an analytical model for the strategy of winning hearts and …the strategy of winning hearts and...

Defence Research and Development Canada Scientific Report

DRDC-RDDC-2020-R064

August 2020

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An analytical model for the strategy of winning hearts and minds

P. Bao U. Nguyen DRDC – Centre for Operational Research and Analysis

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DRDC-RDDC-2020-R064 i

Abstract

We propose using epidemic models to simulate a complex warfare scenario that includes the strategy of

winning hearts and minds. Two epidemic models are compared to a Lanchester model, which simulates

exchanges of fire. The first epidemic model, Susceptible-Infected (SI) has a known closed form and an

analytical solution. The second epidemic model, Susceptible-Infected-Recovered (SIR) has no known

closed form solution. Its solution is approximated using successive transformations that are simple, closed

form and analytical. We show that the approximation reproduces nearly perfectly the exact numerical

results of the SIR model. In addition, we illustrate the variability of a military operation by introducing

noise in the SI model and show how this affects the outcomes of a war.

Significance to defence and security

This Scientific Report documents a model of the strategy of winning hearts and minds. To develop the

model we make use of epidemiology. Two epidemic models are used: the Susceptible-Infected (SI) model

and the Susceptible-Infected-Recovered model (SIR).

We interpret the process of infections as the process of making allies. This allows the blue force to win

the war even when it is initially outnumbered by the red force. This happens if the blue force can gather

sufficient allies (infections) from the local population. The strategy of winning hearts and minds has been

employed by the military as far as warfare exists. Recently, the United States (US) employed this strategy

in Iraq, in Afghanistan and in Vietnam. However, models of such a strategy with sufficient complexity

have not been known in the open literature. The model presented here may be the first of its kind. The

result is probabilistic and includes noise to account for unexpected events in a military operation such as

lack of information, internal fightings, morale of the troops, etc. The result is also analytical which allows

for quick assessments and provides insights into the parameters: neutralization probability, blue force

size, red force size and infection rate, etc.

ii DRDC-RDDC-2020-R064

Résumé

Nous proposons d’utiliser des modèles épidémiques pour simuler un scénario de guerre complexe

comprenant la stratégie visant à conquérir les cœurs et les esprits. Deux modèles épidémiques sont

comparés à un modèle de Lanchester, qui simule des échanges de tirs. Le premier modèle épidémique,

Susceptible-Infecté (SI), a une forme fermée connue et une solution analytique. Le second modèle

épidémique, Susceptible-Infecté-Rétabli (SIR) n’a pas de solution connue sous forme fermée. Une

solution approximative est obtenue par des transformations successives qui sont simples, sous forme

fermée et analytiques. Nous montrons que l’approximation reproduit presque parfaitement les résultats

numériques exacts du modèle SIR. De plus, nous illustrons la variabilité d’une opération militaire en

introduisant du bruit dans le modèle SI et nous montrons comment ce bruit influe sur les résultats

d’une guerre.

Importance pour la défense et la sécurité

Ce rapport documente un modèle de la stratégie de conquête des cœurs et des esprits. Pour établir le

modèle, nous utilisons l’épidémiologie. Deux modèles épidémiques sont utilisés: le modèle

Susceptible-Infecté (SI) et le modèle Susceptible-Infecté-Rétabli (SIR).

Nous interprétons le processus d’infection comme le processus de se faire des alliés, ce qui permet à la

force bleue de gagner la guerre même si, au début, la force rouge était plus nombreuse. La force bleue

peut être victorieuse si elle parvient à rassembler suffisamment d’alliés (infections) auprès de la

population locale. La stratégie de conquête des cœurs et des esprits a été employée par les militaires

depuis que la guerre existe. Récemment, les États-Unis ont utilisé cette stratégie en Iraq, en Afghanistan

et au Vietnam. Cependant, les modèles de cette stratégie suffisamment complexes ne se retrouvent pas

dans les sources publiées. Le modèle présenté ici est peut-être le premier du genre. Le résultat est

probabiliste et inclut le bruit pour tenir compte des événements inattendus dans une opération militaire

tels que le manque d’information, les combats internes, le moral des troupes, etc. Le résultat est

également analytique, ce qui permet des évaluations rapides et donne un aperçu des paramètres:

probabilité de neutralisation, taille de la force bleue, taille de la force rouge et taux d’infection, etc.

