# COVID-ABS: An Agent-Based Model of COVID-19 Epidemic to Simulate Health and
Economic Effects of Social Distancing Interventions

Petrônio C. L. Silva  Paulo V. C. Batista  Hélder S. Lima  Marcos A. Alves
Frederico G. Guimarães  Rodrigo C. P. Silva  Grupo de Pesquisa em Ciência de
Dados e Inteligência Computacional - $\\{ci\partial ic\\}$ Instituto Federal
do Norte de Minas Gerais (IFNMG), Brazil  Machine Intelligence and Data
Science (MINDS) Laboratory, Federal University of Minas Gerais, Brazil
Department of Electrical Engineering, Universidade Federal de Minas Gerais
(UFMG), Brazil  Graduate Program in Electrical Engineering - Universidade
Federal de Minas Gerais - Av. Antônio Carlos 6627, 31270-901, Belo Horizonte,
MG, Brazil  Department of Computer Science, Universidade Federal de Ouro Preto
(UFOP), Brazil

###### Abstract

The COVID-19 pandemic due to the SARS-CoV-2 coronavirus has directly impacted
the public health and economy worldwide. To overcome this problem, countries
have adopted different policies and non-pharmaceutical interventions for
controlling the spread of the virus. This paper proposes the COVID-ABS, a new
SEIR (Susceptible-Exposed-Infected-Recovered) agent-based model that aims to
simulate the pandemic dynamics using a society of agents emulating people,
business and government. Seven different scenarios of social distancing
interventions were analyzed, with varying epidemiological and economic
effects: (1) do nothing, (2) lockdown, (3) conditional lockdown, (4) vertical
isolation, (5) partial isolation, (6) use of face masks, and (7) use of face
masks together with 50% of adhesion to social isolation. In the impossibility
of implementing scenarios with lockdown, which present the lowest number of
deaths and highest impact on the economy, scenarios combining the use of face
masks and partial isolation can be the more realistic for implementation in
terms of social cooperation. The COVID-ABS model was implemented in Python
programming language, with source code publicly available. The model can be
easily extended to other societies by changing the input parameters, as well
as allowing the creation of a multitude of other scenarios. Therefore, it is a
useful tool to assist politicians and health authorities to plan their actions
against the COVID-19 epidemic.

###### keywords:

COVID-19 , Agent-based Simulation , epidemic models , SEIR

††journal: Chaos, Solitons & Fractals

##  1 Introduction

The Coronavirus disease 2019 (COVID-19) pandemic is an ongoing outbreak,
caused by severe acute respiratory syndrome coronavirus 2 (so-called
SARS‑CoV‑2). The outbreak was identified in Wuhan, China, in December 2019
[1]. The World Health Organization (WHO) declared the outbreak a Public Health
Emergency of International Concern on January 30th 2020, and a pandemic on
March 11th. In Brazil, the first confirmed case was on February 25th 2020,
when a man from São Paulo tested positive for the virus. Since then, Brazil
has been severely affected. As of June 26th 2020, the country reached more
than 1,220,000 confirmed cases and more than 55,000 deaths by COVID-19,
according to official data by the Brazilian Ministry of Health.

In addition to the public health crisis, the coronavirus has impacted all
aspects of life, politics, education, economy, social, environment and
climate. It is also having an unprecedented impact on global supply chains and
production. The only known effective course of action to fight the disease
outbreak is to implement highly restrictive social distancing measures on the
population, as reported by a number of studies and systematic reviews [2, 3,
4]. Many countries are implementing such interventions with different degrees
of success.

Given the complexity of the societies, it is hard to predict the implications
of such actions in the short and medium terms [5]. Therefore, modeling and
simulating the COVID-19 epidemic is a relevant and helpful way to understand
the spread of the disease and the epidemiological effects of social distancing
interventions. For this purpose, many studies in the literature have developed
or adapted equation-based models to simulate the COVID-19 epidemic, using the
Susceptible-Infected-Recovered (SIR) model or the Susceptible-Exposed-
Infected-Recovered (SEIR) model to characterize the dynamics, see references
in Section 2. Nonetheless, agent-based models have also been proposed for this
goal, see for instance [6, 7] and other studies discussed with more detail in
Section 2.

In this paper, we develop an Agent-based Model (ABM) to simulate the dynamics
of the COVID-19 epidemic and the epidemiological and economic effects of
social distancing interventions. The proposed ABM aims to emulate a closed
society living on a shared environment, consisting of agents that represent
people, houses, businesses, the government and the healthcare system, each one
with specific attributes and behaviors.

A society living over a territory is a complex and dynamic system. Such
systems have many interacting variables, present nonlinear behavior and their
properties evolve over time. Their behavior is generally stochastic and may
depend on the initial conditions. It can be affected by neighbor societies
(with different policies and dynamics) and it can show emergence of complex
behaviors and patterns. Agent-Based Simulations (ABS) are a good choice to
simulate such systems, due to their simplicity of implementation and accurate
results when compared with real data [8]. The main goal of ABS is to simulate
the temporal evolution of the system, storing statistics derived from the
internal states of the agents in each iteration and the global behaviors that
emerge due to the interactions between the agents over the iterations. This
approach allows the simulation of systems with intricate nonlinear
relationships, complex conditions and restrictions that may be hard to
describe mathematically. Since in this paper we are interested in simulating
the effects of different social-distancing interventions and other control
measures that affect the behaviors of agents and groups of agents, it is much
easier to simulate these scenarios with an agent-based model. The
epidemiological and economic effects are observed as emerging from the
interactions of the agents in the simulation.

The ABM proposed here not only simulates the epidemic dynamics but also models
the economy in this society of agents, which can help us estimate the economic
impact under different types of interventions. The model (described in Section
3) allows the design of scenarios that correspond to different types of
interventions performed in the society, by changing the simulation
environmental variables and measuring their effects. Therefore, the proposed
ABM becomes a useful tool to assist politicians and health authorities in
planning their actions against the COVID-19 epidemic. The model was
implemented in Python version 3.6 programming language and encapsulated in the
COVID_ABS package, whose source code is available at
https://bit.ly/COVID19_ABSsystem. The source code of all the experiments
reported herein is available at https://bit.ly/covid_abs_experiments.

The main contributions and findings are listed below:

  * 1.

A new SEIR agent-based model to simulate the COVID-19 epidemic using a society
of agents.

  * 2.

Assessment of the economic effects of seven different scenarios with specific
social-distancing interventions, via simulation of COVID-ABS: (1) do nothing,
(2) lockdown, (3) conditional lockdown, (4) vertical isolation, (5) partial
isolation, (6) use of face masks, and (7) use of face masks together with 50%
of adhesion to social isolation. These scenarios and their simulated results
are described in Section 5.

  * 3.

The simulations support the notion that lockdown and conditional lockdown are
the best scenarios in terms of controlling the number of infected and deaths,
which is primary goal. Economical countermeasures and subsidies are required
by the government since this scenario presents the worst economic losses to
the industry with potential unemployment, and recession can be observed during
the lockdown period. Also, to be effective, these scenarios depend on the
ability of the government to enforce the social isolation.

  * 4.

Our simulations present additional evidence that the so-called vertical
isolation simply does not work, although it is the policy advocated by some
governments like the Brazilian
one111https://agenciabrasil.ebc.com.br/en/politica/noticia/2020-04/bolsonaro-
brazil-must-not-be-informed-through-panic.

  * 5.

The scenario combining the use of masks and partial isolation of the
population could be a good compromise and it is more realistic for
implementation in terms of social cooperation. The infection curve is
flattened and the economy has smoother effects than the scenarios with
lockdown.

The rest of the paper is organized as follows: Section 2 provides a brief
review of related work with focus on the mathematical modeling for epidemics
and some recent papers related to the SARS-CoV-2. Section 3 details the
proposed agent-based system modeling. Section 4 describes the experimental
methodology and Section 5 shows the simulations results and some discussions
related to the pandemic. Section 6 concludes the paper and gives future
directions.

##  2 Related Work

Since WHO announced the Coronavirus Disease 2019, the scientific community has
been working hard to investigate SARS-CoV-2 epidemiological dynamics. Some
works used the SIR model to characterize the COVID-19 dynamics [9, 10, 11,
12]. However, more precise simulations usually used an approach based on the
SEIR model [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28,
29, 30]; given that this disease has a known incubation period [31]. Some
authors added new states to refine the model, for instance, super-spreaders
[32] or isolated [19, 14, 27, 26, 24, 23, 20, 18, 16, 13], hospitalized [19,
14, 27, 23, 20, 13], and asymptomatic infected [28, 23, 21, 20, 18].

We note that equation-based models to simulate the epidemic represent the
majority among those proposed in the literature. Nonetheless, some papers with
agent-based models have also been proposed for it. For a discussion about ABM
and its advantages over equation-based models, we refer the reader to [8, 33].
In the report released by Ferguson et al. [6], an individual-based simulation
model was used to explore scenarios for COVID-19 in GB and USA and the impact
of non-pharmaceutical interventions on the healthcare demand. Bossert et al.
[7] developed an agent-based model combining socio-economic and traffic data
to analyze COVID-19 spreading in a South Africa city under social isolation
scenarios. The prediction suggests that lockdown strategy is useful to
mitigate the disease. Another study using an ABM also analyzed several
scenarios and highlighted that with 90% of the population in isolation, it is
possible to control the disease within 13 weeks when joined with effective
case isolation and international travel restrictions, considering the
Australian context [34]. An appealing characteristic of agent-based modeling
is the easiness to simulate different scenarios. For instance, the scenario
that considers universal use of masks integrated with social distance is the
recommended one to control the pandemic according to Braun et al. [35] and Kai
et al. [36]. Given the flexibility of the agent-based approach, previous works
have employed this method to simulate specific topics in the COVID-19 context,
such as testing policies [37], strategies for reopening public buildings [38],
hypothetical effective treatments [39], and a spatio-temporal strategy for
vaccination [40].

Few works in the literature used agent-based models to simulate the economic
impacts of the COVID-19. For instance, Inoue and Todo [41] quantified that a
possible one month lockdown in Tokyo would lead to a total production loss of
5.3% in Japanese annual gross domestic product (GDP). Dignum et al. [42]
proposed a tool to analyze the health, social, and economic impacts of the
pandemic when the government implements a number of interventions, such as
closing schools, requiring that employees work at home, and providing subsidy
for the population.

