On the Dynamic of Country Development

In this work, we study the dynamics of the development of a country P. We find that this dynamic is governed by forces called development forces. There are three different types: natural force, exterior force, and interior force of development. They all have three components: human, economic and political. Depending on the intensity of these three forces, the resilience to natural and negative exterior forces, the level of development of P varies between two consecutive periods. This study allows us to conclude that development relies essentially on the interior force.


Introduction
Development is a very wide concept with many aspects.Todaro (1977) supported that development is concerned with the economic and political processes necessary for affecting rapid structural and institutional transformation of entire societies in a manner that will influence more economic progress to the broadest segments of their populations.The French economist (Perroux, 1961) argued that the aspects of development of a country are social, technical and infrastructures, economic and cultural.Sen (1999b) defines development as freedom, a suitably broad definition that incorporates not only economic indicators but also freedoms like human and political rights, social opportunities, transparency guarantees and protective security.These different definitions of a country's development show that development is a change of society towards better conditions and is based on economic, social and political aspects.Yet these conditions are subjected to different forces which can alter considerably in positive or negative ways the economy, social and political situation of one given country P.These forces are first the result of its geographical, geological and climatic situation and which leads to natural disasters.Second, they can come from other either countries or partners or international organizations, during this study we consider sanctions.And they can even be a force from the country itself viewed by its economic, social and political situation.
Natural disasters are the result of the interaction between natural hazards (floods, earthquakes, tsunamis, volcanic eruptions, . . . ) and social systems (cities, schools, hospitals, roads, . . . ).Impacts of natural disasters depend on the vulnerability and resilience of a society.Nowadays the losses caused by natural disasters keep increasing ( Mileti & Joseph Henry, 1999).That can become a real drag on a country's development, according to Mark Malloch Brown, (Rapport mondial, 2004), natural disasters exert considerable pressure on development (Note 1).They significantly reduce the chances of achieving the Millennium Development Goals, and in particular halve extreme poverty by 2015.The global economic loss caused by natural disasters was more than US $ 1.8 trillion between 1995 and 2015, and US $ 262 billion in Europe alone (CRED & UNISDR, 2015).Stephane Hallegatte even advances the idea that natural disasters plunge 26 million people into poverty (Note 2).According to him, if there were no natural disasters for a year, the population living below the poverty line would be reduced by 26 million out of a total of 700 million people currently.
Similarly, sanctions hinder the growth and development of a country P. Employed first in 1919 by the U.S. President Woodrow Wilson during his speech in Indianapolis, sanctions are used since the early 1960s to project power or influencing another government's behavior without resorting to military conflict, to destabilize a hostile regime and hasten its demise instead of deploying force to target country to withdraw its troops from border skirmishes and desist from military adventures.Since then, several other foreign policy goals have been pursued, including efforts to protect human rights and promote democracy, stem nuclear proliferation, and fight international terrorism (Hufbauer, Schott, & Elliott, 1999a).According to the sanctions literature, the cost of sanctions against a target country is supposed to result in maximum economic damage in order to coerce the target state to alter its policies in favor of the sender states (Hufbauer, Schott, Elliott, &B. Oegg, 2007 andDizaji &van Bergeijk, 2013).(Neuenkirch & Neumeier, 2015a) found that on average, the imposition of UN sanctions decreases the target state's annual real per capita GDP growth rate by more than 2 percentage points (pp) and about 13.4% declines in GDP (Note 3).Their effects do not stop with the economic recessions but also on humanitarian crisis and a serious social impoverishment (Allen & Lektzian, 2013;Garfield &Santana, 1997 andDaponte &Garfield, 2000).
Finally, every country, whatever the type of the country's regime, either aristocratic or democratic or autocratic, has always its leaders.By their action, these leading people shape political, human and economic situation.Leaders can improve the situation like the case of some Asian countries as well as worsen it like the case of many African countries.On one hand, the impressive economic success achieved by the East Asian newly industrializing countries (NIcs)-Taiwan, South Korea, Singapore and Hong-Kong since the 1960s is due to the role play by the state (Gore, 1996), macroeconomic performance, political and institutional stability (Jaffrelot, 2008), public investment in infrastructure and human capital, the strategy of external openness, growth driven by exports, the competitive market of factors of production, the flexibility of economic policies and the speed of adjustment to external shocks (World Bank, 2008), a high level of schooling, a relatively egalitarian income distribution and a demographic transition early.On the other hand, there are many African countries struggling to get out of poverty traps and misery.Africa faces many problem: human and social (the high youth unemployment rate, the prevalence of precarious (UN Commission ?conomique pour l'Afrique, UNEP et FAO, ONUDI, & UNDP, 2015); low quality education of public school, health system problem (IFC, 2016), food insecurity and undernourishment, (Banque Africaine de Développement, & UNDP CEA, 2017), economic (the weight of debt, mismanagement of the economy and unsustainable external debt, manufacturing decreasing, (Zamfir, 2016) and politic (military conflicts, institutional weaknesses, the effects of insecurity and political uncertainty, (UNISDR, 2008).
Despite of the complexity of development itself, all of these situation show that many forces can impact development's level.Then we have two questions: what are forces that influence the development's dynamics of one country?And how they impact development's level?We hypothesize that there are different types of forces which governs development and that they impacts it negatively or positively.To answer these questions, the paper is structured as follows: section two introduces the different types of development forces.Section three discusses the links between development and these different forces.Then Section four establishes one development equation.And finally the last section is about Discussion and Conclusion.We adopt different method during the elaboration of this theory: documentation, inductive method and the theory of dynamical system.
The first section introduces development forces and their characteristic.