DRDC-RDDC-2020-R064 iii

Table of contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Significance to defence and security . . . . . . . . . . . . . . . . . . . . . . . . . i

Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Importance pour la défense et la sécurité . . . . . . . . . . . . . . . . . . . . . . . ii

Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Lanchester models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 The Susceptible-Infected (SI) model . . . . . . . . . . . . . . . . . . . . . . . 4

4 The Susceptible-Infected-Recovered (SIR) model . . . . . . . . . . . . . . . . . . 6

5 Probabilistic rate of infection . . . . . . . . . . . . . . . . . . . . . . . . . 14

6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

List of symbols/abbreviations/acronyms/initialisms . . . . . . . . . . . . . . . . . . 25

iv DRDC-RDDC-2020-R064

List of figures

Figure 1: Number of individuals and units as a function of time. . . . . . . . . . . . . . . 5

Figure 2: /df dt as a function of f for the exact case and for the quadratic approximation.. . . . 9

Figure 3: /df dt as a function of f for the exact case, for the quadratic approximation and for

the asymmetric approximation. . . . . . . . . . . . . . . . . . . . . . . 10

Figure 4: I as a function of time for the exact case, for the quadratic approximation and for the

asymmetric approximation. . . . . . . . . . . . . . . . . . . . . . . . . 10

Figure 5: I as a function of time for the exact case, for the time-shifted approximation, for the

asymmetric approximation and for the quadratic approximation. . . . . . . . . . 12

Figure 6: S as a function of time for the exact case, for the time-shifted approximation, for the


Figure 7: R as a function of time for the exact case, for the time-shifted approximation, for the


Figure 8: Number of susceptible individuals and infected individuals as a function of time. . . 16

Figure 9: Number of susceptible individuals and infected individuals as a function of time (with

and without noise). . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Figure 10: Number of susceptible individuals as a function of time (with and without noise). . . 17

Figure 11: Number of infected individuals as a function of time (with and without noise). . . . 18

Figure 12: Number of recovered individuals as a function of time (with and without noise). . . 18

Figure 13: Number of SIR individuals as a function of time (with and without noise). . . . . . 20

DRDC-RDDC-2020-R064 1

1 Introduction

In 1895, French General Louis Hubert Gonzalve Lyautey was the first person to use the expression

“hearts and minds” in his strategy to counter the Black Flags rebellion along the Indochina-Chinese

border [1]. General Lyautey referred to a strategy that appealed to the local population for support.

Subsequently, the same phrase was used to describe a potential strategy for winning other conflicts in

other times and places. In 1952, British General Sir Gerald Templer said that victory in the Malayan war

“lies not in pouring more soldiers into the jungle, but in the hearts and minds of the Malayan people” [2].

In the 1960s, US President Lyndon Baines Johnson inspired the United States (US) to conduct a “hearts

and minds” campaign in Vietnam. The United States engaged mainly the South Vietnamese people to

help overthrow the North Vietnamese [2]. More recently, the United States and its allies spent billions of

US dollars on the strategy of winning “hearts and minds” in Afghanistan and in Iraq [3].

History clearly suggests the importance of the “hearts and minds” strategy in warfare. However, there are

not many mathematical models that allow analysis of the effectiveness of this strategy. Unlike attrition

models, where red force and blue force exchange fire, the “hearts and minds” models, if any, are not

readily available and/or are not well understood.

In this Scientific Report, we propose the use of epidemic models to examine the time evolution as well as

the outcomes of such a strategy. We will consider two epidemic models: the Susceptible-Infected model

(SI) and the Susceptible-Infected-Recovered (SIR) model [4]. The complexity of the two epidemic

models will be compared to attrition models, known as Lanchester models [5].

Technically, the strategy of winning hearts and minds aims to gather political support from the local

population. However, we take the liberty of assuming that the strategy of winning hearts and minds could

increase the blue force strength, taking as an example the way in which the South Vietnamese people and

their army worked with the United States towards a common goal.

Generally, the basis of a scientific theory is its falsifiability. Therefore, the validity and falsifiability

of modelling the strategy of winning hearts and minds using epidemic models will ultimately rely on

data. We will endeavor to analyze historical data in the future. With data, we can modify the model

accordingly, if needed. There will be many options available as there is a whole spectrum of epidemic

models in the literature. It is also possible that the correct model lies in differential equations that are

not in epidemic models. However, the idea of modelling the strategy of winning hearts and minds remains

unchanged.

We are aware that attrition models such as Lanchester’s equations have limitations. But they are still

useful, since attrition models describe important ideas. The same applies to epidemic models and

analytical models in general. Usually, to include details of a war, we would need a simulation.

Statement of originality. The main contributions are two-fold. First, an accurate approximation of an

analytical solution to the SIR model is derived. Second, we consider how making allies can be interpreted

in terms of infection. These two steps allow us to model the idea of winning hearts and minds in an

attrition model. To the best of our knowledge, this has never been done.

2 DRDC-RDDC-2020-R064

2 Lanchester models

Lanchester models are well known: there are hundreds of papers on Lanchester models in the literature on

topics ranging from directed fire, to area fire, to guerilla warfare, etc. [5]. Here, we describe one of the

simplest Lanchester models: area fire that corresponds to a linear law.

Denoting r as the number of red (hostile) units and b as the number of blue (friendly) units, the area fire Lanchester model consists of two linear differential equations:

0

0

db br

dt b

dr br

dt r

(1)

where t is the time in days; and is the attrition rate of the blue force due to the fires of the red force

and similarly is the attrition rate of the red force due to the fires of the blue force.