In this work, we use a SEIR agent-based model to simulate the health and
economic impacts of the COVID-19 epidemic. We perform analysis to seven
possible scenarios: (1) do nothing, (2) lockdown, (3) conditional lockdown,
(4) vertical isolation, (5) partial isolation, (6) use of face masks, and (7)
use of face masks together with 50% of social isolation. We use data from
Brazil for all scenarios considered but the proposed agent-based model is
fully parameterized and can be easily transferred to other contexts given that
corresponding data is provided.

##  3 COVID-ABS: Proposed Agent-based System Modeling

The proposed agent-based approach aims to emulate a closed society living on a
shared finite environment, composed of humans, which are organized in
families, business and government, which interact with each other. This
characterization is trying to cover the main elements of the society. The
agents, their attributes and possible actions are described in Table 1.

A1: Person
---
Description |  $A1$ is the main type of agent. Its dynamic position varies according to the environment and may be associated with $A2$, or not (homeless) and $A3$, or not (unemployed).
Attributes |  Position (dynamic), Age, House ($A2$), Employer ($A3$), Epidemiological status, Infection status, Wealth, Income and Social Stratum
Actions |  Walk freely (daily), Go home (daily), Go to work (daily), Personal contact (hourly), Business contact (hourly), Go to the hospital
A2: Houses
Description |  $A2$ represent the families. They share a house and financial bills.
Attributes |  Position (static), Social stratum, Housemates (group of $A1$), Wealth, Incomes and Expenses
Actions |  Homemate check-in (daily), Accounting (monthly)
A3: Business
Description |  $A3$ are the economical agents, e.g. industries, shops or markets. It interacts with $A1$ by paying a salary or selling a product.
Attributes |  Position (static), Social stratum, Employees (group of $A1$), Wealth, Incomes and Expenses
Actions |  Accounting (monthly), Business contact (hourly)
A4: Government
Description |  $A4$ is a singleton agent that receives taxes from $A2$ and $A3$, provide funds to $A5$ and insurance for homeless and unemployed $A1$.
Attributes |  Position (static), Wealth
Actions |  Accounting (monthly)
A5: Healthcare System
Description |  $A5$ is also a singleton, which represents the health system that ideally should be able to serve the entire population.
Attributes |  Position (static), Wealth
Table 1: Types of agents and their attributes and actions.

The model is an iterative procedure, with $T$ representing the number of
iterations. The model takes an input parameter set $P$, listed in Table 2, and
produces a response $Y_{t}$ (observable variables), related to epidemiological
or economic effects of the pandemic. Its internal state $\Theta_{t}$
($t=1\ldots T$) consists of the union of the internal states of the agents
$\theta^{i}_{t}$, where $i=1\ldots n$ and $n$ is the number of agents, such
that $\Theta_{t}=\bigcup_{i=1}^{n}\theta^{i}_{t}$.

Variable |  Domain/ Unit |  Current value |  References and Observations
---|---|---|---
Social and Demographic
$\alpha_{1}$ \- Height |  ${\mathbb{N}^{+}}$ |  $500$ |  Defined empirically. Each unit corresponds to 7 meters.
$\alpha_{2}$ \- Width |  ${\mathbb{N}^{+}}$ |  $500$ |  Defined empirically. Each unit corresponds to 7 meters.
$\alpha_{3}$ \- Population size |  ${\mathbb{N}^{+}}$/people |  $300$ |  Defined empirically.
$\alpha_{4}$ \- Age |  $[0,100]$ |  $\beta(2,4)$ |  [43]
$\alpha_{5}$ \- Average family size |  ${\mathbb{N}^{+}}$/people |  $3$ |  [44]
$\alpha_{6}$ \- Mobility |  ${\mathbb{N}^{+}}$ |  $10$ |  Defined empirically. Each unit corresponds to 7 meters.
$\alpha_{7}$ \- Homeless rate |  $[0,1]$ |  $0.0005$ |  [45]
Epidemiological
$\beta_{1}$ \- Contagion distance |  ${\mathbb{R}^{+}}$ |  $1$ |  [46]
$\beta_{2}$ \- Contagion probability |  $[0,1]$ |  $0.9$ |  [46]
$\beta_{3}$ \- Incubation time |  ${\mathbb{N}^{+}}$/days |  $5-6$ |  [47, 48]
$\beta_{4}$ \- Transmission time |  ${\mathbb{N}^{+}}$/days |  $8-10$ |  [49]
$\beta_{5}$ \- Recovering time |  ${\mathbb{N}^{+}}$/days |  $20$ |  [50]
$\beta_{6}$ \- Hospitalization rate per age |  $[0,1]$ |  Table 6 |  [46]
$\beta_{7}$ \- Severe cases rate per age |  $[0,1]$ |  Table 6 |  [46]
$\beta_{8}$ \- Death rate per age |  $[0,1]$ |  Table 6 |  [46]
$\beta_{9}$ \- % initial infected |  $[0,1]$ |  $0.01$ |  Defined by the authors.
$\beta_{10}$ \- % initial immune |  $[0,1]$ |  $0.01$ |  Defined by the authors.
$\beta_{11}$ \- Critical limit of the Health System |  $[0,1]$ |  $0.05$ |  The proportion of ICU beds to the population
Economical
$\gamma_{1}$ \- Income distribution |  |  Table 4 |  [51, 52]
$\gamma_{2}$ \- Proportion of businesses |  ${\mathbb{R}^{+}}$ |  0,01875 |  Considering the number of businesses per 100k inhabitants [53]
$\gamma_{3}$ \- Total GDP |  ${\mathbb{R}^{+}}$/R$ |  $1.000.000,00$ |  Defined by the authors.
$\gamma_{4}$ \- Public GPD rate |  $[0,1]$ |  $0.01$ |  Defined by the authors.
$\gamma_{5}$ \- Business GPD rate |  $[0,1]$ |  $0.05$ |  Defined by the authors.
$\gamma_{6}$ \- Personal GPD |  $[0,1]$ |  $0.04$ |  $\gamma_{6}=1-\%A4-\%A3$
$\gamma_{7}$ \- Minimum income |  ${\mathbb{R}^{+}}$/R$ |  $900,00$ |
$\gamma_{8}$ \- Minimum expenses |  ${\mathbb{R}^{+}}$/R$ |  $600,00$ |
$\gamma_{9}$ \- Unemployment rate |  $[0,1]$ |  $0.12$ |  [54]
$\gamma_{10}$ \- Proportion of informal businesses |  $[0,1]$ |  $0.40$ |  Informal economy [54, 55]
$\gamma_{11}$ \- EAP age group |  |  |  $16<\text{EAP}<65$
Table 2: Definitions of the parameters of the proposed ABS model

The model is described in Algorithm 1. The initialization of internal states
in line 2, discussed in subsection 3.2, creates the agents. The simulation
dynamics starts in line 3, discussed in subsection 3.3, and depends on the
type of the agent, the parameter $P$ and the current iteration $t$ (discrete
time). As mentioned before, each type of agent has its own set of actions in
different time frames (hourly, daily, weekly or monthly).

$P$ the parameter set, $T$ the number of iterations

$\Theta_{0}\leftarrow$ initialize($P$)

for $t\leftarrow 1\textrm{ to }T$ do

for all agent $a_{i}\in\Theta_{t}$ do

$\theta^{i}_{t}\leftarrow a_{i}$.execute_actions($t,P,\Theta_{t}$)

if type of $a_{i}=A1$ then

for all agent $a_{j}\in\Theta_{t}~{}|~{}i\neq j$ do

if distance($a_{i},a_{j}$) $\leq\delta$ then

$a_{i}$.contact($a_{j}$)

end if

end for

end if

end for

$Y_{t}\leftarrow$ summarize($\Theta_{t}$)

$\Theta_{t+1}\leftarrow\bigcup_{i=1}^{n}\theta^{i}_{t}$

end for

Algorithm 1 General procedure of the proposed agent-based approach

At each iteration, it checks if there was contact between any pair of agents.
A contact happens when the distance between any two agents is less than or
equal to a threshold $\delta$ defined in $P$. The contact can be
epidemiological (if the agents are of type A1) or economical (A1 and A3). The
computation of the distance between each pair of agents per iteration makes
the asymptotic complexity of the method equal to $O(n^{2})$, where $n$ is the
number of agents.

Variable |  Description
---|---
Epidemiological
$S_{t}$ |  Percentage of Susceptible agents in population
$I_{t}$ |  Percentage of Infected agents in population
$I^{A}_{t}$ |  Percentage of Infected Asymptomatic agents in population
$I^{H}_{t}$ |  Percentage of Infected Hospitalized agents in population
$I^{S}_{t}$ |  Percentage of Infected Severe agents in population
$R_{t}$ |  Percentage of Recovered and Immune agents in population
$D_{t}$ |  Percentage of Dead agents in population
Economical
$W^{A1}_{S,t}$ |  Percentage of Gross Domestic Product owned by the people (A1 agents) at time $t$ under scenario $S$
$W^{A3}_{S,t}$ |  Percentage of Gross Domestic Product owned by businesses (A3 agents) at time $t$ under scenario $S$
$W^{A4}_{S,t}$ |  Percentage of Gross Domestic Product owned by government (A4 agent) at time $t$ under scenario $S$
Table 3: Response Variables

###  3.1 Parameter estimation

Some parameters in Table 2 were empirically estimated, in a way that the
epidemiological response variables present in Table 3 correspond to those
produced by a SEIR model. For that purpose the Epidemic
Calculator222https://gabgoh.github.io/COVID/ was employed – this is an open-
source SEIR implementation and visual tool for epidemic simulations. The
initial percentage of infected ($\beta_{9}$) and immune ($\beta_{10}$) agents
were chosen in order to represent the complete epidemic dynamics.

The population size parameter, $\alpha_{3}$, is particularly concerning
because it affects the execution time of the simulation. On the other hand,
the Population density, defined as $\alpha_{3}/(\alpha_{1}\alpha_{2})$,
follows the population density of the area under study which is 24 people per
km2. The mobility parameter $\alpha_{6}$ was empirically estimated as the
average range that a person walks randomly in his free time.