Development Forces
Let's assume P a country.

Definition 1 We denote by development force all action which can increase or decrease the development's level of P.
Proposition 2 There are three types of development forces: natural force N, negative exterior force E and interior force I.
Proof.Let P be a country.Firstly P is located on the globe, it is exposed to all the natural hazards that results from its geographical, geological and climatic condition.Hence the existence of the natural force manifested by the hazards conditioned and the vulnerability of P which leads to natural disasters.Secondly, there are more than 180 countries in the world.These countries interact with each other through trade, partnerships, bilateral and/or multi-lateral relations, conventions, Charters and agreements.These different relationships produce power struggles and interdependencies between countries.Failure to respect these agreements leads to sanctions from signatory countries and/or International Organizations.Thus, if we consider two countries P and P ′ , P ′ can start actions against P which can affect badly the economic, social and political status of P. Hence the existence of negative exterior force of P from foreign countries.And third, the economic, political and social situation of P are the results of the leading way of the established regime.Hence the existence of the interior force of P.
First, let's see the characteristics of the natural force.

Natural Force
Definition 3 Hazards (geophysical, atmospheric, or hydrological event, rainfall, tropical storms, flooding, and earthquake) are events triggered by natural forces, but they only turn into disasters if people are exposed to the hazard and are not resilient to fully absorbing the impact without damage to life or property causing substantial damages, disruption, and perhaps casualties and leaving the affected communities unable to function normally.A disaster implies some combination of losses, in human, physical, and financial capital, and a reduction in economic activity such as income generation, investment, consumption, production, and employment in the real economy, (Behrman & Srinivasan, 1995).Natural disasters is also the product of the combined effects of the intensity of the hazards and the social vulnerability (O' Keefe, Westgate, & Wisner, 1976).
Definition 4 Let P be a country.A natural force of P is a set of hazards conditioned by the geographical, geological, biological and meteorological situation that alter the three components of development (human, economic and politic (Rakotoarivelo & Ravelonirina, 2017).
Proposition 5 The natural force of P is mainly composed by hazards.
Proof.Immediate according to the Definition 4.
Globalisation makes all countries interconnected, interlinked and interdependent.These interactions usually come from different agreements like political, economic, military, trade, . . . .This leads to the weakness and vulnerabilty of some countries on foreign country.Let's see the negative exterior force.