This system of differential equations has an analytical and closed form solution [5] that is shown below:

0 0

0 0

2

0

1/2

0 0

2

0

1/2

0 0

1

1

1

t

t

b t

b e

r t

r e

(2)

where 0b is the initial number of blue units, 0r is the initial number of red units and 0 is the superiority

parameter 0

1 . Analysis of the solution provides a criterion defined through the superiority parameter

that predicts the outcome of the exchange of fires between the blue force and the red force. The

superiority parameter is defined as:

0

0

0

b

r (3)

If 0

1 , the blue force will eventually win. That is, the number of red units will eventually become zero

while there will be some blue units remaining. If 0

1 , the opposite will happen. That is, the red force

will eventually win.


If 0

1 , neither side wins and the solution becomes simply:

0 0

1

1

b t r t

b r t

(4)

For convenience, we will refer to the area fire (linear law) model simply as the Lanchester model for the

remainder of this Report. The Lanchester model assumes implicitly that each force is aware of only the

general area in which the opposing force is located and fire is allocated into this area, hence the term

area fire.

We note that when fire is spread uniformly against the opposite force, we maximize the probability of

neutralizing all units of the opposite force [6]. This holds for a continuous allocation of weapons. For a

discrete allocation, as is the case of missile defence, for example, spreading weapons as evenly as

possible will also maximize the probability of neutralizing units of the opposite force [6].

In the next section we will describe the SI epidemic model. We will then compare the SI model to the

Lanchester model.


3 The Susceptible-Infected (SI) model

In epidemiology, the SI model describes lifelong infections such as herpes [7]. Once an individual is

infected, that individual stays infected. The SI model is defined through a set of differential equations that

are nearly identical to those of the Lanchester equations, with the exception of a change of sign and the

coupling parameter as shown below:

dSSI

dt

dISI

dt

(5)

where S is the number of susceptible individuals who could be infected; I is the number of infected

individuals; is the rate of infection; and t is time in days. The population is constant, i.e., S I N .

Hence, / / / 0dS dt dI dt dN dt . One could consider S as the red force and I as the blue force. We

observe that the SI differential equations are very similar to the Lanchester differential equations with two

exceptions:

1. / 0dI dt unlike / 0db dt ; and,

2. The coupling constants in the Lanchester model are different: and while there is only one

coupling constant in the SI model: .

Mathematically, if one can solve the Lanchester equations, then one can solve the SI equations. Indeed,

the solution to the SI equations [4] can be written as:

0

01

Nt

Nt

NI eI t

N I e

S t N I t

(6)

To illustrate, we plot the number of susceptible individuals, the number of infected individuals, the

number of blue units and the number of red units as a function of time in Figure 1. We assume that

0 0, 0.25b I , 0 0, 0.75r S ,

0 0

, , 0.1b r

and 10N . Due to normalization, /I I N and /S S N , the

actual value of N does not affect I and S as a function of time.

In the Lanchester model, both blue force and red force decrease as a function of time. In the SI model, red

force (susceptible individuals) also decreases as a function of time. However, blue force (infected

individuals) increases as a function of time. This is consistent with SI differential equations where

/ 0dI dt SI as , , 0S I .


If we interpret the infected individuals as the allies from the local population in addition to the blue force

then this shows the impact of the strategy of winning hearts and minds. That is, the blue force (friendly

force) increases with time while the red force (hostile force) decreases with time.

Figure 1: Number of individuals and units as a function of time.

In the Lanchester model, the red force (dashed curve) wins against the blue force (dashed curve) due to

the fact that the red force 0 0.75r outnumbers the blue force 0 0.25b , while their weapon

effectiveness is the same. However, in the SI model, the outcome is the opposite even with the same

initial conditions. Despite the fact that the blue force is outnumbered by the red force, with local support,

the blue force can still win the war. This outcome is contingent on the assumptions in the SI model. That

is, the red force strength keeps decreasing with time while the blue force keeps increasing. We will

elaborate more on this issue in the next section. This outcome shows the significance of having allies. So,

if the strategy of winning hearts and minds is successful, we can reverse the outcome of a war in a

seemingly impossible scenario where the red force is three times the blue force with identical weapon

effectiveness from both sides.

We observe that, in both models, the trends are monotonic. That is, the blue force either increases or

decreases. The same trend is seen with the red force.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

Nu

mb

er

of

ind

ivid

ual

s an

d u

nit

s

Time (days)

Number of susceptible individuals, infected individuals, blue units, red units as a function of time

Susceptible

Infected

Red

Blue


4 The Susceptible-Infected-Recovered (SIR) model

In the SI model, there are no competing effects—i.e., effects that raise the number of units and effects that

lower the number of the same units—as there should be in any complex scenario of epidemics or warfare.