The Total GDP, parameter $\gamma_{3}$, and the percentage rates by kind of
agent ($\gamma_{4},\gamma_{5}$ and $\gamma_{6}$) are abstractions of closed
local economy. The minimum income $\gamma_{7}$ represents the minimum net
salary, the nominal income after taxes, and the minimum expense $\gamma_{8}$
represents the approximate market value of a basic needs grocery pack.

###  3.2 Initialization

The simulation is performed in a squared bi-dimensional environment shared by
all types of agents. $Ai$ agents, $i\in\\{2,3,4,5\\}$, are randomly
initialized inside this environment given by Equation (1).

$\centering
Ai_{pos}=\left\\{\begin{array}[]{ll}x\sim\mathcal{U}(0,\alpha_{1})\\\
y\sim\mathcal{U}(0,\alpha_{2})\end{array}\right.\@add@centering$ (1)

where $\mathcal{U}(a,b)$ is a sample from a uniform distribution in the
interval $[a,b)$.

Agents $A1$ are initialized in their $A2$ location, following Equation (2),
where $\sigma_{k}$ is the variability of the position inside the house. For
homeless agents, Equation (1) is used.

$A1_{pos}=A2_{pos}+\mathcal{N}(0,\sigma_{k})$ (2)

where $\mathcal{N}(\mu,\sigma)$ is a sample from a normal distribution with
mean $\mu$ and standard deviation $\sigma$.

The number of $A1$ agents is controlled by the variable population size, that
is $|A1|=\alpha_{3}$. The number of houses ($A2$ agents) is calculated using
Equation (3) considering the average family size $\alpha_{5}$:

$|A2|=\left\lceil\frac{\alpha_{3}}{\alpha_{5}}\right\rceil$ (3)

The number of $A3$ agents is calculated according to Equation (4), considering
the population size $\alpha_{3}$, the proportion of formal and informal
businesses, $\gamma_{2}$ and $\gamma_{10}$, respectively.

$|A3|=\lceil\alpha_{3}\gamma_{2}+\alpha_{3}\gamma_{10}\rceil$ (4)

When a person, $A1$ type, is created, it is assigned to a randomly chosen
house, type $A2$, or it is considered homeless according to the Homeless Rate,
$\alpha_{7}$. Parameter $\gamma_{9}$ defines the probability of an $A1$ to be
unemployed. If a person is employed and belongs to Economical Active
Population (EAP) (controlled by $\gamma_{11}$) an employer is randomly chosen
among the available $A3$s. A single instance of $A4$ and $A5$ agents are
created.

The age distribution of $A1$ agents is given by $\alpha_{4}$ parameter, such
as $A1_{age}\sim\beta(2,5)$ as explained in [56], where $\beta(a,b)$ is the
beta distribution with shape parameters $a,b$.

The social stratum of $A1,A2$ and $A3$ is represented by the income
distribution $\gamma_{1}$, listed in the Table 4, meaning the slice of the
wealth represented by the GDP parameter $\gamma_{3}$. The social stratum of
agents is sampled such that $Ai_{stratum}\sim\mathcal{U}(1,5)$, for
$i=\\{1,2,3\\}$. The total wealth of the simulation, represented by
$\gamma_{3}$, is shared among agents, according to public, business and
personal percentages defined by $\gamma_{4}$, $\gamma_{5}$ and $\gamma_{6}$.

Quintile | Social Stratum |  | % of GDP
---
Share
| Cummulative
---
% of GDP Share
Q1 | Most Poor | 3.62 | 3.62
Q2 | Poor | 7.88 | 11.50
Q3 | Working Class | 12.62 | 24.17
Q4 | Rich | 19.71 | 43.88
Q5 | Most Rich | 56.12 | 100.00
Table 4: Income distribution ($\gamma_{1}$). Adapted from World Bank [52]

After the creation of all agents the simulation model starts its iteration
loop, which represents the time dynamics, explained in the next section.

###  3.3 Simulation Dynamics

Each iteration represents one hour when the agents are invoked to perform
actions that depend on their type and behaviors, as shown in Figure 1 for $A1$
agents, and more detailed in subsection 3.3.1. During its movement, an $A1$
agent may get in the proximity with other $A1$, $A2$ or $A3$ agents.
Subsection 3.3.2 presents the possibility of contagion that can happen through
contact between two $A1$ agents. Finally, subsection 3.3.3 presents the
economic relationships between agents, caused by contact of $A1$ and $A3$
agents (commercial transactions), payment of taxes for the government ($A4$
agent), labor relationships between $A3$ and $A1$ agents and house expenses
between $A2$ and $A1$ agents.

Figure 1: A1 agent activity cycle

####  3.3.1 $A1$ Mobility Patterns

The distribution of $A1$ agents’ work, rest and leisure hours is shown in
Table 5 and it is based on the Universal Declaration of Human Rights [57].
Basically, it is the standard deviation of a Gaussian distribution with
average $\mu=0$, representing the variability of the movement amplitude of
$A1$ in its free time or, in other words, how far the agent can walk from its
actual position.

Start Time | End Time | Activity | Action
---|---|---|---
0 | 8 | Rest |  |  If $A1$ is not homeless:
---
Go home (Equation (5))
Otherwise:
Walk freely (Equation (7))
8 | 12 | Job |  |  If $A1$ is not unemployed:
---
Go to work (Equation (6))
Otherwise:
Walk freely (Equation (7))
12 | 14 | Lunch |  Walk Freely (Equation (7))
14 | 18 | Job |  |  If $A1$ is not unemployed:
---
Go to work (Equation (6))
Otherwise:
Walk freely (Equation (7))
18 | 0 | Recreation |  Walk freely (Equation (7))
Table 5: A1 agent movement routines considering a full day and different
activities

The actions “Go home”, “Go to work” and “Walk freely” occur according to the
Equations (5), (6) and (7). Besides these ordinary actions, all the agents
that are infected and have infection severity equal to hospitalization or
severe execute the “Go to hospital” action, according to the Equation (8). All
the dead agents have their positions set to zero.

$\displaystyle A1_{pos}=A2_{pos}+\mathcal{N}(0,\sigma_{k})$ (5) $\displaystyle
A1_{pos}=A3_{pos}+\mathcal{N}(0,\sigma_{k})$ (6) $\displaystyle
A1_{pos}=A1_{pos}+\mathcal{N}(0,\alpha_{6})$ (7) $\displaystyle
A1_{pos}=A5_{pos}+\mathcal{N}(0,\sigma_{k})$ (8)

where $\sigma_{k}=0.01$ is the random noise variance for “Go to…” actions, and
the mobility parameter $\alpha_{6}$ is the random noise variance for “Walk
freely” action, representing the amplitude of movement the A1 agents have in
their free time.

####  3.3.2 Contagion Spreading

COVID-19 is a highly contagious disease. According to the Report 3 of the
Imperial College London “on average, each case infected 2.6 (uncertainty
range: 1.5–3.5) other people up to 18th January 2020” [58]. Following the SEIR
model, in each simulation, there is an initial percentage of infected and
immune people ($\beta_{9}$ and $\beta_{10}$, respectively), and the remaining
population consists of susceptible individuals. There is also a Dead status,
since part of the population dies due to the disease and its complications
[59].

The possibility of contagion happens by the interaction of the agents by
proximity or contact. Hence, the higher the mobility of a person, the greater
the probability that he/she approaches an infected person and gets infected.
Each simulation considers a contagion distance threshold $\beta_{1}$, which is
the minimal distance that two agents have to be to occur the viral
transmission, and a probability of contagion $\beta_{2}$ in case of contact.

The model of the medical condition evolution of the infected agents follows
[60, 61]. Once an agent is infected, it can be in one of these sub-states: a)
asymptomatic, which includes mild symptoms without hospitalization, b)
hospitalization and c) severe, used in cases of hospitalization in intensive
care unit (ICU). These states and their transitions are illustrated in Figure
2.

Figure 2: Epidemiological and infection state diagram for A1 agents based in
SEIR model, with the corresponding population response variables and
parameters of their transition probabilities

The evolution of the medical condition is stochastic and follows the
probabilities summarized in Table 6, represented by the parameters
$\beta_{6}$, $\beta_{7}$ and $\beta_{8}$, respectively. The hospitalization
cases require medical infrastructure, which is limited. It varies from country
to country, but is always less than the total population. In each simulation,
a critical limit $\beta_{11}$ is considered, it represents the percentage of
the population that the healthcare system is capable to handle simultaneously.
As a consequence, if the number of hospitalizations and severe cases increase
above this limit, there are no beds in hospitals to manage the demand.

| Age-group
---
(years)
|  $\beta_{6}$ \- % symptomatic cases
---
requiring hospitalization
|  $\beta_{7}$ \- % hospitalised cases
---
requiring critical care
|  $\beta_{8}$ \- Infection Fatality
---
Ratio
0 - 9 | 0.100 | 5.000 | 0.002
10 - 19 | 0.300 | 5.000 | 0.006
20 - 29 | 1.200 | 5.000 | 0.030
30 - 39 | 3.200 | 5.000 | 0.080
40 - 49 | 4.900 | 6.300 | 0.150
50 - 59 | 10.200 | 12.200 | 0.600
60 - 69 | 16.600 | 27.400 | 2.200
70 - 79 | 24.300 | 43.200 | 5.100
80+ | 27.300 | 70.900 | 9.300

Table 6: Rates of medical conditions considering hospitalized ($\beta_{6}$)
and severe ($\beta_{7}$) and death ($\beta_{8}$) cases grouped by age. Adapted
from Ferguson et al. [6]

####  3.3.3 Economic Transactions

The secondary goal of this study is to simulate the impact caused in the
economy by the different types of mobility restrictions [62, 63, 64, 65]
imposed by the authorities.

Figure 3 shows the transactions by which agents exchange wealth in the
simulation. The economic dynamics follows seasonal routines that also depend
on the type of the agent.

Figure 3: Economic relationships between agents.