Negative Exterior Force
In the literature, scholars distinguish two types of sanctions used in international relations: positive and negative sanctions.Positive sanctions take the form of actual or promised rewards (humanitarian aid, tariff's reduction or tariff's abolition and so on) whereas negative sanctions are the use or threatened use of punishments.
In this paper, we consider only the negative sanctions.
Definition 6 US President Woodrow Wilson defines a country's sanctions as a peaceful, silent and deadly remedy that no nation can resist.Sanctions involve several actions such as tariffs, export controls, import restrictions, travel bans, freezing assets, reduction or removal of foreign aid and severing of diplomatic relationships (Allen, 2008).To sanction one country P is to interrupt diplomatic and/or economic relations (Garfield, Devin, & Fausey, 1995).
Remark 1 -The expressions "sender" and "target" denote respectively "the country that imposes sanctions and the country that receives the economic punishment", (Caruso, 2003).
-If the sanctions are enacted by one state, then we speak of unilateral sanctions and if they are taken by several states or by a regional or international organization, they are said to be defined by multilateral sanctions.
There are different types of sanctions according (Garfield et al., 1995): economic sanctions; financial sanctions: restrictions on the funding of the target country, direct payments investment restrictions, restrictions (or control) relating to banking transactions; trade sanctions: embargo, boycott, blocus; smart sanctions; advertising sanctions: sportive sanctions, cultural sanctions and diplomatic sanctions.
According to (Eyler, 2007), sanctions obey this following diagram (cf. Figure 1) : Note.This figure is from (Eyler, 2007).It shows that any sanction aims at touching and modifying the policy adopted by the sanctioned country.But to reach that point, it goes through economic and human repercussions.
Definition 7 Given P a country and P ′ another country or an organization (Note 4).A negative exterior force of P is all actions made by P ′ toward P which have negative effects which produce negative effects on all three components of a P sanctioned country development.
Proposition 8 The negative exterior force of P is the combination of sanctions from P ′ .
Proof.Immediate from the Definition 7.
The negative exterior force does not take into account if the sanctions are effective or not (Note 5).It measures all the impacts on the human, economic and political caused on the target country P.
Every country have its own force, let's see the characteristic of interior force.

Interior Force
Definition 9 (Rakotoarivelo & Ravelonirina, 2017) A determining factor is a sector and/or pillar area of the development as well as of the non development of a country.

Definition 10 Given a country P. An interior force of P, all sectors and/or area on which the development of P relies on.
Proposition 11 Interior force is composed by the ten determining factors and is subdivided into three groups: human force, economic force and political force, (Rakotoarivelo & Ravelonirina, 2017).
Proof.Obvious from the Definition 10.
Natural and negative exterior forces are not from P, then P can have the capacity to resist and to minimize their effects.We define this capacity as "resilience".

Resilience
Holling 1973, first described resilience as a measure of the persistence of systems and of their ability to absorb change and disturbance and still maintain the same relationships between populations or state variables.In operationalizing the concept of regional economic resilience, (Hill, Wial, & Wolman 2008), argue that resilience is the ability of a region . . . to recover successfully from shocks to its economy that either throw it off its growth path or have the potential to throw it off its growth path but do not actually do so.If applied to economic sanctions, this definition would only label a target state resilient if it were able to bounce back to the level of economic output and stability it experienced prior to the actual imposition of a given sanctioned regime.
The following section studies the link between these forces and the development.

Natural Force
Let N t denotes the value of the natural force between two periods t and t + 1.