This is a shortfall of the SI model. To remedy this shortfall, we propose to examine the

Susceptible-Infected-Recovered (SIR) model. On the one hand, the SIR model can be used to describe

diseases such as measles, where an individual may recover from being infected and remain immune

afterwards. On the other hand, the SIR model cannot model diseases such as tuberculosis in which

infected individuals may never recover. The SIR model assumes a time scale short enough so that there

are no natural births and deaths but includes deaths from the diseases. Such diseases often arise in cycles

of outbreaks. It is actually not difficult to model natural births and deaths but for now, we prefer to focus

on demonstrating the idea of modelling the strategy of hearts and minds. In terms of warfare, the effect of

an infection implies that an ally could have a change of heart and become an enemy. The SIR model

originates from the work of [8].

The SIR model can be expressed by a system of differential equations shown below:

dSaSI

dt

dIaSI bI

dt

dRbI

dt

(7)

where t is time in days; S is the number of susceptible individuals; I is the number of infected individuals; and R is the number of recovered individuals. We interpret S as the red force, I as the blue

force and R as the force removed permanently from the war, i.e., those who are killed or those who have

no interest in either side. The expression of /dI dt includes both the effect of attrition due to fighting

with red force bI and the effect of hearts and minds due to the conversion of the red force to blue

force aSI . Although we state that t is time in days, the time unit chosen could be adapted to the scenario. It could be months or years, especially in the case of a long term war such as the Vietnam War.

This interpretation of the SIR model assumes that the red force does not increase, unlike the blue force.

Admittedly, this is a limitation of the model and remains an aspect for future investigation. There are no

known closed form solutions for the SIR model in the literature. However, there are two exact solutions

obtained through parametrization: one by Harko [9] and the other by Miller [10][11]. Independently and

unaware of Miller’s solution, we have also found the same solution (as described briefly in [12][13]). Due

to the parametrization, these solutions are not considered as closed form solutions, making them more

inefficient to compute than closed form solutions.

Instead of solving the SIR model exactly, we will provide an accurate approximation using simple and

successive linear transformations that yield a closed form and analytical solution. [12][13] show that to

solve the SIR model, it is equivalent to solve the differential equation below:


01 af

dfbf S e

dt (8)

where

0

0t

f t I t dt (9)

with initial conditions:

0

0

0

0

0 0

0

0 0

0 0

0

1

0

f I t dt

dfI I

dt

S S

S I

R

(10)

/df dt has two roots [14]:

/0

1

/0

2

1 11,

1 10,

a b

a b

aSf f W e

b a b

aSf f W e

b a b

(11)

where W are Lambert functions. There are two branches to Lambert functions for a real variable

x : 1,W x and 0,W x , [15]. [12] shows that 2 0f and 1 0f . It is also shown that /df dt is a convex function in f .

This leads to the first approximation:

0 1 21af

qbf S e c f f f f (12)


where

2

2

1 2

0

2 2

1 2

0

1

f

af

o

fq

df f f f f bf S e

c

df f f f f

(13)

and /df dt is approximated as a quadratic equation. [12] shows that the quadratic approximation yields

the following results:

2

2 1

1

/

t

t

f ef

f f e (14)

where

1 2

0qc f f (15)

and,

2

1 2 1 2

2

2 1

'

t

t

qc f f e f fI t f t

f f e

(16)

0

af tS t S e (17)

R t bf t (18)

For illustration, we assume that 3/ 2a , 1/ 3b , 0 0.99S , 0 0.01I and 00R . It is seen in Figure 2

that the quadratic equation reproduces the shape of an upside down cup of the exact equation. However,

unlike the quadratic equation, the exact equation is not symmetrical. This leads to the second

approximation [13]:

1 1

0 1 21 afbf S e c f f f f (19)


Figure 2: /df dt as a function of f for the exact case and for the quadratic approximation.

The parameter induces the asymmetry of /df dt . is selected so that the maximum of the left hand

side (LHS) and the maximum of the right hand side (RHS) of the above equation match at

* 0ln / /f f a S b a . There is no ambiguity in this process, since both functions are convex and as a

result they have exactly one maximum each.

[13] shows that:

*

1 2

1 2

2 f f f

f f (20)

0

1 1* *

1 2

1 / 1 ln /b a a S bc

f f f f

(21)

2 1

2 1

21

c f f uI

u (22)

while

0

af tS t S e (23)

R t bf t (24)

where

1/

2 1u c f f t A (25)

1

2 2 1

1 1fA

c f f f

(26)

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3d

f/d

t f

df/dt as a function of f

Exact

Quadratic


For illustration, we plot /df dt in Figure 3 below for three cases: the exact function, the quadratic

function and the asymmetric equation modelled by the parameter . It is seen that the asymmetric

equation reproduces the asymmetry of the exact function.

Figure 3: /df dt as a function of f for the exact case, for the quadratic approximation

and for the asymmetric approximation.

Figure 4: I as a function of time for the exact case, for the quadratic approximation

and for the asymmetric approximation.

For illustration, we plot I t in Figure 4 as a function of time for the three cases: the exact results

obtained numerically from Mathematica [14], the quadratic approximation and the asymmetric

approximation. It is seen that the asymmetric approximation is closer to the exact results than the

quadratic approximation: both in terms of the maximum of I and its location. Yet, the asymmetric

approximation still does not reproduce the exact results especially at the maximum of I . This leads to the

third approximation.