The “business contact” action happens hourly, when an A1 agent in its free
time gets in contact with an A3, and occurs the transference of wealth from A1
to A3. These economic transactions are the most sensitive to the A1 agents
mobility (the more the agents move, the more they spend) and affects the A3
agent income. In pandemic times, that can happen in almost all scenarios,
since the population tends to leave their houses just to buy essential items
or to solve a problem which could not have been solved over the Internet. The
values exchanged in “business contact” depend on the social stratum of the A1
agents, and the higher the quintile the higher the spending following the
wealth distribution $\gamma_{1}$. In each day, the wealth of A2 and A3 agents
is decreased by its minimal fixed expenses, proportional to the sum of the
expenses of housemates and employees, respectively.

The “accounting” actions happen monthly for A2, A3 and A4 agents. Accounting
is the payment of taxes from A2 and A3 agents to A4, and it represents the
major income of A4. During accounting, A3 agents also pay salaries to their A1
employees determined in the initialization by the social stratum. Finally, A2
agents transfer money to a random A3 agent, representing supplier payments.

The accounting of the government agent, A4, transfers funds to A5 agent,
equivalent to its fixed expenses and the daily expenses of the hospitalized
agents. Eventually, the A4 agent pays aids for unemployed and homeless A1
agents.

Considering the periodicity of the economic transactions, it is necessary to
execute at least one complete cycle (720 iterations) in order to execute all
economic transactions at least once.

###  3.4 Discussion

The proposed model tries to apprehend the complexity of the social,
epidemiological and economical relationships, but without being simplistic.
Then, the complexity of the model reflects the complexity of the system being
modelled although on a smaller scale.

Previous versions of the model did not consider social constraints and
routines and, despite the good performance of the epidemiological response
variables, it could not accurately represent the economic dynamics. The
introduction of social constructs, as families and businesses, and periodical
routines, such as working hours, free time and bed time, brought more
feasibility to the simulation and improved the performance of the economic
response variables when compared with the real world values.

These social constructs and periodic routines are hard-coded, although they
can be adapted by users. Major flexibility is provided by the parameter set
that can be adjusted to represent from a single community to a complete
country.

In the next section, the experimental methodology is discussed, as well as the
performance metrics and their evaluation.

##  4 Experimental Methodology

To evaluate the proposed approach, seven different scenarios that reflect
adopted and/or hypothetical social distancing interventions have been
formulated. The proposed ABS model was implemented in Python version 3.6
programming language and encapsulated in the COVID_ABS package, whose source
code is available at https://bit.ly/covid_abs_experiments and
https://bit.ly/COVID19_ABSsystem.

Each scenario simulates the impact of a given social distancing policy,
considering the values of the parameters in Table 2, on the response
variables, summarized in Table 3. For each scenario 35 executions were
performed, each one with $T=1,440$ iterations. Since each iteration
corresponds to one hour, each execution covers exactly $2$ months and one
complete accounting cycle for houses, government and business, with one salary
and tax payment, which occur in the 30th day of the month. The monthly
“accounting” event is important for $A2$, $A3$ and $A4$ agents due to its
severe cash impact and wealth transfers among agents.

The main goal of social interventions is to minimize the death curve $D_{t}$.
It is directly related to flattening the infection curve $I_{t}$, in order to
keep the hospitalization $I^{H}_{t}$ and severe $I^{S}_{t}$ cases below the
critical limit of the healthcare system $\beta_{11}$. Flattening the $I_{t}$
curve means minimizing the infection peak $I_{P}$, defined in Equation (9),
and extending the time $T_{IP}$ spent to reach this peak, defined in Equation
(10).

$\displaystyle I_{P}=\max\\{\;I_{t}\;|\;t=1\ldots T\;\\}$ (9) $\displaystyle
T_{IP}=\min\\{\;t\;|\;I_{t}=I_{P}\;\\}$ (10)

To compare the scenarios, the response variables $D_{t}$, $I_{t}$ will be
considered, condensed in the metrics $I_{P}$ and $T_{IP}$.

The economical analysis aims to assess the evolution of wealth, represented by
the $W^{*}_{S,t}$ response variables. To allow the economic comparison among
scenarios with respect to the same reference, a baseline scenario, $B$,
without a pandemic was designed. It is meant to isolate the economic dynamic
and can be used to assess the impacts of the different interventions in the
economy.

For comparison among scenarios, the increase in wealth, $\Delta W^{i}_{S}$,
for the group of agents, $i\in\\{A1,A3,A4\\}$, in scenario $S$ is computed as
follows:

$\Delta W^{i}_{S}=\frac{W^{i}_{S,T}-W^{i}_{B,T}}{W^{i}_{B,T}}$ (11)

where $W^{i}_{B,T}$ is the wealth of the group of agents, $i$, at the final
simulation time step, $T$, of $B$.

In the following section, the scenarios are defined, their simulation results
are presented and compared, and the main findings discussed.

##  5 Results and Analysis

This section shows the results for seven scenarios, chosen to represent the
major interventions adopted or defended by governments. In this study, the
parameters are based on data from Brazil. Nevertheless, in order to encourage
further studies and/or applications, it is possible to easily transfer the
model to other societies by changing the social, demographic and economical
parameters listed in Table 2 and creating a multitude of other scenarios
adapted to the new regions.

The meaning of each scenario, its parameters and dynamics of the response
variables are discussed below.

###  5.1 Baseline ($B$): No Coronavirus Pandemic

This scenario simulates the economic behavior without a pandemic. It is used
as baseline for comparison with all the other scenarios. To generate this
scenario, set parameters to $\beta_{9}\leftarrow 0$ and $\beta_{10}\leftarrow
1$.

Despite the economic result of this simulation, it is artificial data and
consequently it does not represent the reality of any country. Nonetheless, we
argue that it is based on projections before the pandemic outbreak. See, for
instance, the references regarding each parameter listed in Table 2.

The evolution of GDP is illustrated in Figure 4. The GDP indicates a recession
chart where the population ($A1$) and government ($A4$) is losing wealth and
the businesses ($A3$) are floating at the equilibrium point (when the incomes
and expenses are equal). Initially the $A3$ are profiting but, in the
accounting day, the profits are settled by the labor and tax expenses. The
baseline scenario is consistent with the economic predictions of stagnation in
Brazil.

Figure 4: Daily averaged response variables for $B$.

###  5.2 Scenario 1: Do Nothing

This scenario represents what could happen if politicians decided not to take
any actions to avoid the increase of the number of people infected by the
SARS-CoV-2 virus. Usually, this decision only targets the economic point of
view. Figure 5 shows the epidemiological and economical average curves of this
scenario and their variances. It can be seen that the economic curves look
closer to the ones of the baseline, confirming the economic motivation of
keeping the environment without interventions.

However, when the contagion curve $I_{t}$ is considered, it is possible to
note how the Healthcare System critical limit $\beta_{11}$ was trespassed,
pushing the death curve $D_{t}$ up. The high number of lost lives makes this
the most catastrophic scenario, despite its economic resemblance with $B$.

Figure 5: Daily averaged response variables for scenario “Do Nothing”

###  5.3 Scenario 2: Lockdown

This scenario represents the complete social isolation, following the WHO
recommendations, during a well defined date range. In this scenario, all $A1$
agents are kept in their houses, and the “walk freely” and “go to work”
routines are suppressed. Also $\alpha_{6}\leftarrow 1$, reducing the mobility
amplitude of all $A1$ even the homeless, as discussed in Section 3.3.1. The
lockdown is unconditional, meaning that from $t=0$ to $T$, all the
restrictions are applied.

This scenario is highly conservative in healthcare terms, and the main goal is
to save as many lives as possible by minimizing viral spreading. In the
impossibility of effective testing, the entire population stays in lockdown
for a predefined period of time. Broadly speaking, the infected agents only
have contact with their housemates and the $I_{t}$ (and especially
$I^{S}_{t}$) stays below the healthcare critical limit $\beta_{11}$, and the
deaths $D_{t}\leftarrow 0$, meaning that the healthcare system could handle
effectively all cases, using its available resources333Considering the given
population size in the simulation..

Considering the economic point of view, see Figure 6, this scenario is the
worst for the industry because the $A1$ agents cannot generate wealth, but
keep receiving their labor incomes444In our simulation, people can not get
fired, which means that the onus of keeping them at home is for the company.
In practice this may generate unemployment.. $A3$ does not have income, but
keeps paying taxes to $A4$ and labor expenses to $A1$. In this scenario, after
two months, the businesses lost 20% of its GDP share, see $W^{A3}_{S,T}$ in
Figure 6.

The key point for the success of lockdown policy is staying at home
(voluntarily or under laws). Economical countermeasures to its harm can also
be adopted by $A4$, as tax exemptions and universal income, in order to
minimize the wealth losses. In the impossibility of implementing this
scenario, another one that considers protective and distance measures should
be evaluated.

Figure 6: Daily averaged response variables for Scenario 2

###  5.4 Scenario 3: Conditional Lockdown

This scenario imposes the same restrictions on $A1$ mobility presented in
scenario 2, but conditionally. In the system, when the infection curve grows
above a certain threshold, $I_{t}\geq 0.05$, the lockdown restrictions are
activated, being released when $I_{t}\leq 0.05$.

As we can see in Figure 7, the viral spreading represented by the infection
curve $I_{t}$ is controlled, not allowing the explosion of $D_{t}$ curve.
Economically, recession can be observed during the lockdown period,
$W^{i}_{3,t}$ lower than $W^{i}_{B,t}\forall i$, but as soon as the
restrictions are released the business performance is recovered.
$W^{A3}_{3,t}$ remains below $W^{A3}_{B,t}$ but above the complete lockdown
curve $W^{i}_{2,t}$.

Less conservative than scenario 2 (and also less efficient in terms of
$D_{t}$), this scenario was implemented in New Zealand [66], and it depends on
an effective healthcare system that is capable of carrying out the necessary
tests in the population, granting reliability in $I_{t}$ estimates and, as in
scenario 2, the governmental ability to enforce the social isolation.

Figure 7: Daily averaged response variables for Scenario 3

###  5.5 Scenario 4: Vertical Isolation

Vertical isolation is the name given to the social intervention policy where
the known infected people and the known risk groups – elderly and people with
pre-existent diseases – are kept in social isolation, whereas young people and
adults are allowed to work regularly. This policy has, for instance, been
advocated by the Brazilian president555See
https://agenciabrasil.ebc.com.br/en/politica/noticia/2020-04/bolsonaro-brazil-
must-not-be-informed-through-panic \- Acessed: June 03, 2020, and
https://www.bbc.com/814portuguese/internacional-52043112 \- Acessed: June 03,
2020.

