recordsLinked to original sources

arXiv2024

Design and Financial Analysis of a Health Insurance Based on an SIH-Type Epidemic Model

We present a design and financial analysis of a health insurance based on an SIH-type epidemic model. Specifically, we first construct the model in a continuous form, study its dynamical properties, and formulate the financial quantities involved in our insurance. Subsequently, we discretise the model using the forward Euler method, study the dynamical properties of the resulting discrete model, and formulate discrete analogues of the above financial quantities. We conduct a numerical simulation using two sets of parameter values, each representing a disease-free and an endemic scenario, which reveals that in the latter scenario, the insurance's gross premium is higher, the insurer's minimum loss-preventing start-up capital is lower, and the insurer's total profit is higher, compared to the corresponding values in the former scenario. Finally, through a sensitivity analysis, we show that in both scenarios, the disease's basic reproduction number, the gross premium, the minimum start-up capital, and the total profit depend most sensitively on the population's natural death coefficient, the disease's incidence coefficient, the hospitalisation benefit, and the premium surcharge percentage allocated to profit, respectively.

math.DS↗

arXiv2024

The Dynamics of One-Dimensional Quasi-Affine Maps

We study the dynamics of the one-dimensional quasi-affine map $x\mapsto \left\lfloor λx +μ\right\rfloor$, providing a complete description of the map's periodic points, and of the limit points of every $x\in\mathbb{R}$ under the map, for all real parameter values. Specifically, we establish the existence of regions of parameter values for which the map possesses $n$ fixed points for all $n\in\mathbb{N}_0\cup \{\infty\}$, an explicit formula for the number of 2-cycles possessed by the map, and the $ω$-limit set of any $x\in\mathbb{R}$ under the map, which, depending on the parameter values, is either a singleton of a fixed point, a 2-cycle, $\{-\infty,\infty\}$, $\{\infty\}$, or $\{-\infty\}$.

math.DS↗

arXiv2023

Combating New COVID-19 Variants in Indonesia: Is Accelerating Four-Dose Vaccinations Sufficient?

As new COVID-19 variants continue to emerge globally, we develop an SISI-type mathematical model to determine whether the transmission risks arising when such a new variant enters Indonesia can be mitigated by solely accelerating the country's ongoing four-dose vaccination programme. We begin by determining the model's basic reproduction number, as well as the model's equilibria and their stability. Subsequently, employing parameter values representing the country's situation as of March 20, 2023, we conduct a numerical sensitivity analysis in two simulated cases corresponding to two different levels of the new variant's transmission. The results show that, a satisfactory mitigation relying solely on vaccinations necessitates a drastic acceleration in the low-transmission case, and proves unachievable in the high-transmission case. Accordingly, we recommend that the acceleration of the ongoing four-dose vaccinations be carried out in conjunction with other intervention measures, such as improvements of the vaccine's efficacy and the disease's recovery rate.

math.DS↗

arXiv2023

Codimension-Two Bifurcations of an SIR-Type Model for COVID-19 and Their Epidemiological Implications

We study the codimension-two bifurcations exhibited by a recently-developed SIR-type mathematical model for the spread of COVID-19, as its two main parameters -- the susceptible individuals' cautiousness level and the hospitals' bed-occupancy rate -- vary over their domains. We use AUTO to generate the model's bifurcation diagrams near the relevant bifurcation points: two Bogdanov-Takens points and two generalised Hopf points, as well as a number of phase portraits describing the model's orbital behaviours for various pairs of parameter values near each bifurcation point. The analysis shows that, when a backward bifurcation occurs at the basic reproduction threshold, the transition of the model's asymptotic behaviour from endemic to disease-free takes place via an unexpectedly complex sequence of topological changes, involving the births and disappearances of not only equilibria but also limit cycles and homoclinic orbits. Epidemiologically, the analysis confirms the importance of a proper control of the values of the aforementioned parameters for a successful eradication of COVID-19. We recommend a number of strategies by which such a control may be achieved.

math.DS↗

arXiv2022

A Format for a Plagiarism-Proof Online Examination for Calculus and Linear Algebra Using Microsoft Excel

As educational systems move from onsite to online due to the COVID-19 pandemic, teachers face the difficulty of designing online examination formats which minimise opportunities for dishonesty. In this paper, we expose our design of such a format: a protected Microsoft Excel spreadsheet containing short-answer questions, which was implemented in a calculus module taught by us. This format allows examiners to randomise questions with the aim that each student receives each question with different numerical details, making plagiarism impossible, while keeping the marking effort very low.

math.HO↗

arXiv2022

Mathematical Analysis of an Epidemic Model for COVID-19: How Important Is the People's Cautiousness Level for Eradication?