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3

df/

dt

f

df/dt as a function of f

Exact

Quadratic

Asymmetric

0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40

I

Time (days)

I as a function of time

Exact

Quadratic

Asymmetric


We know from above that I t is maximal when * 0ln / /f f a S b a . This means that I t is

maximal when:

*

*

0 01

f

e af

dft t

bf S e (27)

In the asymmetric model, I t is maximal when [13]:

*

*

*

1 1

0 1 2

f

ut t A

c

df

c f f f f

(28)

where

* 1

1u (29)

We define

* *

s e st t t (30)

which is the difference between the location of the maximum of the exact solution and the location of the

maximum of the asymmetric approximation. We use st to shift the location of the maximum of the

asymmetric approximation so that it coincides with the location of the maximum of the exact solution.

That is,

s sI t I t t (31)

We do the same for S t

and R t

i.e.,

s sS t S t t (32)

s sR t R t t (33)


For illustration, we plot I t for the four cases (Figure 5): the exact result, the quadratic approximation,

the asymmetric approximation and the time-shifted approximation. In order of increasing accuracy when

compared to the exact result: the quadratic approximation is the furthest from the exact result followed by

the asymmetric approximation, followed by the time-shifted approximation. It is evident that the

time-shifted approximation reproduces nearly perfectly the exact result.

Figure 5: I as a function of time for the exact case, for the time-shifted approximation, for the

asymmetric approximation and for the quadratic approximation.

One may wonder why the asymmetric approximation does not reproduce the exact result when the curves

of /df dt for the two cases are nearly the same (Figure 3). The reason lies in Equations (27–28). Even

when

1 1

0 1 2/ 1 afdf dt bf S e c f f f f , the integrals (areas under the integrands) may still

differ slightly:

*

*

*

1 1

0 1 2

*

0 01

f

f

e af

dft

c f f f f

dft

bf S e

(34)

Similarly, we plot S t below (Figure 6) and R t below (Figure 7) for the four cases.

0

0.1

0.2

0.3

0.4

0.5

0 10 20 30 40

Nu

mb

er

of

infe

cte

d u

nit

s

Time (days)

Number of infected units (blue force) as a function of time

Exact

Time-shifted

Asymmetric

Quadratic


Figure 6: S as a function of time for the exact case, for the time-shifted approximation, for the


Figure 7: R as a function of time for the exact case, for the time-shifted approximation, for the


By analyzing Figures 5, 6, and 7, the exact (and time-shifted) results show that the red force is virtually

eliminated by Day 10. At that time, fifteen percent of the population is blue while the remaining is

recovered (either killed or not participating in the war). There are two interpretations for the time beyond

Day 10. First, we consider that the war is over by Day 10 and stop the calculation by Day 10. Second, we

consider that the peace is restored at the end, i.e., the number of recovered units approaches one: either

those individuals are killed or no longer participating in the war.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

Nu

mb

er

of

susc

ep

tib

le in

div

idu

als

Time (days)

Number of susceptible individuals (red force) as a function of time

Exact

Time-shifted

Asymmetric

Quadratic

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

Nu

mb

er

of

susc

ep

tib

le in

div

idu

als

Time (days)

Number of recovered individuals (neutral force) as a function of time

Exact

Time-shifted

Asymmetric

Quadratic


5 Probabilistic rate of infection

Unlike the exact sciences such as physics or chemistry where experiments can be very precise e.g., the

measurement of the magnetic moment of an electron is known to 7.6 parts in 1310 , [16]; there are many unexpected events in a military operation due to incomplete intelligence information or unforeseen

weather or illness/moral issues among the soldiers. Therefore, it is necessary to account for these

unpredicted incidents when we model attrition. One typical way to do so is to introduce noise in the

simulation of attrition models. For illustration, we assume that the noise [17], obeys a Gaussian density

distribution. That is,

2 2/ 2

2

xe

f x (35)

where 2 is the variance of the Gaussian density distribution. We consider for example the SI model. We

let the rate of infection to be x where ,x . Since the range of x is a finite interval unlike the

infinite domain of the Gaussian density distribution , , we need to normalize the density

distribution, i.e.,

2 2/ 2

2

x

n

ef x f x

N

(36)

where the subscript n stands for noise and

2 2/ 2

2

xe

N dx (37)

Hence

2 2

2 2

/ 2

/ 2

x

n x

ef x

dx e (38)

This means that the rate of infection is x has a density distribution nf x . The corresponding

probability decreases as x increases. Hence, the most likely rate of infection is . Probability

theory dictates how to determine the expected values of any metrics that involve the rate of infection.

Note that this is not simply a sensitivity analysis as there is a probability attached to each value of the

rate of infection.


The expected number of infected/susceptible units averaged over the noise can be written as

0

01

Nt

Nt

x

n nx

n n

NI e

N I eI t dx f x

S t N I t

(39)

We can do the same for the Lanchester model. However, we restrict the scenario to0 0

,b r

for

simplicity.