In terms of the proposed model, over $65$, below $18$ years old and
symptomatic regardless of the age stay at home.

The assumption of this policy is that all the people outside the risk groups
would not develop the severe cases of the disease. This assumption was proved
to be fragile and this policy showed to be ineffective by [67]. The results
shown in Figure 8 are in accordance with the literature [67] and produced
almost the same epidemiological and economical results of Scenario 1, i.e.,
the same results of doing nothing.

Figure 8: Daily averaged response variables for Scenario 4

###  5.6 Scenario 5: Partial Isolation

In the scenarios with lockdown (2 and 3), the mobility of all agents must be
restricted, requiring restrictive public policies enforced by the government.
When these policies are non-existent or are not taken seriously by the entire
population, partial isolation levels are reached. The partial isolation level
$IL\in[0,1]$ means the percentage of the population that is fulfilling the
isolation, while the remaining $1-IL$ is not.

Then, it is possible to define that in the lockdown $IL\geq 0.9$, considering
that essential services and a few industries can not stop in order to avoid
supply breakdown. On the other hand, the scenarios 0 and 1 have $IL\leq 0.1$,
and the scenario 4 has $IL\approx 0.2$, because of the age distribution and
the definition of risk groups.

This scenario aims to assess the effects of intermediate $IL$s. It was
simulated by randomly choosing agents $A1$ with probability $IL\leftarrow 0.5$
to stay at home.

Observing the results in Figure 9, although the $I_{t}$ curve is flattened
when compared with scenarios $B$ and 4, it is still less efficient than
scenarios 2 and 3. Notice the $D_{t}$ still grows exponentially before
reaching the peak. For the economic perspective, this scenario behaves
similarly to the baseline. These metrics offer evidence that $IL\leftarrow
0.5$ is not enough for effective epidemiological control, and a level of
isolation greater than that is recommended.

The impact of different isolation levels can be seen in Figures 10 and 11, for
$IL\in[0.3,0.9]$, which represents the response of epidemiological and
economical curves for increasing $IL$. In Figure 10 it is possible to see how
the infection curve $I_{t}$ flattens as the isolation level increases from no
isolation towards lockdown. Figure 11 shows that as the value of $IL$
increases, wealth loss of the A3 agents is higher, represented by
$W^{A3}_{S,t}$ curve, showing the importance of agent’s mobility in the
economy.

Figure 9: Daily averaged response variables for Scenario 5 Figure 10:
Infection curves by varying values of partial isolation level ($IL$). Figure
11: $W^{A3}_{S,t}$ curves by varying values of partial isolation level ($IL$).

###  5.7 Scenario 6: Use of Face Masks

Evidence was found about the use of masks and gloves as measures against viral
spreading [68]. This scenario represents the policy of mandatory usage of face
masks and physical distancing, but without imposing restrictions on the
mobility of agents.

This scenario was implemented by reducing the contagion distance
$\beta_{1}=0.5$ and the contagion rate $\beta_{2}=0.3$ as the effect of using
masks and physical distancing. Figure 12 shows a flatter $I_{t}$ curve when
compared to scenario 5 while still keeping economic performance close to $B$.
Notice, however, that $D_{t}$ is significantly higher when compared with
scenarios 2 and 3.

Figure 12: Daily averaged response variables for Scenario 6

###  5.8 Scenario 7: Use of Face Masks and 50% of Social Isolation

This scenario combines the policies used in the scenarios 5 and 6, granting
the necessary use of face masks plus partial isolation of the population. This
scenario was implemented by using $\beta_{1}=0.5$, $\beta_{2}=0.3$ and
$IL=0.5$.

Figure 13 shows the dynamics of this scenario. Although the $D_{t}$ is still
above the values of scenarios 2 and 3, it presents less resistance from the
general population. The $I_{t}$ is flattened, and the economy, despite the
downturn, suffers less than it would in scenarios with lockdown. This scenario
has already been discussed in [68] with similar results.

Figure 13: Daily averaged response variables for Scenario 7

###  5.9 Comparisons Among the Scenarios

The $I_{t}$ curves (averages) of each scenario are shown in Figure 14. There,
the effects of each intervention policy in flattening the curve can be
observed and contrasted. The epidemiological effectiveness of the scenarios
are shown in Figure 15, which compares the infection peak $I_{P}$ reached in
each case, the number of days $T_{IP}$ to reach the peak $I_{P}$ and the max
number of deaths $D_{t}$ (as a proportion of the population).

As expected, scenarios 2 and 3 have the best epidemiological values followed
by scenario 7.

Figure 14: Infection evolution for the several scenarios Figure 15: Death
evolution for the several scenarios

Figure 16 shows the economic result of each scenario for the agent types $A1$,
$A3$ and $A4$. Assuming that businesses are not firing anyone, from the point
of view of the citizen, scenarios 2 and 3 are not economically damaging. On
the other hand, the same scenarios are the worst from the business
perspective. At this point, it is important to explain that the expenses of
government in our simulation are related with the costs of the healthcare
system. Thus, in scenarios with a high number of deaths, such as scenarios 1
and 4, the cost of maintaining the healthcare system is increased which
demands an increase of public expenses.

Figure 16: Economical result of each scenario compared to Scenario 0 by
response variable

Figure 17 shows the scatter plots of the wealth increase (with respect to the
baseline) of each type of agent by the percentage of deaths in the
populations. It can be seen that, from a life preservation perspective, there
is no better policy than the lockdown (scenario 2). Furthermore, in the
simulated model, scenario 2 Pareto dominates666Given a set of criteria, Pareto
optimality can be defined as a situation where no individual criterion can be
better off without making at least one other criterion worse off. Given an
initial situation, a Pareto improvement is a new situation where there will be
gains in all criteria. A situation is called Pareto dominated if it has a
Pareto improvement. Finally, a situation is called Pareto optimal if no change
could lead to an improvement in all the objectives. all the scenarios for both
people an government. On the other hand, it represents the worst case,
financially, for businesses.

In the impossibility of enforcing a lockdown (discarding scenarios 2 and 3),
which may happen in underdeveloped countries, the best solution is represented
by scenario 7. From the remaining Pareto optimal solutions for businesses, it
is the one with lowest number of deaths. It also becomes the best solution for
government and people in both wealth and number of deaths.

Figure 17: Percentage of deaths versus percentage of GDP variation

##  6 Conclusion

The COVID-19 pandemic brought to humankind many challenges, including the
demand for new medical treatments, social policies and economical approaches.
The fast response of the scientific community to deal with coronavirus was
divided into studies of the epidemiological aspects, proposals of new
treatments and diagnostic tools and new models to forecast the viral
spreading, including SIR and SEIR models among others. Nonetheless, few
studies focused on looking at the pandemics as a governmental policy-making
problem. With this viewpoint, although the epidemiological aspects are
priority, the social and economical aspects can not be neglected.

The present work proposed an Agent-Based Model (ABM) that simulates the
epidemiological and economical effects of COVID-19 pandemic in a closed
society, whose results can be generalized for wider contexts and used by
governmental rulers to prospect social policies and assess its potential
effectiveness in real scenarios.

The model was encapsulated in the free and open source software library COVID-
ABS, which contains 29 epidemiological, social, demographic and economic input
parameters, and 10 output response variables. New features can be designed and
the library can be easily extended to other scenarios.

In a wider perspective, the proposed approach can be used as a decision-
support system for the governments and scientific community. Policy-makers can
design scenarios and evaluate the effectiveness of social interventions
through different simulations, and analyse how the $P$ parameters, in the time
horizon of $T$, can affect the response variables $\Theta_{t}$.

Seven different scenarios were elaborated to reflect specific social
interventions. Lockdown and conditional lockdown were the best evaluated
scenarios in preserving lives. These scenarios present a slower evolution of
the epidemic, a smaller number of infections and deaths. Given the
impossibility of implementing lockdown policies, the scenario with 50% of
social isolation with using masks and physical distancing was the best
approach in the preservation of lives. On the other hand, the vertical
isolation scenario is totally ineffective and resembles the “Do nothing”
scenario.

The results showed that COVID-ABS approach was capable to effectively simulate
social intervention scenarios in line with the results presented in the
literature. Also, the results showed that policies adopted by some countries,
for instance US, Sweden and Brazil, are ineffective when the objective is to
preserve lives. Governments that chose to preserve the economy by not using
severe isolation policies, fatally reached a situation with a high cost in
human lives, and still embittered economic losses. The evidence provided by
the simulation model shows that there is a false dichotomy between healthcare
and the economy. In the scenarios where it was tried to save the economy by
not taking hard social isolation policies, consequently, the social costs
ended up impacting negatively into the economy.

COVID-ABS is an open software and can be easily extended and customized. Also,
new scenarios can be designed, taking into consideration the specificities of
each region under study. Future research aims to improve the model by
implementing mechanisms to close and open companies as well as allowing people
to get fired. In addition, it will be integrated with optimization libraries,
for automatic scenario creation, and multi-criteria decision making tools that
could help governmental crisis committees to plan and manage the social
policies to mitigate the COVID-19 effects.

## Acknowledgements

Petrônio Silva, Paulo Batista and Helder Seixas would like to thank the
financial support given by the Instituto Federal do Norte de Minas Gerais,
Brazil.

Marcos A. Alves declares that this work has been supported by the Brazilian
agency CAPES.

Frederico Gadelha Guimarães would like to thank the support given by the
Brazilian Agencies CNPq (grant no. 306850/2016-8) and FAPEMIG.

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# Dirichlet Series Expansions of p-adic L-Functions

Heiko Knospe, Lawrence C. Washington<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We study $p$-adic $L$-functions $L_{p}(s,\chi)$ for Dirichlet characters
$\chi$. We show that $L_{p}(s,\chi)$ has a Dirichlet series expansion for each
regularization parameter $c$ that is prime to $p$ and the conductor of $\chi$.
The expansion is proved by transforming a known formula for $p$-adic
$L$-functions and by controlling the limiting behavior. A finite number of
Euler factors can be factored off in a natural manner from the $p$-adic
Dirichlet series. We also provide an alternative proof of the expansion using
$p$-adic measures and give an explicit formula for the values of the
regularized Bernoulli distribution. The result is particularly simple for
$c=2$, where we obtain a Dirichlet series expansion that is similar to the
complex case.