Proposition 12 If the intensity of hazard is high (resp. low), then the value of the natural force is also high (resp. low).
Proof.Natural force is principally composed by hazards and so if the intensity of hazards varies then the value of natural force change as well.
Proposition 13 Between two periods t and t + 1, the higher (rep.lower) the human, economic and politic damages caused by a natural force, the more (rep.less) the development of a country is affected.
Proof.This proof is divided in three parts, we assess the impacts in three components: human, economic and politic.

Economic components:
The principal impacts of natural force on economy is that it leads on average to negative effects on GDP.It is due to the destruction of capital stock and the disruption of economic activities and production, (Hochrainer, 2009a(Hochrainer, , 2009b;;Benson &Clay, 2004, andCharvériat, 2000).Moreover, the reconstruction after an intense hazard affects public revenue, (Benson & Clay, 2004).
Political components: Natural force have indirect effects on politic.Indeed, depending on the way on how government manage (failure or successful) disaster have political effects on the government measured as political support during the next elections (Uslaner & Yamamura, 2016;Dodds, 2008Dodds, , 2015;;Brändström, 2016, andBytzek, 2008).And so, this can increase the risk of anti-government domestic political activities, anti-government demonstrations, riots, revolutions, intrastate conflict, (Choudhury, 2013).2008).Second, Malawi experienced fiscally chaos, in the period of the two droughts in the early 1990s, there was increase of 23 percent of the deficit, declined of GDP, increase of 30 percent of expenditure between 1992-1993 and 1994-1995), decrease of revenue (9 percent in 1992-1993 and again 11 percent in 1993-1994 ).Third (Benson, Twigg, & Rossetto, 2007), studied the impacts of natural disasters on poverty and found theses results: in El Salvador, the two earthquakes in 2001 led to an estimated 2.6-3.6 per cent increase in poverty.In Honduras, the percentage of poor households increased from 63.1 per cent in March 1998 to 65.9 per cent in March 1999 as a consequence of Hurricane Mitch in October 1998.
In Aceh, Indonesia, the 2004 tsunami is estimated to have increased the proportion of people living below the poverty line from 30 per cent to 50 per cent.During the 2011 earthquake and tsunami in Japan, scholars identified decreasing levels of trust in government following the disaster (Uslaner & Yamamura 2016).
Fourth, we see the case of Honduras in Figure 2 Figure 2. Observed GDP in Honduras with events vs. projected growth without events Note.This graph is from (Zapata, 2008;World Bank, 2007).Zapata (2008) uses a model based projection, IIASA projects growth statistically based on pre-disaster observed GDP to make this prediction.And this show, how disaster can alter the GDP.