20 0

20 0

2

0 1

2

2

0 1

2

1

1

1

nn

nn

nn n

x b r xx

n

nn n

x b r xx

n

xb t b dx f x

x e

xr t r dx f x

x e

(40)

where

0 00 0

0

0

n

b x r b

r x b rx

(41)

which is independent of x in this scenario.

Figure 8 shows the number of susceptible individuals and infected individuals as a function of time. The

“ ” and “ ” denote the number of individuals when the rate of infection is equal to and respectively where /2 . All of the other parameters are kept the same as those assumed in Figure 1.

When assuming / the difference between S and I is increased/decreased. The change is

significant e.g., at the time equal to approximately 10, the difference between S and I varies from 20 percent to 90 percent. This can modify or accelerate the outcome of a war if we consider the SI model or an epidemic model in general as a warfare model.


Figure 8: Number of susceptible individuals and infected individuals as a function of time.

Figure 9: Number of susceptible individuals and infected individuals as a function of time

(with and without noise).

Figure 9 shows the number of susceptible individuals and infected individuals as a function of time with

and without noise. The numbers of individuals with noise are expected values obtained by integrating the

number of individuals over the range of noise and compounded with the density distribution, as shown in

Equation (39). It is seen that with noise the number of susceptible individuals decreases while the number

of infected individuals increases. This could be interpreted that blue force is favored when noise is

accounted for.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

Nu

mb

er

of

ind

ivid

ual

s

Time (days)

Number of susceptible individuals and infected individuals as a function of time

Susceptible

Susceptible+

Susceptible-

Infected

Infected+

Infected-

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

Nu

mb

er

of

ind

ivid

ual

s

Time (days)

Number of susceptible individuals and infected individuals (with and without noise) as a function of time

Susceptible

Susceptible (noise)

Infected

Infected (noise)


We do the same for the SIR model. Figure 10 shows the number of susceptible individuals as a function

of time. Figure 11 shows the number of infected individuals as a function of time. Figure 12 shows the

number of recovered individuals as a function of time.

All of the parameters remain the same as those in Section 4 with the exceptions that the “ ” suffix

means that / 2a a aa a and / 2b b bb b . Similarly “ ” suffix means that

/ 2a a aa a and / 2b b bb b . 2a and

2b are the variances of the Gaussian noises

added to the coupling parameters a and b . Note that we add the noise to parameter a with an opposite sign to the noise for parameter b . We do so to obtain the furthest effects on S , I andR as parameter a and parameter b induce opposite effects. That is, a increases I while b decreases I as seen in Equation (7). If

we add noises with same sign, e.g., aa a and bb b then this “ ” scenario will yield effects

that are bound by the “ ” scenario and the “ ” scenario.

Figure 10 shows that the “ ” scenario lowers the number of susceptible individuals while the

“ ” scenario raises the number of susceptible individuals generally. Figure 11 shows that the “ ”

scenario raises the number of infected individuals while the “ ” scenario lowers the number of infected

individuals generally. Figure 12 shows that the “ ” scenario raises the number of recovered individuals

while the “ ” scenario lowers the number of recovered individuals generally. Clearly the changes to S , I andR due to noises are significant. For example, the difference in S at time equal to 10 days is almost 80 percent. This could reverse the outcome of a war.

Figure 10: Number of susceptible individuals as a function of time (with and without noise).

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

Nu

mb

er

of

ind

ivid

ual

s

Time (days)

Number of susceptible individuals as a function of time with and without noise

Susceptible+-

Susceptible

Susceptible-+


Figure 11: Number of infected individuals as a function of time (with and without noise).

Figure 12: Number of recovered individuals as a function of time (with and without noise).

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

Nu

mb

er

of

ind

ivid

ual

s

Time (days)

Number of infected individuals as a function of time with and without noise

Infected+-

Infected

Infected-+

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

Nu

mb

er

of

ind

ivid

ual

s

Time (days)

Number of recovered individuals as a function of time with and without noise

Recovered+-

Recovered

Recovered-+


The expected values to S , I andR when averaged over the noise density distributions can be written as:

, ,

, ,

1

b a

n a b

b a

b a

n a b

b a

n n n

S t dx dy S x y t f x f y

I t dx dy I x y t f x f y

R t S t I t

(42)

where n stands for noise and

2 2

2 2

2 2

2 2

/ 2

/ 2

/ 2

/ 2

a

a

b

b

x

a a x

a

y

b b y

b

ef x

dx e

ef y

dy e

(43)

We choose 1

12

so that when we add noises to parameters a and b i.e., a x and b y the

coupling parameters a x and b y will not be zeroes. This is essential since 1f and 2f in Equation (11)

depend on 1 / a x and 1/ b y . Additionally, this will maintain the ratio

0

1 / 3 3 / 2/ 2 20.99

/ 2 3 / 2 1 / 2 3

b y b bS

a x a a so that the number of infected individuals will increase with

time initially (when time is equal to zero: / 0dI dt in Equation (7)).