We construct an SIR-type model for COVID-19, incorporating as a parameter the susceptible individuals' cautiousness level. We determine the model's basic reproduction number, study the stability of the equilibria analytically, and perform a sensitivity analysis to confirm the significance of the cautiousness level. Fixing specific values for all other parameters, we study numerically the model's dynamics as the cautiousness level varies, revealing backward transcritical, Hopf, and saddle-node bifurcations of equilibria, as well as homoclinic and fold bifurcations of limit cycles with the aid of AUTO. Considering some key events affecting the pandemic in Indonesia, we design a scenario in which the cautiousness level varies over time, and show that the model exhibits a hysteresis, whereby, a slight cautiousness decrease could bring a disease-free state to endemic, and this is reversible only by a drastic cautiousness increase, thereby mathematically justifying the importance of a high cautiousness level for resolving the pandemic.

math.DS↗

arXiv2022

From Pandemic to a New Normal: Strategies to Optimise Governmental Interventions in Indonesia Based on an SVEIQHR-Type Mathematical Model

There are five different forms of intervention presently realised by the Indonesian government in an effort to end the COVID-19 pandemic: vaccinations, social restrictions, tracings, testings, and treatments. In this paper, we construct an SVEIQHR-type mathematical model for the disease's spread in the country, which incorporates as parameters the rates of the above interventions, as well as the vaccine's efficacy. We determine the model's equilibria and basic reproduction number. Using the model, we formulate strategies by which the interventions should be realised in order to optimise their impact. The results show that, in a disease-free state, when the number of new cases rises, the best strategy is to implement social restrictions, whereas in an endemic state, if a near-lockdown policy is undesirable, carrying out vaccinations is the best strategy; however, efforts should be aimed not primarily towards increasing the vaccination rate, but towards the use of high-efficacy vaccines.

math.DS↗

arXiv2021

A Design of Governmental Policies for the Eradication of COVID-19 in Jakarta Using an SIR-Type Mathematical Model

Using a discretised version of our recently-developed SIR-type mathematical model for the spread of COVID-19, we construct a design of governmental policies for the eradication of the disease in the province of DKI Jakarta, Indonesia, taking as a basis the actual data of mid-2021. The design takes the form of a precise, quantitative method to determine the appropriate level(s) of restrictions on community activities (PPKM) which should be enforced in the province on any given day, based on the current values of the disease's effective reproduction number and the hospitals' bed-occupancy rate.

math.DS↗

arXiv2021

The Akiyama Mean-Median Map Has Unbounded Transit Time and Discontinuous Limit

Open conjectures state that, for every $x\in[0,1]$, the orbit $\left(x_n\right)_{n=1}^\infty$ of the mean-median recursion $$x_{n+1}=(n+1)\cdot\mathrm{median}\left(x_1,\ldots,x_{n}\right)-\left(x_1+\cdots+x_n\right),\quad n\geqslant 3,$$ with initial data $\left(x_1,x_2,x_3\right)=(0,x,1)$, is eventually constant, and that its transit time and limit functions (of $x$) are unbounded and continuous, respectively. In this paper we prove that, for the slightly modified recursion $$x_{n+1}=n\cdot\mathrm{median}\left(x_1,\ldots,x_{n}\right)-\left(x_1+\cdots+x_n\right),\quad n\geqslant 3,$$ first suggested by Akiyama, the transit time function is unbounded but the limit function is discontinuous.

math.DS↗

arXiv2020

The Prime-Power Map

We introduce a modification of Pillai's prime map: the prime-power map. This map fixes $1$, divides its argument by $p$ if it is a prime-power $p^k$, otherwise subtracts from its argument the largest prime-power not exceeding it. We study the iteration of this map over the positive integers, developing, firstly, results parallel to those known for the prime map. Subsequently, we compare its dynamical properties to those of a more manageable variant of the map under which any orbit admits an explicit description. Finally, we present some experimental observations, based on which we conjecture that almost every orbit of the prime-power map contains no prime-power.

math.DS↗

arXiv2019

On the Unboundedness of the Transit Time of Mean-Median Orbits

The transit time of mean-median orbits ---the time it takes for an orbit to become stationary--- has been conjectured to be finite but unbounded over the rationals. Through a study of some near-regular structures in these orbits, we construct two non-trivial sequences of initial sets of increasing size for which the transit time grows linearly and quadratically, respectively, with the size of the set.

math.DS↗

arXiv2019

Geometrical Properties of the Mean-Median Map

We study the mean-median map as a dynamical system on the space of finite sets of piecewise-affine continuous functions with rational coefficients. We determine the structure of the limit function in the neighbourhood of a distinctive family of rational points, the local minima. By constructing a simpler map which represents the dynamics in such neighbourhoods, we extend the results of Cellarosi and Munday (arXiv:1408.3454v1 [math.CO]) by two orders of magnitude. Based on these computations, we conjecture that the Hausdorff dimension of the graph of the limit function of the set $[0,x,1]$ is greater than 1.

math.DS↗