Negative Exterior Force
Let E t denotes the value of the negative exterior force between t and t + 1.
Proposition 15 If the intensity of the sanctions is high, then the value of the negative exterior force is also high.
Proof.Immediate because negative exterior force is mainly composed by sanctions.
Proposition 16 If negative exterior force increases, thus the politic, economic and human components level decrease.
Proof.This proof is divided in three parts, we assess the impacts in three components: human, economic and politic.
Economics components: Negative exterior force as sanctions aim to induce maximum economic damage in order to coerce the target state to modify its policies in favor of the sender states (Hufbauer et al., 2007 andDizaji &van Bergeijk, 2013).And so, it inevitably causes adverse economic outcomes on national currency, poverty, GDP, trade, government consumption, employment and increase inflation (Peksen & Son, 2015;Neuenkirch & Neumeier, 2015a, 2015b;and Peksen & Drury, 2010).
Human components: Negative exterior force causes indirectly many humanitarian impacts especially embargo and blocus.
In (Garfield, Devin, & Fausey, 1995;Garfield & Santana, 1997;and Gibbons & Garfield, 1999) we find that sanctions are correlated with more humanitarian problems.Others like (Ali Mohamed & Shah, 2000;Daponte & Garfield, 2000;Heine-Ellison, 2001;Neuenkirch & Neumeier, 2016) find that it induces infant mortality and reduces life expectancy.These effects are due to shortage in supplies and commodities necessary to subsistence and in health system due to the restriction in import, export and the withdrawal of foreign and aid ( Hufbauer et al., 2007 andHeine-Ellison, 2001).The imposition of sanctions increases state-sponsored repression and suggests that these sanctions contribute to worsening humanitarian conditions of the civilian population, (Wood, 2008).
Political components: Economic sanctions may cause more violence, oppression, restriction in the target country, deteriorating democratic freedom and civil rights, (Peksen & Drury, 2010) and worsen human rights conditions in the target country, (Drury & Li, 2006;and Peksen & Drury, 2009).Economic sanctions also lead to political instability in sanctioned country by destabilizing the target leadership, (Marinov, 2005) and inciting more violence in the forms of protests and riots against the established regimes, (Allen, 2008;Peksen & Drury, 2010).(Thyne 2006), found that diplomatic statements criticizing the target alone can signal support for opposition groups within another country.(Allen 2008), showed that foreign economic pressure, especially against more democratic states, increases the level of violent and anti-violent protests against the government.Proof.According to Proposition 16, between t and t + 1, the development level is inversely proportional to the value of E t .So we take one function which completes this condition.
Example 2 First for Iran, sanctions led to: 9% of decreasing in GDP in the two year sending March 2014, declination about 56% of Iranian Rial from January 2012 to January 2014, acceleration of inflation between 50% and 70% during 2011-2013, (Lew & Jacob, 2016).Secondly for Cuba sanctions induced: decrease in GDP from 35% to 50%, decline of exports and imports by 80% and 73% respectively within that same timeframe, fall from 70% in 1989 to less than 30% in 1993 in the use old installed capacity, (ECLAC, 2001).
And finally, Figure 3 shows impact of the duration of sanctions:

Interior Force
Let I t denote the value of the interior force between t and t + 1.
Proposition 18 If interior force increases, thus the human, economic and politic components level increase.
Proof.According to Proposition 11, we see that interior force is composed by these three components.

Proposition 19 At time t, if development interior force is high, then interior force's value i t increases, and the development level increases.
Proof.As interior force is composed by three components, we then evaluate its intensity according them: Human components: We evaluate human force on four levels: education, health, employment and culture.Firstly, (Bils &Klenow, 2000 andHanushek &Kimko, 2000) find a positive correlation between a quality of education and economic growth; (Townsend, 2000) finds that it is the key issue to improve living standards and reduce social, income inequality and poverty.Secondly, health is essential to economic growth because it leads to improvements in income and have a positive impacts development, as can see from these different works (Workd Health Organisation, 1996 andBils &Klenow, 2000).Thirdly, (Seyfried, 2005;Boltho &Glyn, 1995 andSwane &Vistrand, 2006) find that there is a relationship between employment, economic growth and it reduces poverty, (Islam, 2004 andHull, 2009).Lastly, culture leads to development because it is a vehicle for economic development and social cohesion, (UNESCO, 20014).
Proof.From Proposition 19, we can deduce that development is proportional to the value of I t .Hence the equation.
Example 3 Interior force with positive value: China has lifted over 450 million people out of poverty since 1979, this is due to the rapid economic growth between 1985 and 2001, (Lin, 2003).Mozambique records also rapid reduction in poverty associated with growth over a shorter period.From 1996 to 2002, the economy grew by 62 per cent and the proportion of people living in poverty declined from 69 per cent to 54 per cent, (Arndt, James, & Simler , 2006).And for interior force with negative value: political instability is one of cause which can lead to negative value of this force.
Political instability has direct impacts on economy, (Fosu, 1992) when studying Sub-Saharan Africa countries found that it causes more than 33 percent decrease of GDP growth over the 1960-1986 period.Also (Alesina et al., 1996) find a very robust relation between political instability and GDP per capita growth while studying 113 countries for the period 1950-1982.
The last section is about the dynamic of development.