Figure 13 shows the number of susceptible individuals, the number of infected individuals and the

number of recovered individuals without noise and averaged over noise as a function of time. It is shown

that the difference between the case (susceptible individuals for example) without noise and the

corresponding case (susceptible individuals with noise) averaged over noise is less than 10 percent. The

maximal number of infected individuals with noise is less than the one without noise. This means then

when noise is accounted for, the strategy of winning hearts and minds is less effective than when noise is

not accounted for. Similarly, the number of susceptible individuals without noise is generally less than the

number with noise. Also, the number of recovered individuals without noise is generally greater than the

number with noise. Therefore, noise favors red force in the SIR model.


Figure 13: Number of SIR individuals as a function of time (with and without noise).

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

Nu

mb

er

of

ind

ivid

ual

s

Time (days)

Number of individuals (different types) with and without noise as a function of time

Susceptible

Susceptible (noise)

Infected

Infected (noise)

Recovered

Recovered (noise)


6 Discussion

First, these exact and analytical solutions require us to perform integrals such as Equation (34) repeatedly

and numerically, making them less efficient than closed form solutions provided by the three

approximations: the quadratic approximation, the asymmetric approximation and the time-shifted

approximation.

Second, all of the parameters needed for the three approximations have closed form formulas and are

simple to compute, with the exception of *t for the time-shifted approximation. However, the *t calculation requires us to perform two numerical integrals only once, at the beginning of the analysis.

Third, with closed form solutions we can easily determine the characteristics of the solution, such as how

they evolve with time; when they increase or decrease; when they reach a maximum or a minimum; and

what happens when time gets large (asymptotic limits).

In terms of warfare, we show that if the strategy of winning hearts and minds is successful then even with

an initial disadvantage (outnumbered by a factor of three to one), the blue force can still win the war. This

would be impossible with a Lanchester equation but is feasible with the SI model. However, this

possibility arises due to an assumption of the SI model embedded in the system of differential equations

that defines the model. This leads to the investigation of the SIR model.

The SIR model adds complexity to the SI model. Here, the infected units (blue force and allies) can

increase and then decrease. This means that allies of the blue force could have a change of heart with

time. This characteristic occurs in our daily lives. Today’s allies can become tomorrow’s enemies. To

avoid this eventuality, it is to the advantage of the blue force to win the war before this turning point.

Admittedly, this could happen to the red force as well. That is, the red force could influence the blue force

and their allies to switch to the red side. Therefore, this interpretation of the SIR model assumes that the

number of blue force and their allies switching to the red side is limited. In reality, switching sides could

happen both to the blue force and to the red force. This is an issue that we will investigate in the near

future. Potentially, this issue can be rectified by modifying the system of differential equations that

defines the epidemic models.

Additionally, as alluded to in Section 5, military operations are usually conducted with incomplete

information and with many unexpected events. As is often the case, we model the variability of

the scenarios through noise in the warfare models. We show that noise does indeed affect the timing

of a war. The war can end earlier or later than expected, which evidently influences financial costs,

economies, casualties, world peace etc. Even though we did not show it: noise can also influence

the outcome of a war, whether blue force wins or loses. Hence, modelling noise adds reality to

warfare models.

This Report provides an analytical model of warfare that includes the strategy of winning hearts and

minds, which is not always available in the literature. It shows the surprising impact of this strategy. The

Report also gives a glimpse into multi-faction conflicts (Susceptible, Infected and Recovered units). We

note that Ref [18] describes a simulation that examines multi-faction analyzes including effects of

strategic campaigns. An example of three faction conflicts is also analyzed by [19].


It is hoped that this model will help in the planning of resources and policies that include the strategy of

winning hearts and minds. In the future, data on infection and recovery rates, etc., should be collected so

that they can be input into the model. With such data, we can validate/invalidate and test the model. We

have hypothesized that the SIR model can simulate both the effect of attrition and the effect of hearts and

minds. In reality, the exact expressions (if they existed) of the differential equations could be different

from those of the SIR model. However, the idea remains unchanged. To the blue force: the effect of

attrition is negative and the effect of hearts and minds is positive. The validation lies in the principle of

the falsifiability of the data.

We suggest that it is time to expand the curriculum for traditional defence analyzes to include epidemic

models in addition to the Lanchester models already in use.


References

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090130_vietnam_study.pdf. Last accessed Jan 2020.

[3] Sexton, R. (06 Jan, 2017) Did U.S. non-military aid win hearts and minds in afghanistan? Yes and no. The Washington Post. https://www.washingtonpost.com/news/monkey-cage/wp/2017/01/06/did-u-s-

nonmilitary-aid-win-hearts-and-minds-in-afghanistan-yes-and-no/?utm_term=.c152677432a1. Last

accessed Mar 2019.

[4] Smith, R. (2008) Modelling disease ecology with mathematics. American Institute of Mathematical Sciences: 14–30.

[5] Przemieniecki, J. S. (2000) Mathematical methods in defense analyses. 3rd edition, AIAA education series: 82–87.