###### 2020 Mathematics Subject Classification:

Primary: 11R23. Secondary: 11R42, 11S80, 11M41

##  1\. Introduction

Let $p$ be a prime, let $q=p$ if $p$ is odd and $q=4$ if $p=2$, and let $\chi$
be a Dirichlet character of conductor $f$. A $p$-adic $L$-function
$L_{p}(s,\chi)$ for a Dirichlet character $\chi$ is a $p$-adic meromorphic
function and an analogue of the complex $L$-function. For powers of the
Teichmüller character $\omega$ of conductor $q$, one obtains the $p$-adic zeta
functions $\zeta_{p,i}=L_{p}(s,\omega^{1-i})$, where $i=0,\,1,\,\dots,\,p-2$
($i=0,1$ if $p=2$). It is well known that $L_{p}(s,\chi)$ is identically zero
for odd $\chi$. $p$-adic $L$-functions have a long history and the primary
constructions going back to Kubota-Leopoldt [6] and Iwasawa [3] are via the
interpolation of special values of complex $L$-functions. It can also be shown
that $p$-adic $L$-functions are in fact Iwasawa functions.

It is well known that for $Re(s)>0$,

$(1-2^{1-s})\zeta(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{s}}$

and, more generally, if $c\geq 2$ is an integer,

$(1-\chi(c)c^{1-s})L(s,\chi)=\sum_{n=1}^{\infty}\chi(n)\frac{a_{c,n}}{n^{s}},$

where $a_{c,n}=1-c$ if $n\equiv 0\text{ mod}\ c$ and $a_{c,n}=1$ if
$n\not\equiv 0\text{ mod}\ c$. In the following, we derive similar, but
slightly different, expansions for $p$-adic $L$-functions.

An explicit formula for $L_{p}(s,\chi)$ is given in [9] (Theorem 5.11): let
$F$ be any multiple of $q$ and $f$. Then $L_{p}(s,\chi)$ is a meromorphic
function (analytic if $\chi\neq 1)$ on $\\{s\in\mathbb{C}_{p}\ |\
|s|<qp^{-1/(p-1)}\\}$ such that

(1) $L_{p}(s,\chi)=\frac{1}{F}\frac{1}{s-1}\sum^{F}_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}\chi(a)\langle
a\rangle^{1-s}\sum_{j=0}^{\infty}\binom{1-s}{j}\left(\frac{F}{a}\right)^{j}B_{j}.$

In Section 2, we will use formula (1) to derive a Dirichlet series expansion
of $L_{p}(s,\chi)$.

$p$-adic $L$-functions can be also be defined using distributions and
measures. Let $\chi$ have conductor $f=dp^{m}$ with $(d,p)=1$. Choose an
integer $c\geq 2$, where $(c,dp)=1$. Then there is a measure $E_{1,c}$ on
$(\mathbb{Z}/d\mathbb{Z})^{\times}\times\,\mathbb{Z}_{p}^{\times}$ (the
regularized Bernoulli distribution) such that

(2) $-(1-\chi(c)\langle
c\rangle^{1-s})L_{p}(s,\chi)=\int_{(\mathbb{Z}/d\mathbb{Z})^{\times}\times\,\mathbb{Z}_{p}^{\times}}\chi\omega^{-1}(a)\langle
a\rangle^{-s}\ dE_{1,c}$

(see [9] Theorem 12.2). In Section 3, we give an explicit formula for the
values of $E_{1,c}$ and derive the Dirichlet series expansion from (2).

The expansion is particularly simple for $c=2$, and this parameter can be used
for $p\neq 2$ and Dirichlet characters with odd conductor. For this case we
obtain similar results as in [1], [2], and [4]. In Section 4, we provide
examples for different parameters $c$.

##  2\. Expansions of $p$-adic $L$-Functions

First, we derive an approximation of $L_{p}(s,\chi)$ that is close to the
original definition of Kubota-Leopoldt (see [6]).

For $r\in\mathbb{C}_{p}^{\times}$ we write $\delta(r)$ for a term with
$p$-adic absolute value $\leq|r|$.

######  Proposition 2.1.

Let $p$ be a prime number, $\chi$ an even Dirichlet character of conductor
$f$, and $F$ a multiple of $q$ and $f$. For $s\in\mathbb{C}_{p}$ with
$|s|<qp^{-1/(p-1)}$, we have

(3) $L_{p}(s,\chi)=\frac{1}{F}\frac{1}{s-1}\sum^{F}_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}\chi(a)\langle a\rangle^{1-s}+\delta(F/qp).$

###### Proof.

We use formula (1) above and look at the series
$\sum_{j=0}^{\infty}\binom{1-s}{j}\left(\frac{F}{a}\right)^{j}B_{j}$. The
first two terms are $1+(1-s)\frac{-F}{2a}$. We claim that the $p$-adic
absolute value of the other terms ($j\geq 2$) is less than or equal to
$|(s-1)F^{2}/qp|$. To this end, we note that $|1/j!|\leq p^{(j-1)/(p-1)}$ and

$\left|\binom{1-s}{j}\right|\leq|1-s|\,p^{(j-1)/(p-1)}(qp^{-1/(p-1)})^{j-1}=|1-s|\,q^{j-1}$

since we assumed that $|s|<qp^{-1/(p-1)}$. Since $|F|\leq\frac{1}{q}$,
$|a|=1$, and $|B_{j}|\leq p$, we obtain

$\left|\binom{1-s}{j}\left(\frac{F}{a}\right)^{j}B_{j}\right|\leq|1-s|\,q^{j-1}q^{2-j}|F|^{2}\,p=|1-s|\,|F|^{2}\,qp.$

Then (1) implies

$L_{p}(s,\chi)=\frac{1}{F}\frac{1}{s-1}\sum^{F}_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}\chi(a)\langle
a\rangle^{1-s}+\frac{1}{2}\sum^{F}_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}\chi\omega^{-1}(a)\langle
a\rangle^{-s}+\delta(F/qp).$

It remains to show that the second sum can be absorbed into $\delta(F/qp)$. We
have

$\displaystyle\sum^{F}_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}\chi\omega^{-1}(a)\langle a\rangle^{-s}$
$\displaystyle=\sum^{F}_{\begin{subarray}{c}b=1\\\
p\,\nmid\,b\end{subarray}}\chi\omega^{-1}(F-b)\langle F-b\rangle^{-s}$
$\displaystyle=-\sum^{F}_{\begin{subarray}{c}b=1\\\
p\,\nmid\,b\end{subarray}}\chi\omega^{-1}(b)\langle b-F\rangle^{-s}$
$\displaystyle=-\sum^{F}_{\begin{subarray}{c}b=1\\\
p\,\nmid\,b\end{subarray}}\chi\omega^{-1}(b)\langle
b\rangle^{-s}+\delta(F/q).$

The last step can be justified by noting that

$\frac{\langle b-F\rangle^{-s}}{\langle
b\rangle^{-s}}=\left(1-\frac{F}{b}\right)^{-s}=1+\sum_{k=1}^{\infty}\binom{-s}{k}\left(\frac{-F}{b}\right)^{k}=1+\delta(F/q),$

since $|s|<qp^{-1/(p-1)}$ (this is the same estimate as earlier, without the
presence of the Bernoulli number). This proves the proposition. ∎

######  Remark 2.2.

For $F=fp^{n}$ and $n\rightarrow\infty$, formula (3) gives the original
definition of $L_{p}(s,\chi)$ by Kubota and Leopoldt (see [6]).

######  Remark 2.3.

Suppose that $p\neq 2$. Then the error term in the above Proposition (as well
as in the following Theorem 2.4) can be improved to
$\delta(F/p^{2-(p-2)/(p-1)})$. First we note that $B_{j}=0$ for odd $j\geq 3$.
By the von Staudt–Clausen Theorem (see [9] 5.10), we have for even $j\geq 2$:
$|B_{j}|=p$ iff $(p-1)\mid j$, and otherwise $|B_{j}|\leq 1$. Furthermore,
$|1/j!|=p^{(j-S_{j})/(p-1)}$, where $S_{j}$ is the sum of the digits of $j$,
written to the base $p$ (see [5]). Since $j\equiv S_{j}\text{ mod}\ (p-1)$,
$j\equiv 0\text{ mod}\ (p-1)$ is equivalent to $S_{j}\equiv 0\text{ mod}\
(p-1)$. We conclude that $|B_{j}|=p$ yields $S_{j}\geq p-1$ and $|1/j!|\leq
p^{(j-1)/(p-1)}p^{-(p-2)/(p-1)}$. This implies the above error term. We also
see that this error term cannot be further improved.$\hfill\lozenge$

Now we give the Dirichlet expansion of $L_{p}(s,\chi)$. For $m\in\mathbb{N}$,
we denote by $\\{x\\}_{m}$ the unique representative of $x\text{ mod}\
m\mathbb{Z}$ between $0$ and $m-1$.