Value of the Natural Force
The natural force is composed of several different hazards and each of them have their own scale of measurement, they must be normalized on the same scale so that we can evaluate and compare them by the value of the parameter n t .For doing so, we categorize the hazards using 0 to 10 scale.

Value of the Intensity
We use scale according to the type of hazards to measure their value: -Earthquake: we use the Richter scale to measure it.
-Cyclone, Typhoon, Hurricane: we measure them according to the velocity of the wind and to the cyclonic basins of the ocean.
-Drought: the Standardized Precipitation Index (SPI) is used here to measure drought (Appendix, Table A1).

Value of the Resilience
To classify countries according to their resilience, we use the classification made by GOAL (Note 7) (Appendix Table A2).According to this classification, we adopt the value of the resilience to the natural force r n t as follow: -If P is a country with minimal resilience then 0 ≤ r n t ≤ 2.
-If P is a country with low resilience then 2 < r n t ≤ 4.
-If P is a country with medium resilience then 4 < r n t ≤ 6.
-If P is a resilient country 6 < r n t ≤ 8.
-If P is a country with high resilience then 8 < r n t ≤ 10.
Figure 4.A hypothetical trajectory of resilient and less resilient community (Modified from Zhang, 2006) Note.This figure is from (Mayunga, 2007).It represents two trajectories of communities over time through four phases, pre-disaster, disaster, restoration, and recovery.It shows that more resilient country experienced less disaster impacts and short term recovery.Whereas, less resilient country take more time for recovering.The other category of resilience is between these two levels.

Value of the Intensity Negative Exterior Force
We evaluate the value of the negative exterior force as follows (cf.The value of n t is the sum of the different sanctions used by senders to the target country.

Value of the Resilience
To evaluate the resilience to negative exterior force noted r e t , which is the capacity of the targeted country to offset especially goods, products restricted and financial losses, and to continue the economy activity (export and imports), we adopt the following classification (cf.Example 4 In 2014, Iran's Supreme Leader Ayatollah Khamenei proclaimed that the government would "turn every threat into an opportunity . . .".And so iranian diverted trade products (exports non-oil products) and established new relationship in markets (Asia, Asia, Africa and Latin America).In 1994, Cuba began resilient to sanctions by growing and diversifying its imports and exports markets (ECLAC): exports rise from 10% in 1990 to 37% in 1997.

Value of Interior Force Intensity
The interior force is mainly composed of the ten determining factors f j , j ∈ {1, . . ., 10} and the Table A4 (Appendix), shows the components of each f j .
Remark 2 The component with the sign "+" indicates that this component increases the level of development whereas "-" diminishes it.
To calculate the value of the intensity of the interior force, each component of each determining factors is evaluated on percentage (%).After, we use the following formula: where f j : determining factor, j ∈ {1, . . ., 10} and n j = card(c k j ) t , c j the k th components of the j th determining factor between t and t + 1, and n t k : the percentage of c k j between t and t + 1.If the components c j have a sign (-), then its value is multiplied by (-).And if the value of c j preceded by sign (+) is negative, then c j is also negative.
And we compute the value of the interior force, by using the formula: These following components are exceptions, because they do not have data on percentage.And so, we have to convert them: -Strong diplomacy: regional integration, international integration, number of ambassador discharged, deal win-win, deal satisfying international norms and condition.We take 100% the number of all regional and international organization, then the value of regional and international of P depends to the number(%) of regional and international integrated by P.
-Good governance: WGI Worldwide Governance Indicators by the World Bank in a scale of -2.5 to 2.5, we convert this index into percentage by adopting the following value, −2.5 corresponds to 0% and 2.5 to 100%.
-Liberty: we use the index Human Freedom Index by Cato Institute, on a scale of 0 to 10.We convert this scale by the following conversion, 0 corresponds to 0% and 1 to 100%.
-Rule of law: Rule of law index by WJP on a scale of 0 to 1 (1 strongest adherence to the rule of law).We convert this by the following conversion, 0 corresponds to 0% and 1 to 100% (Note 8).
-Culture: annual rate of daily newspaper variations (%), number of literary books (% of total), number of literary books (% of total), (Appendix Table A4), all of this indicators are from (UNESCO, 2014).
-Innovation: Global Index of Innovation by Cornell University, INSEAD and WIPO (Note 9).
It can be seen that variation in the intensity of the interior force influences the development of a country the most.Thus, the mastery of this force is essential for the development.