[6] Soland, R. M. (1987) Optimal Terminal Defense Tactics when Several Sequential Engagements Are Possible. Operations Research 35: 537–542.

[7] Institue for Disease Modeling. (2019) SI and SIS models. https://www.idmod.org/docs/hiv/model-si.html. Last accessed May 2020.

[8] Kermack, W. O., and McKendrick, A. G. (1927) A Contribution to the Mathematical Theory of Epidemics. Proceedings of the Royal Society A. 115 (772): 700–721.

[9] Harko, T., Lobo, F. S. N., and Mak, M. K. (2014) Exact analytical solutions of the Susceptible-Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and

birth rates. Applied Mathematics and Computation. 236: 184–194.

[10] Miller, J. C. (2012) A note on the derivation of epidemic final sizes. Bulletin of Mathematical Biology, 74, Section 4.1.

[11] Miller, J. C. (2017) Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes. Infectious Disease Modelling, 2, Section 2.1.3.

[12] Nguyen, B. U. (2017) Modelling cyber vulnerabilities using epidemic models. 7th international conference on simulation and modelling methodologies, technologies and applications.

[13] Nguyen, B. U. (2018) A simultaneous cyber-attack and a missile attack. 7th international conference on simulation and modelling methodologies, technologies and applications.

[14] Mathematica [Technical Computing]. (2011) Wolfram Research Inc. https://www.wolfram.com/mathematica/. Last accessed Jan 2020.

https://csis-website-prod.s3.amazonaws.com/s3fspublic/legacy_files/files/media/csis/pubs/090130_vietnam_study.pdf.https://csis-website-prod.s3.amazonaws.com/s3fspublic/legacy_files/files/media/csis/pubs/090130_vietnam_study.pdf.https://www.washingtonpost.com/news/monkey-cage/wp/2017/01/06/did-u-s-nonmilitary-aid-win-hearts-and-minds-in-afghanistan-yes-and-no/?utm_term=.c152677432a1https://www.washingtonpost.com/news/monkey-cage/wp/2017/01/06/did-u-s-nonmilitary-aid-win-hearts-and-minds-in-afghanistan-yes-and-no/?utm_term=.c152677432a1https://www.idmod.org/docs/hiv/model-si.htmlhttps://www.wolfram.com/mathematica/


[15] Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., and Knuth, D. E. (1996) On the Lambert W Function. Advances in Computational Mathematics Volume 5 Issue 4: 329–359.

[16] Odom, B., Hanneke, D., D'Urso, B., and Gabrielse, G. (2007) New measurement of the electron magnetic moment using a one-electron quantum cyclotron. Physical Review Letters Vol 97, 030802.

[17] Smith, S. W. (1997) The Scientist & Engineer's Guide to Digital Signal Processing, Chapter 2. California Technical Publishing, San Diego.

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List of symbols/abbreviations/acronyms/initialisms

LHS Left Hand Side

RHS Right Hand Side

SI Susceptible-Infected

SIR

US

Susceptible-Infected-Recovered

United States

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3. TITLE (The document title and sub-title as indicated on the title page.)

An analytical model for the strategy of winning hearts and minds

4. AUTHORS (Last name, followed by initials – ranks, titles, etc., not to be used)

Nguyen, P. B. U.

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August 2020

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Epidemic Models; Differential Equations; Wargame; Risk Analysis; Probability; Attrition Models; Strategy of Winning Hearts and Minds

13. ABSTRACT (When available in the document, the French version of the abstract must be included here.)

We propose using epidemic models to simulate a complex warfare scenario that includes the strategy of winning hearts and minds. Two epidemic models are compared to a Lanchester model, which simulates exchanges of fire. The first epidemic model, Susceptible-Infected (SI) has a known closed form and an analytical solution. The second epidemic model, Susceptible-Infected-Recovered (SIR) has no known closed form solution. Its solution is approximated using successive transformations that are simple, closed form and analytical. We show that the approximation reproduces nearly perfectly the exact numerical results of the SIR model. In addition, we illustrate the variability of a military operation by introducing noise in the SI model and show how this affects the outcomes of a war.

Nous proposons d’utiliser des modèles épidémiques pour simuler un scénario de guerre complexe comprenant la stratégie visant à conquérir les cœurs et les esprits. Deux modèles épidémiques sont comparés à un modèle de Lanchester, qui simule des échanges de tirs. Le premier modèle épidémique, Susceptible-Infecté (SI), a une forme fermée connue et une solution analytique. Le second modèle épidémique, Susceptible-Infecté-Rétabli (SIR) n’a pas de solution connue sous forme fermée. Une solution approximative est obtenue par des transformations successives qui sont simples, sous forme fermée et analytiques. Nous montrons que l’approximation reproduit presque parfaitement les résultats numériques exacts du modèle SIR. De plus, nous illustrons la variabilité d’une opération militaire en introduisant du bruit dans le modèle SI et nous montrons comment ce bruit influe sur les résultats d’une guerre.

an analytical model for the strategy of winning hearts and …the strategy of winning hearts and...

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