######  Theorem 2.4.

Let $p$ be a prime number, $\chi$ be an even Dirichlet character of conductor
$f$, and $F$ a multiple of $q$ and $f$. Let $c>1$ be an integer satisfying
$(c,F)=1$. For $a\in\mathbb{Z}$, define

$\epsilon_{a,c,F}=\frac{c-1}{2}-\\{-aF^{-1}\\}_{c}\,\in\left\\{-\frac{c-1}{2},\,-\frac{c-1}{2}+1,\,\dots,\
\frac{c-1}{2}\right\\}.$

Then we have for $s\in\mathbb{C}_{p}$ with $|s|<qp^{-1/(p-1)}$ the formula

$-(1-\chi(c)\langle
c\rangle^{1-s})L_{p}(s,\chi)=\sum^{F}_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}\chi\omega^{-1}(a)\langle
a\rangle^{-s}\epsilon_{a,c,F}+\delta(F/qp).$

###### Proof.

Use (3) with $cF$ in place of $F$, and subtract $\chi(c)\langle
c\rangle^{1-s}$ times (3) with $F$, to obtain

(4) $\displaystyle\begin{split}(1-\chi(c)\langle
c\rangle^{1-s})L_{p}(s,\chi)&=\frac{1}{cF}\frac{1}{s-1}\sum^{cF}_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}\chi(a)\langle a\rangle^{1-s}\\\
&-\frac{1}{F}\frac{1}{s-1}\sum^{F}_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}\chi(ac)\langle
ac\rangle^{1-s}+\delta(F/qp).\end{split}$

Let $0<a_{0}<F$ with $(a_{0},\,p)=1$. Since we assumed $(c,\,F)=1$ and $p\mid
F$, there is a unique number of the form $a_{0}c$ with $0<a_{0}c<cF$ and
$(a_{0}c,\,p)=1$ in each congruence class modulo $F$ relatively prime to $p$.
The first sum in (4) can be written as

$\displaystyle\frac{1}{cF}\frac{1}{s-1}\sum^{F}_{\begin{subarray}{c}a_{0}=1\\\
p\,\nmid\,a_{0}\end{subarray}}\chi(a_{0}c)\langle
a_{0}c\rangle^{1-s}\left(\sum_{\begin{subarray}{c}a=1\\\ a\equiv a_{0}c\text{
mod}\ F\end{subarray}}^{cF}\left\langle
1+\frac{a-a_{0}c}{a_{0}c}\right\rangle^{1-s}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{cF}\frac{1}{s-1}\sum^{F}_{\begin{subarray}{c}a_{0}=1\\\
p\,\nmid\,a_{0}\end{subarray}}\chi(a_{0}c)\langle
a_{0}c\rangle^{1-s}\left(\sum_{\begin{subarray}{c}a=1\\\ a\equiv a_{0}c\text{
mod}\ F\end{subarray}}^{cF}1+(1-s)\frac{a-a_{0}c}{a_{0}c}\right)+\delta(F/q).$

Note that $|\frac{a-a_{0}c}{a_{0}c}|\leq|F|$, so this is the same type of
estimate used in the proof of Proposition 2.1. Subtracting the second sum in
(4) yields

$\displaystyle(1-\chi(c)$ $\displaystyle\langle c\rangle^{1-s})L_{p}(s,\chi)$
$\displaystyle=\frac{-1}{cF}\sum^{F}_{\begin{subarray}{c}a_{0}=1\\\
p\,\nmid\,a_{0}\end{subarray}}\chi(a_{0}c)\langle
a_{0}c\rangle^{1-s}\left(\sum_{\begin{subarray}{c}a=1\\\ a\equiv a_{0}c\text{
mod}\ F\end{subarray}}^{cF}\frac{a-a_{0}c}{a_{0}c}\right)+\delta(F/qp)$
$\displaystyle=\frac{-1}{c}\sum^{F}_{\begin{subarray}{c}a_{0}=1\\\
p\,\nmid\,a_{0}\end{subarray}}\chi\omega^{-1}(a_{0}c)\langle
a_{0}c\rangle^{-s}\left(\sum_{\begin{subarray}{c}a=1\\\ a\equiv a_{0}c\text{
mod}\ F\end{subarray}}^{cF}\frac{a-a_{0}c}{F}\right)+\delta(F/qp).$

We compute the inner sum. Let $b=\\{a_{0}c\\}_{F}$. Then
$a_{0}c=b+\\{-F^{-1}b\\}_{c}\,F$, since the latter sum is congruent to $b$
modulo $F$ and congruent to $0$ modulo $c$. If $a$ satisfies $a\equiv
a_{0}c\text{ mod}\ F$ and $0<a<cF$, then $a=b+jF$ with $0\leq j<c$. Hence

$\sum_{\begin{subarray}{c}a=1\\\ a\equiv a_{0}c\text{ mod}\
F\end{subarray}}^{cF}\frac{a-a_{0}c}{F}=\sum_{j=0}^{c-1}(j-\\{-F^{-1}b\\}_{c})=c\
\epsilon_{b,c,F}\ .$

Since $b\equiv a_{0}c\text{ mod}\ F$, we have $\chi\omega^{-1}(b)\langle
b\rangle^{-s}=\chi\omega^{-1}(a_{0}c)\langle a_{0}c\rangle^{-s}+\delta(F/q)$
by the same estimate as earlier, so

$-(1-\chi(c)\langle
c\rangle^{1-s})L_{p}(s,\chi)=\sum^{F}_{\begin{subarray}{c}b=1\\\
p\,\nmid\,b\end{subarray}}\chi\omega^{-1}(b)\langle
b\rangle^{-s}\epsilon_{b,c,F}+\delta(F/qp).$

This completes the proof. ∎

We can take the limit of $F=fp^{n}$ as $n\rightarrow\infty$ and obtain:

######  Corollary 2.5.

Let $p$ be a prime number, $\chi$ an even Dirichlet character of conductor
$f$, and $c>1$ an integer satisfying $(c,pf)=1$. Then we have for
$s\in\mathbb{C}_{p}$ with $|s|<qp^{-1/(p-1)}$,

$-(1-\chi(c)\langle
c\rangle^{1-s})L_{p}(s,\chi)=\lim_{n\rightarrow\infty}\sum^{fp^{n}}_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}\chi\omega^{-1}(a)\frac{\epsilon_{a,c,fp^{n}}}{\langle
a\rangle^{s}}.$

The next Theorem shows that a finite number of Euler factors can be factored
off in a similar way as in [8], where a weak Euler product was obtained. The
main statement is that the remaining Dirichlet series has the expected form,
similar to the complex case.

######  Theorem 2.6.

Let $p$ be a prime number and let $\chi$ be an even Dirichlet character of
conductor $f$. Let $S$ be any finite (or empty) set of primes not containing
$p$ and set $S^{+}=S\cup\\{p\\}$. Let $F$ be a multiple of $q$, $f$ and all
primes in $S$. Let $c>1$ be an integer satisfying $(c,F)=1$. Then we have for
$s\in\mathbb{C}_{p}$ with $|s|<qp^{-1/(p-1)}$ the formula

$\displaystyle-(1-\chi(c)\langle c\rangle^{1-s})\cdot\prod_{l\in
S}(1-\chi\omega^{-1}(l)\langle l\rangle^{-s})\ \cdot\ L_{p}(s,\chi)=$
$\displaystyle\sum^{F}_{\begin{subarray}{c}a=1\\\
(a,S^{+})=1\end{subarray}}\chi\omega^{-1}(a)\frac{\epsilon_{a,c,F}}{\langle
a\rangle^{s}}\ +\ \delta(F/qp).$

###### Proof.

We prove the statement by induction on $|S|$. By Theorem 2.4, the formula is
true for $S=\varnothing$. Now assume the formula is true for $S$, and $l\neq
p$ is a prime with $l\notin S$ and $(c,l)=1$. It suffices to prove the
following formula:

(5) $\displaystyle\begin{split}(1-\chi\omega^{-1}(l)\langle
l\rangle^{-s})\sum^{F}_{\begin{subarray}{c}a=1\\\
(a,S^{+})=1\end{subarray}}\chi\omega^{-1}(a)\langle
a\rangle^{-s}\epsilon_{a,c,F}\ =\\\ \sum^{lF}_{\begin{subarray}{c}a=1\\\
(a,S^{+}\cup\\{l\\})=1\end{subarray}}\chi\omega^{-1}(a)\langle
a\rangle^{-s}\epsilon_{a,c,lF}\ +\ \delta(F/qp).\end{split}$

Note that $|1-\chi\omega^{-1}(l)\langle l\rangle^{-s}|\leq 1$ and $|lF|=|F|$,
so we can keep the error term. We can use $lF$ in place of $F$ and write the
left side of (5) as

(6) $\displaystyle\sum^{lF}_{\begin{subarray}{c}a=1\\\
(a,S^{+})=1\end{subarray}}\chi\omega^{-1}(a)\langle
a\rangle^{-s}\epsilon_{a,c,lF}\ -\sum^{F}_{\begin{subarray}{c}a=1\\\
(a,S^{+})=1\end{subarray}}\chi\omega^{-1}(la)\langle
la\rangle^{-s}\epsilon_{a,c,F}+\delta(F/qp).$

Now we have

$\epsilon_{la,c,lF}=\frac{c-1}{2}-\\{-la(lF)^{-1}\\}_{c}=\frac{c-1}{2}-\\{-aF^{-1}\\}_{c}=\epsilon_{a,c,F}.$

Thus (6) is equal to

$\displaystyle\sum^{lF}_{\begin{subarray}{c}a=1\\\
(a,S^{+})=1\end{subarray}}\chi\omega^{-1}(a)\langle
a\rangle^{-s}\epsilon_{a,c,lF}\ $
$\displaystyle-\sum^{F}_{\begin{subarray}{c}a=1\\\
(a,S^{+})=1\end{subarray}}\chi\omega^{-1}(la)\langle
la\rangle^{-s}\epsilon_{la,c,lF}+\delta(F/qp)$
$\displaystyle=\sum^{lF}_{\begin{subarray}{c}a=1\\\ (a,S^{+})=1\\\
l\,\nmid\,a\end{subarray}}\chi\omega^{-1}(a)\langle
a\rangle^{-s}\epsilon_{a,c,lF}\ +\ \delta(F/qp),$

which shows equation (5). ∎

######  Remark 2.7.

What happens if $S$ contains more and more primes? It is well known that the
Euler product does not converge $p$-adically. There are infinitely many primes
$l$ with $\chi\omega^{-1}(l)=1$, and we have for $l\neq p$,

$1-\langle l\rangle^{-s}=-\sum_{j=1}^{\infty}\binom{-s}{j}(\langle
l\rangle-1)^{j}.$

For $|s|<qp^{-1/(p-1)}$, the $p$-adic absolute value of each term of the above
series is less than

$(qp^{-1/(p-1)})^{j}p^{(j-1)/(p-1)}q^{-j}=p^{-1/(p-1)}<1.$

Hence the product of Euler factors $(1-\chi\omega^{-1}(l)\langle
l\rangle^{-s})^{-1}$ does not converge as $S$ picks the primes one after the
other. We also see that the $p$-adic Dirichlet series over numbers relatively
prime to $S$ converges to $0$ as $S$ expands to include all primes,
$n\rightarrow\infty$ and $F$ being a multiple of $p^{n}$, $f$ and the primes
in $S$.