Equation of Development With Negative Sanction
Theorem 21 The development level of a country P at t + 1 depends on the level reached at time t and the value of the three development forces between t and t + 1.So we have the following equation: Then, we have where ξ(t) is a "noise" term.
NB: Noise can come from another force or phenomena that are not predicted by the deterministic model, or it can from the irregularity, unpredictable development process or by observational process.
Proof.The development at time t + 1 depends on its level at t and the value of development's forces.Thus, we have We add the noise term ξ(t) to measure all of unpredictable phenomena which affect development apart of theses forces.
And according to Theorems 14, 17 and 20, we find that development is pushed downward by natural and negative exterior forces and pushed upward by interior force.So, the equation ( 8) becomes Theorem 22 So the equation ( 6) becomes Considering the natural and exterior resilience capacity, we obtain the final equation: Proof.The deterministic part of the equation ( 9) is obtained by using equations ( 1), ( 2) and (3).Also for the equation ( 10), is got by using the principle that the resilience reduces the impacts of natural and negative exterior forces.And then we add the noise term in both equations ( 9) and ( 10).
Corollary 23 The development level of P at t + 1 increases (resp.decreases, in equilibrium) if the interior force produced is greater (resp. lower, equal) than the sum of natural and negative exterior forces.
Proof.Immediate according to the equation (6).
Remark 3 If between the period t to t + 1, there is no natural force and/or the negative exterior force E t , then n t = r n t = e t = r e t = 0.And if between this period there are more than one hazards or sanctions, then we add up their values.
In the following, we focus only on the deterministic part of the equation.And let denotes by f the function associated to the equation D t+1 − D t = 0.
Theorem 24 If i t > 0, so D t+1 increases between two consecutive periods t and t + 1.In other words, for a given country, the level of development depends essentially on the value of the interior force.
Proof.Let denotes by f (x) = i t x 2 + B t x 2 , where B t = (b ′ t − r n t ) + (e ′ t − r e t ) the function associated with the equation D t+1 − D t = 0.And the derivative of f shows that if i t > 0, then f ′ (x) > 0. And so, D t+1 > D t .
Theorem 25 If i t < 0, then D t+1 increases if the value of D n is between the root of f and decreases otherwise.
Proof.The study of the function f associated with the equation where B t = (n t − r n t ) + (e t − r e t ).
x 1 and x 2 are stable if −1 < i t < 0. However, with this value of a t , x 1 and x 2 are not defined because β t > 0 for all period t.So we can conclude the non-stability of the fixed points.

Discussion and Conclusion
From this work, we conclude that between t and t + 1, each country P is subjected to: hazards conditioned by its geographical and geological situation; different actions that comes from another country P ′ and its own force.These three different forces act on the human, economic and political components of P called development forces: natural force n t , negative exterior force and interior force i t .Depending of the intensity of n t , e t and i t , they influence the level of development D t+1 of P.
For the natural force, we find that the main impacts of this force on development are mostly the human and economical aspects and these impacts leads to political consequences.The socio-economic impacts depend intrinsically on the intensity of the hazards, the nature of the hazards, the vulnerability and the resilience of P. Every country experiences natural force, but they are different on the level of resilience.This capacity is very important because it allows to recover and minimize rapidly the economic and human impacts of natural disaster.Resilience r n t of P plays an important role in development.Based on the studies we have seen previously, if the relative resilience is high, it significantly decreases its impacts and thus allows for the continuity of all the activities.In addition, a strong resilience between t and t + 1 ensures a sustained and increasing level of development regardless of the intensity of these forces.
For the negative exterior force of P, it arises from the different interactions between two countries or more countries and affects particularly the political and economic sovereignty of one of the two countries.The negative exterior force e t worsens the situation of P. Constituted by sanctions, this force aims to make economic pressure to induce political change.This kind of force has many adverse economic impacts that which extends to human components.So, it deteriorates economic, human and politics situation of P.Even though, P can not delete directly sanctions, it can be resilient to this force.The resilience here, allows P to find solution that can reduce the economic impacts of sanction by finding new trade partner, substituting the goods and financial aid.
For the interior force i t of P, it is the driving force of P. Evaluating by the economic, political performance and the social situation, it can have negative and positive intensity.Negative intensity corresponds to interior force that exacerbate the development of P, and positive improves it.Further result shows that development of P depend essentially on i t .No matter, how high natural and negative exterior force are, if the interior force is high, then development is not too much affected.But if it is not, then the development of P considerably decreases.
This paper shows as well that D t+1 continuously increases if the interior force i t unceasingly increases and a very high resilience of P. Another result is that D t+1 is greater (resp.lower, equal) than D t between t and t + 1 if intensity of i t is greater (reps.lower, equal) to the sum of n t and e t .Therefore, to increase the level of development, the interior force i t must be greater than the other two forces.Then, without the forces n t and e t , the development is linear and strictly increasing ot decreasing depending of the value of i t .Further result is about the dynamic of the development, we find that it has two fixed points that are not stable.And so, the development is not stable, that is, there is no particular level to reach and where P can stay indefinitely.In addition, the non-stability of fixed points implies that there is not a level where each country tends to reach.And even if the country reaches the level of the fixed-point value between two consecutive periods t and t + 1, it can decrease or increase immediately the period after.
Development requires the control of the three development forces.For the natural force, as it can not be avoided, it is necessary to introduce into politics the resilience of each country and minimize the damage from as much as possible.
For the negative exterior force, the impacts of sanction can be reduced by a strong diplomacy and international politics, diversification of bilateral and multilateral relations and adjustment of politic.And for the interior force, development involves a high value of this force which depends on: sustained economic growth, stable political situation and better social environment.With this work, we have more perspectives: we can add vulnerability of P in the equation and see the dependency between vulnerability and resilience of P to better model the reality.As exterior force have many aspects, we can explore another sorts of exterior force like: aid, grants, globalisation.All of them have their own particularity and its effects on the development.And finally, we also expect for future research to focus on the non linear noise term of the equation ( 9), investigate the effects of whether additive or multiplicative or other type of noise on the dynamic of the system and then study how types of noise alter the behavior of the country development.

Figure 1 .
Figure 1.Sanction time-line and sanction effectiveness continuum

Figure 3 .
Figure 3. Impact of sanctions on poverty over time Theorem 14 Between t and t + 1, if the value of natural force increases N t , therefore the development level D t decreases Here are some illustrative examples about the impacts of natural force on development.First in Mozambique, World Bank has constructed 487 schools over 20 years, but more than 500 schools are destroyed by floods in 2000, (UNDP, t : intensity of hazards between t and t + 1, n t ∈ [0, 10].Proof.According to Proposition 13, the development level is inversely proportional to the value of N t .So we choose one function which satisfied this condition.Example 1 Theorem 17 Between t and t + 1, if the negative exterior force value E t increases, thus development level decreases D t .

Table 1 ) : Table 1 .
Value of negative exterior force

Table 2 ) : Table 2 .
Value of resilience to negative exterior forcePercentage of loss covered or substituted by P Value of r e t +∞[, where B t = (b ′ t − r n t ) + (e ′ t − r e t ) > 0 and i t < 0.