##  3\. Regularized Bernoulli Distributions

Let $p$ be a prime number and let $d$ be a positive integer with $(d,p)=1$.
Define $X_{n}=(\mathbb{Z}/dp^{n}\mathbb{Z})$ and $X=\varprojlim
X_{n}\cong\mathbb{Z}/d\mathbb{Z}\times\mathbb{Z}_{p}$. Let $k\geq 1$ be an
integer. Then the Bernoulli distribution $E_{k}$ on $X$ is defined by

$E_{k}(a+dp^{n}X)=(dp^{n})^{k-1}\frac{1}{k}B_{k}\left(\frac{\\{a\\}_{dp^{n}}}{dp^{n}}\right),$

where $B_{k}(x)$ is the $k$-th Bernoulli polynomial and $B_{k}=B_{k}(0)$ are
the Bernoulli numbers (see [5], [7]). For $k=1$, one has
$B_{1}(x)=x-\frac{1}{2}$. Choose $c\in\mathbb{Z}$ with $c\neq 1$ and
$(c,dp)=1$. Then the regularization $E_{k,c}$ of $E_{k}$ is defined by

$E_{k,c}(a+dp^{n}X)=E_{k}(a+dp^{n}X)-c^{k}E_{k}\left(\left\\{\frac{a}{c}\right\\}_{dp^{n}}+dp^{n}X\right).$

One shows that the regularized Bernoulli distributions $E_{k,c}$ are measures
(see [7]). In the following, we consider only $k=1$; the cases $k\geq 2$ are
similar.

######  Theorem 3.1.

Let $p$ be a prime, $c,\,d\in\mathbb{N}$, and $c\geq 2$ such that $(c,dp)=1$.
Let $X$ be as above, and let $E_{1,c}$ be the regularized Bernoulli
distribution on $X$. For $a\in\\{0,1,\dots,dp^{n}-1\\}$, we have

$E_{1,c}(a+dp^{n}X)=\frac{c-1}{2}-\\{-a(dp^{n})^{-1}\\}_{c}=\epsilon_{a,c,dp^{n}}.$

###### Proof.

By definition,

$E_{1,c}(a+dp^{n}X)=E_{1}(a+dp^{n}X)-cE_{1}(c^{-1}a+dp^{n}X)=\frac{a}{dp^{n}}-\frac{1}{2}-c\left(\frac{\\{c^{-1}a\\}_{dp^{n}}}{dp^{n}}\right)+\frac{c}{2}.$

We give the standard representative of $c^{-1}a\text{ mod}\ dp^{n}$:

$\\{c^{-1}a\\}_{dp^{n}}=\frac{\\{-a(dp^{n})^{-1}\\}_{c}\ dp^{n}+a}{c}$

Note that the numerator is divisible by $c$, since $\\{-a(dp^{n})^{-1}\\}_{c}\
dp^{n}\equiv-a\text{ mod}\ c$. Hence the quotient is an integer between $0$
and $dp^{n}-1$. Furthermore, the numerator is congruent to $a$ modulo
$dp^{n}$, and so the quotient has the desired property. We obtain

$E_{1,c}(a+dp^{n}X)=\frac{a}{dp^{n}}+\frac{c-1}{2}-\frac{\\{-a(dp^{n})^{-1}\\}_{c}\
dp^{n}+a}{dp^{n}}=\frac{c-1}{2}-\\{-a(dp^{n})^{-1}\\}_{c}$

which is the assertion. ∎

Now the Dirichlet series expansion in Corollary 2.5 follows from Theorem 3.1
and the integral formula (2).

##  4\. Expansions for different regularization parameters

We look at the coefficients $\epsilon_{a,c,dp^{n}}$ for different parameters
$c$ and the resulting Dirichlet series expansions. The following observation
follows directly from the definition.

######  Remark 4.1.

The sequence of values $E_{1,c}(a+dp^{n}X)=\epsilon_{a,c,dp^{n}}$ for $a=$
$0$,$1$, $2$, $\dots$, $dp^{n}-1$ is periodic with period $c$. The sequence
begins with $\frac{c-1}{2}$ and continues with a permutation of
$\frac{c-3}{2},\dots,-\frac{c-1}{2}$. If we restrict to values of $n$ such
that $dp^{n}$ lies in a fixed congruence class modulo $c$, then the values do
not change as $n\rightarrow\infty$. $\lozenge$

The measure $E_{1,c}$ and the Dirichlet series expansion are particularly
simple for $c=2$. Note that we assumed that $d$ and $p$ are odd in this case.
If $a$ is even, then $\\{-a(dp^{n})^{-1}\\}_{2}=0$ and

$E_{1,2}(a+dp^{n}X)=\epsilon_{a,2,dp^{n}}=\frac{1}{2}.$

If $a$ is odd, then $-a(dp^{n})^{-1}$ is odd, $\\{-a(dp^{n})^{-1}\\}_{2}=1$
and

$E_{1,2}(a+dp^{n}X)=\epsilon_{a,2,dp^{n}}=-\frac{1}{2}.$

Hence $E_{1,2}$ is up to the factor $\frac{1}{2}$ equal to the following
simple measure:

######  Definition 4.2.

Let $p\neq 2$ be a prime, and let
$X\cong\mathbb{Z}/d\mathbb{Z}\times\mathbb{Z}_{p}$ be as above. Then

$\mu(a+dp^{n}X)=(-1)^{\\{a\\}_{dp^{n}}}$

defines a measure on $X$. We call $\mu$ the alternating measure, since the
measure of all clopen balls is $\pm 1$. $\lozenge$

The corresponding integral is also called the fermionic $p$-adic integral (see
[4]).

Now we obtain the following Dirichlet series expansion from Corollary 2.5.

######  Corollary 4.3.

Let $p\neq 2$ be a prime number, and let $\chi$ be an even Dirichlet character
of odd conductor $f$. Then we have for $s\in\mathbb{C}_{p}$ with
$|s|<p^{(p-2)/(p-1)}$,

$(1-\chi(2)\langle
2\rangle^{1-s})L_{p}(s,\chi)=\lim_{n\rightarrow\infty}\frac{1}{2}\sum^{fp^{n}}_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}(-1)^{a+1}\chi\omega^{-1}(a)\frac{1}{\langle
a\rangle^{s}}.$

For $\chi=\omega^{1-i}$ and odd $i=1,\,\dots,\,p-2$, we obtain the branches of
the $p$-adic zeta function:

$\zeta_{p,i}(s)=L_{p}(s,\,\omega^{1-i})=\frac{1}{1-\omega(2)^{1-i}\langle
2\rangle^{1-s}}\cdot\lim_{n\rightarrow\infty}\frac{1}{2}\sum_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}^{p^{n}}(-1)^{a+1}\omega(a)^{-i}\frac{1}{\langle
a\rangle^{s}}$

######  Remark 4.4.

Dirichlet series expansions of $p$-adic $L$-functions were studied by D.
Delbourgo in [1] and [2]. He considers Dirichlet characters $\chi$ satisfying
$(p,\,2f\phi(f))=1$ and their Teichmüller twists. We obtain the same expansion
for $c=2$ and $\chi=\omega^{1-i}$. However, we require $(c,\,fp)=1$ and use
other methods for the proof.

Similar expansions for a slightly different $p$-adic $L$-function using a
fermionic $p$-adic integral (i.e., $c=2$) were also obtained by M.-S. Kim and
S. Hu (see [4]).$\hfill\lozenge$

######  Example 4.5.

We look at the case $c=3$. The sequence of values $\epsilon_{a,3,dp^{n}}$ is
periodic with period $3$. If $dp^{n}\equiv 1\text{ mod}\ 3$, then the sequence
is $1,\ -1,\ 0,\,\dots$ . If $dp^{n}\equiv 2\text{ mod}\ 3$, then we obtain
the sequence $1,\ 0,\ -1,\,\dots$ .

######  Corollary 4.6.

Let $p$ be a prime number, and let $\chi$ be an even Dirichlet character of
conductor $f=dp^{m}$ such that $(3,\,dp)=1$. If $d\equiv 1\text{ mod}\ 3$,
then define a sequence $\epsilon_{0}=1$, $\epsilon_{1}=-1$, $\epsilon_{2}=0$,
$\dots$ with period $3$. Otherwise, set $\epsilon_{0}=1$, $\epsilon_{1}=0$,
$\epsilon_{2}=-1$ and extend it with period $3$. Then we have for
$s\in\mathbb{C}_{p}$ with $|s|<qp^{-1/(p-1)}$,

$-(1-\chi(3)\langle
3\rangle^{1-s})L_{p}(s,\chi)=\lim_{n\rightarrow\infty}\sum^{dp^{2n}}_{\begin{subarray}{c}a=1\\\
p\,\nmid\,a\end{subarray}}\chi\omega^{-1}(a)\frac{\epsilon_{a}}{\langle
a\rangle^{s}}.$

######  Example 4.7.

For $c=5$, we get a periodic sequence with period $5$ and we have
$\epsilon_{a,5,dp^{n}}=2$ for $a\equiv 0\text{ mod}\ 5$. The next four
coefficients are a permutation of the values $-2$, $-1$, $0$ and $1$,
depending on the class of $dp^{n}\text{ mod}\ 5$.

######  Example 4.8.

Let $c=7$. Then $\epsilon_{0,7,dp^{n}}=3$. Now suppose, for example, that
$dp^{n}\equiv 3\text{ mod}\ 7$. Then $(dp^{n})^{-1}\equiv 5\text{ mod}\ 7$.
This yields the values

$\epsilon_{1,7,dp^{n}}=1,\ \epsilon_{2,7,dp^{n}}=-1,\
\epsilon_{3,7,dp^{n}}=-3,\ \epsilon_{4,7,dp^{n}}=2,\ \epsilon_{5,7,dp^{n}}=0,\
\epsilon_{6,7,dp^{n}}=-2,$

and these are extended with period $7$.

Acknowledgements.  The first author thanks Daniel Delbourgo for hints to his
work and helpful conversations.

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